\usepackage or this document’s own macros. Corpus-wide, \bm occurs 327 times, \Perp is defined by no package and spans 10 documents, and 11,088 of 11,624 undefined occurrences are the source document’s own macros. A row that looks wrong here may be correct, and one that looks right here may not compile. The LaTeX report renders through the document’s own preamble and is the surface to judge from.3979 rows
The page, the confidence and the picture of an inline formula are its HOST LINE's --- a formula has none of its own. A line's confidence is not a formula's.
| Identifier | Page | Conf. | LaTeX source | Rendered | Image |
|---|---|---|---|---|---|
| fong-spivak-seven-sketches_FO0001 | 15 | 1.000 | a | | |
| fong-spivak-seven-sketches_FO0002 | 15 | 1.000 | b | | |
| fong-spivak-seven-sketches_FO0003 | 52 | 0.998 | \mathcal{V} | | |
| fong-spivak-seven-sketches_FO0004 | 114 | 1.000 | \Sigma | | |
| fong-spivak-seven-sketches_FO0005 | 114 | 1.000 | \Pi | | |
| fong-spivak-seven-sketches_FO0006 | 176 | 0.679 | \boldsymbol{\operatorname { M a t }}(R) | | |
| fong-spivak-seven-sketches_FO0007 | 219 | 0.926 | \Omega | | |
| fong-spivak-seven-sketches_FO0008 | 264 | 0.930 | \mathbb{I R} | | |
| fong-spivak-seven-sketches_FO0009 | 13 | 0.999 | f: X \rightarrow Y | | |
| fong-spivak-seven-sketches_FO0010 | 13 | 0.999 | X | | |
| fong-spivak-seven-sketches_FO0011 | 13 | 1.000 | Y | | |
| fong-spivak-seven-sketches_FO0012 | 14 | 1.000 | f | | |
| fong-spivak-seven-sketches_FO0013 | 14 | 1.000 | \mathbb{R} | | |
| fong-spivak-seven-sketches_FO0014 | 14 | 1.000 | x | | |
| fong-spivak-seven-sketches_FO0015 | 14 | 1.000 | f(x) | | |
| fong-spivak-seven-sketches_FO0016 | 14 | 1.000 | f: \mathbb{R} \rightarrow \mathbb{R} | | |
| fong-spivak-seven-sketches_FO0017 | 14 | 0.714 | x \leq y | | |
| fong-spivak-seven-sketches_FO0018 | 14 | 0.714 | f(x) \leq f(y) | | |
| fong-spivak-seven-sketches_FO0019 | 14 | 0.714 | x, y \in \mathbb{R} | | |
| fong-spivak-seven-sketches_FO0020 | 14 | 1.000 | |x-y|=|f(x)-f(y)| | | |
| fong-spivak-seven-sketches_FO0021 | 14 | 1.000 | f(x+y)=f(x)+f(y) | | |
| fong-spivak-seven-sketches_FO0023 | 14 | 0.766 | x, y | | |
| fong-spivak-seven-sketches_FO0024 | 14 | 1.000 | y | | |
| fong-spivak-seven-sketches_FO0025 | 14 | 1.000 | f(x)=x^{2} | | |
| fong-spivak-seven-sketches_FO0026 | 14 | 1.000 | x^{2} \leq y^{2} | | |
| fong-spivak-seven-sketches_FO0027 | 15 | 1.000 | c | | |
| fong-spivak-seven-sketches_FO0028 | 15 | 1.000 | a, b | | |
| fong-spivak-seven-sketches_FO0029 | 15 | 1.000 | \Phi | | |
| fong-spivak-seven-sketches_FO0030 | 16 | 1.000 | A | | |
| fong-spivak-seven-sketches_FO0031 | 16 | 1.000 | B | | |
| fong-spivak-seven-sketches_FO0032 | 16 | 1.000 | A \vee B | | |
| fong-spivak-seven-sketches_FO0033 | 16 | 1.000 | z_{1}, \ldots, z_{n} | | |
| fong-spivak-seven-sketches_FO0034 | 16 | 0.562 | z_{1} | | |
| fong-spivak-seven-sketches_FO0035 | 16 | 1.000 | z_{i} | | |
| fong-spivak-seven-sketches_FO0036 | 16 | 1.000 | z_{i+1} | | |
| fong-spivak-seven-sketches_FO0037 | 16 | 1.000 | z_{n} | | |
| fong-spivak-seven-sketches_FO0038 | 16 | 0.506 | \Phi\left(\odot^{\odot}\right)=\Phi\left({ }^{\odot} \mathscr{C}\right)= | | |
| fong-spivak-seven-sketches_FO0039 | 16 | 0.322 | \Phi\left(\odot^{\odot} \vee\right)=\Phi(\cdot \sigma)= | | |
| fong-spivak-seven-sketches_FO0040 | 16 | 0.322 | \bullet | | |
| fong-spivak-seven-sketches_FO0041 | 17 | 1.000 | A \leq B | | |
| fong-spivak-seven-sketches_FO0042 | 17 | 1.000 | \leq | | |
| fong-spivak-seven-sketches_FO0043 | 17 | 1.000 | A \leq(A \vee B) | | |
| fong-spivak-seven-sketches_FO0044 | 17 | 1.000 | B \leq(A \vee B) | | |
| fong-spivak-seven-sketches_FO0045 | 17 | 0.999 | C | | |
| fong-spivak-seven-sketches_FO0046 | 17 | 0.999 | A \leq C | | |
| fong-spivak-seven-sketches_FO0047 | 17 | 0.999 | B \leq C | | |
| fong-spivak-seven-sketches_FO0048 | 17 | 0.999 | (A \vee B) \leq C | | |
| fong-spivak-seven-sketches_FO0049 | 17 | 0.993 | \{\bullet, *\} | | |
| fong-spivak-seven-sketches_FO0050 | 18 | 1.000 | \mathbb{B}=\{ | | |
| fong-spivak-seven-sketches_FO0051 | 18 | 1.000 | \} | | |
| fong-spivak-seven-sketches_FO0052 | 18 | 1.000 | \not \leq | | |
| fong-spivak-seven-sketches_FO0053 | 18 | 0.637 | B .^{2} | | |
| fong-spivak-seven-sketches_FO0054 | 18 | 1.000 | A, B | | |
| fong-spivak-seven-sketches_FO0055 | 18 | 1.000 | \mathbb{B} | | |
| fong-spivak-seven-sketches_FO0056 | 18 | 0.760 | \vee | | |
| fong-spivak-seven-sketches_FO0057 | 18 | 1.000 | \Phi(A)=\Phi(B)= | | |
| fong-spivak-seven-sketches_FO0058 | 18 | 1.000 | \Phi(A) \vee \Phi(B)= | | |
| fong-spivak-seven-sketches_FO0059 | 18 | 1.000 | \Phi(A \vee B)= | | |
| fong-spivak-seven-sketches_FO0060 | 18 | 1.000 | P | | |
| fong-spivak-seven-sketches_FO0061 | 18 | 1.000 | Q | | |
| fong-spivak-seven-sketches_FO0062 | 19 | 1.000 | a, b, c | | |
| fong-spivak-seven-sketches_FO0063 | 19 | 1.000 | A=\{h, 1\} | | |
| fong-spivak-seven-sketches_FO0064 | 19 | 0.911 | h | | |
| fong-spivak-seven-sketches_FO0065 | 19 | 0.911 | \{h, h, 1, h, 1\} | | |
| fong-spivak-seven-sketches_FO0066 | 19 | 1.000 | x \in X | | |
| fong-spivak-seven-sketches_FO0067 | 19 | 1.000 | h \in A | | |
| fong-spivak-seven-sketches_FO0068 | 19 | 1.000 | 1 \in A | | |
| fong-spivak-seven-sketches_FO0069 | 19 | 1.000 | 0 \notin A | | |
| fong-spivak-seven-sketches_FO0070 | 19 | 1.000 | \varnothing | | |
| fong-spivak-seven-sketches_FO0071 | 19 | 0.999 | \mathbb{N} | | |
| fong-spivak-seven-sketches_FO0072 | 19 | 0.999 | 0,1,2,3, \ldots, 90^{717}, \ldots | | |
| fong-spivak-seven-sketches_FO0073 | 19 | 0.992 | \underline{n} | | |
| fong-spivak-seven-sketches_FO0074 | 19 | 0.992 | n \in \mathbb{N} | | |
| fong-spivak-seven-sketches_FO0075 | 19 | 0.992 | n^{\text {th }} | | |
| fong-spivak-seven-sketches_FO0076 | 19 | 0.992 | n | | |
| fong-spivak-seven-sketches_FO0077 | 19 | 0.992 | 1,2, \ldots, n | | |
| fong-spivak-seven-sketches_FO0078 | 19 | 1.000 | \underline{0}=\varnothing, \underline{1}=\{1\} | | |
| fong-spivak-seven-sketches_FO0079 | 19 | 1.000 | \underline{5}=\{1,2,3,4,5\} | | |
| fong-spivak-seven-sketches_FO0080 | 19 | 0.152 | \mathbb{Z} | | |
| fong-spivak-seven-sketches_FO0081 | 19 | 0.152 | 90^{717}, \ldots | | |
| fong-spivak-seven-sketches_FO0082 | 19 | 1.000 | \pi, 3.14,5 * \sqrt{2}, e, e^{2},-1457,90^{717} | | |
| fong-spivak-seven-sketches_FO0083 | 19 | 1.000 | X \subseteq Y | | |
| fong-spivak-seven-sketches_FO0084 | 19 | 1.000 | \{h\} \subseteq A | | |
| fong-spivak-seven-sketches_FO0085 | 19 | 1.000 | \varnothing:=\{ \} | | |
| fong-spivak-seven-sketches_FO0086 | 19 | 0.808 | \{y \in Y \mid P(y)\} | | |
| fong-spivak-seven-sketches_FO0087 | 19 | 0.808 | y^{\prime} | | |
| fong-spivak-seven-sketches_FO0088 | 19 | 1.000 | \mathbb{N}=\{n \in \mathbb{Z} \mid n \geq 0\} | | |
| fong-spivak-seven-sketches_FO0089 | 19 | 1.000 | \mathbb{N}=\{n \in \mathbb{Z} \mid n \geq 1\} | | |
| fong-spivak-seven-sketches_FO0090 | 19 | 1.000 | \varnothing=\{n \in \mathbb{Z} \mid 1<n<2\} | | |
| fong-spivak-seven-sketches_FO0091 | 19 | 1.000 | X_{1} | | |
| fong-spivak-seven-sketches_FO0092 | 19 | 1.000 | X_{2} | | |
| fong-spivak-seven-sketches_FO0093 | 19 | 1.000 | X_{1} \cup X_{2} | | |
| fong-spivak-seven-sketches_FO0094 | 19 | 0.988 | h, 1 | | |
| fong-spivak-seven-sketches_FO0095 | 19 | 0.988 | h, h, 1, h, 1 | | |
| fong-spivak-seven-sketches_FO0096 | 19 | 0.829 | Z:= | | |
| fong-spivak-seven-sketches_FO0097 | 19 | 1.000 | Z | | |
| fong-spivak-seven-sketches_FO0098 | 19 | 1.000 | Z= | | |
| fong-spivak-seven-sketches_FO0099 | 19 | 1.000 | 3+2:=5 | | |
| fong-spivak-seven-sketches_FO0100 | 19 | 1.000 | \}:=\varnothing | | |
| fong-spivak-seven-sketches_FO0101 | 20 | 1.000 | Y=\{1,2,3,4\} | | |
| fong-spivak-seven-sketches_FO0102 | 20 | 1.000 | X_{1}=\{1,2\} | | |
| fong-spivak-seven-sketches_FO0103 | 20 | 1.000 | X_{2}=\{2,4\} | | |
| fong-spivak-seven-sketches_FO0104 | 20 | 1.000 | X_{1} \cup X_{2}=\{1,2,4\} | | |
| fong-spivak-seven-sketches_FO0105 | 20 | 1.000 | \varnothing \cup X=X | | |
| fong-spivak-seven-sketches_FO0106 | 20 | 0.997 | X_{1} \cap X_{2} | | |
| fong-spivak-seven-sketches_FO0107 | 20 | 1.000 | \{1,2,3\} \cap\{2,5\}=\{2\} | | |
| fong-spivak-seven-sketches_FO0108 | 20 | 1.000 | X_{0}=\varnothing, X_{1}=\{1\}, X_{2}=\{1,2\} | | |
| fong-spivak-seven-sketches_FO0109 | 20 | 0.984 | \bigcup_{n \in \mathbb{N}} X_{n} | | |
| fong-spivak-seven-sketches_FO0110 | 20 | 1.000 | X_{n} | | |
| fong-spivak-seven-sketches_FO0111 | 20 | 1.000 | \bigcup_{n \in \mathbb{N}} X_{n}=\{n \in \mathbb{N} \mid n \geq 1\} | | |
| fong-spivak-seven-sketches_FO0112 | 20 | 1.000 | \bigcap_{n \in \mathbb{N}} X_{n} | | |
| fong-spivak-seven-sketches_FO0113 | 20 | 0.715 | X \times Y | | |
| fong-spivak-seven-sketches_FO0114 | 20 | 1.000 | y \in Y | | |
| fong-spivak-seven-sketches_FO0115 | 20 | 1.000 | X \sqcup Y | | |
| fong-spivak-seven-sketches_FO0116 | 20 | 0.994 | (x, 1) | | |
| fong-spivak-seven-sketches_FO0117 | 20 | 0.994 | (y, 2) | | |
| fong-spivak-seven-sketches_FO0118 | 20 | 1.000 | A:=\{h, 1\} | | |
| fong-spivak-seven-sketches_FO0119 | 20 | 1.000 | B:=\{1,2,3\} | | |
| fong-spivak-seven-sketches_FO0120 | 20 | 1.000 | A \times B | | |
| fong-spivak-seven-sketches_FO0121 | 20 | 1.000 | A \sqcup B | | |
| fong-spivak-seven-sketches_FO0122 | 20 | 1.000 | A \cup B | | |
| fong-spivak-seven-sketches_FO0123 | 20 | 0.999 | R \subseteq X \times Y | | |
| fong-spivak-seven-sketches_FO0124 | 20 | 0.973 | R \subseteq X \times X | | |
| fong-spivak-seven-sketches_FO0125 | 20 | 1.000 | R \subseteq A \times A | | |
| fong-spivak-seven-sketches_FO0126 | 20 | 0.651 | a \star b | | |
| fong-spivak-seven-sketches_FO0127 | 20 | 0.651 | (a, b) \in R | | |
| fong-spivak-seven-sketches_FO0128 | 20 | 1.000 | (5,6) \in R | | |
| fong-spivak-seven-sketches_FO0129 | 20 | 1.000 | 5 \leq 6 | | |
| fong-spivak-seven-sketches_FO0130 | 20 | 0.975 | =, \approx,<,> | | |
| fong-spivak-seven-sketches_FO0131 | 21 | 1.000 | p \in P | | |
| fong-spivak-seven-sketches_FO0132 | 21 | 1.000 | A_{p} \subseteq A | | |
| fong-spivak-seven-sketches_FO0133 | 21 | 1.000 | \left\{A_{p}\right\}_{p \in P} | | |
| fong-spivak-seven-sketches_FO0134 | 21 | 1.000 | A_{p} | | |
| fong-spivak-seven-sketches_FO0135 | 21 | 1.000 | p^{\text {th }} | | |
| fong-spivak-seven-sketches_FO0136 | 21 | 1.000 | a \in A | | |
| fong-spivak-seven-sketches_FO0137 | 21 | 1.000 | \left\{A_{p^{\prime}}^{\prime}\right\}_{p^{\prime} \in P^{\prime}} | | |
| fong-spivak-seven-sketches_FO0138 | 21 | 1.000 | p^{\prime} \in P^{\prime} | | |
| fong-spivak-seven-sketches_FO0139 | 21 | 1.000 | A_{p}=A_{p^{\prime}}^{\prime} | | |
| fong-spivak-seven-sketches_FO0140 | 21 | 1.000 | a, b \in\{11,12,13,21,22,23\} | | |
| fong-spivak-seven-sketches_FO0141 | 21 | 1.000 | a \sim b | | |
| fong-spivak-seven-sketches_FO0142 | 21 | 0.560 | (a, b) | | |
| fong-spivak-seven-sketches_FO0143 | 21 | 0.992 | \sim | | |
| fong-spivak-seven-sketches_FO0144 | 21 | 1.000 | a \sim a | | |
| fong-spivak-seven-sketches_FO0145 | 21 | 0.960 | { }^{a} b \sim a | | |
| fong-spivak-seven-sketches_FO0146 | 21 | 0.960 | a, b \in A | | |
| fong-spivak-seven-sketches_FO0147 | 21 | 1.000 | b \sim c | | |
| fong-spivak-seven-sketches_FO0148 | 21 | 1.000 | a \sim c | | |
| fong-spivak-seven-sketches_FO0149 | 21 | 1.000 | a, b, c \in A | | |
| fong-spivak-seven-sketches_FO0150 | 21 | 1.000 | { }^{a} | | |
| fong-spivak-seven-sketches_FO0151 | 22 | 1.000 | a \in A_{p} | | |
| fong-spivak-seven-sketches_FO0152 | 22 | 1.000 | b \in A_{p} | | |
| fong-spivak-seven-sketches_FO0153 | 22 | 0.976 | X \subseteq A | | |
| fong-spivak-seven-sketches_FO0154 | 22 | 0.884 | x^{\prime} \sim x | | |
| fong-spivak-seven-sketches_FO0155 | 22 | 0.884 | x^{\prime} \in X | | |
| fong-spivak-seven-sketches_FO0156 | 22 | 0.884 | (\sim) | | |
| fong-spivak-seven-sketches_FO0157 | 22 | 0.899 | x \sim y | | |
| fong-spivak-seven-sketches_FO0158 | 22 | 0.899 | x, y \in X | | |
| fong-spivak-seven-sketches_FO0159 | 22 | 1.000 | p \neq q | | |
| fong-spivak-seven-sketches_FO0160 | 22 | 1.000 | A_{q} | | |
| fong-spivak-seven-sketches_FO0161 | 22 | 1.000 | A_{p} \cap A_{q}=\varnothing | | |
| fong-spivak-seven-sketches_FO0162 | 22 | 0.562 | A=\bigcup_{p \in P} A_{p} | | |
| fong-spivak-seven-sketches_FO0163 | 22 | 1.000 | A / \sim | | |
| fong-spivak-seven-sketches_FO0164 | 22 | 1.000 | S | | |
| fong-spivak-seven-sketches_FO0165 | 22 | 1.000 | T | | |
| fong-spivak-seven-sketches_FO0166 | 22 | 1.000 | F \subseteq S \times T | | |
| fong-spivak-seven-sketches_FO0167 | 22 | 1.000 | s \in S | | |
| fong-spivak-seven-sketches_FO0168 | 22 | 1.000 | t \in T | | |
| fong-spivak-seven-sketches_FO0169 | 22 | 1.000 | (s, t) \in F | | |
| fong-spivak-seven-sketches_FO0170 | 22 | 1.000 | F | | |
| fong-spivak-seven-sketches_FO0171 | 22 | 1.000 | F: S \rightarrow T | | |
| fong-spivak-seven-sketches_FO0172 | 22 | 1.000 | F(s)=t | | |
| fong-spivak-seven-sketches_FO0173 | 22 | 1.000 | s \mapsto t | | |
| fong-spivak-seven-sketches_FO0174 | 22 | 1.000 | t | | |
| fong-spivak-seven-sketches_FO0175 | 22 | 1.000 | \{s \in S \mid F(s)=t\} | | |
| fong-spivak-seven-sketches_FO0176 | 22 | 1.000 | s_{1}, s_{2} \in S | | |
| fong-spivak-seven-sketches_FO0177 | 22 | 1.000 | F\left(s_{1}\right)=t | | |
| fong-spivak-seven-sketches_FO0178 | 22 | 1.000 | F\left(s_{2}\right)=t | | |
| fong-spivak-seven-sketches_FO0179 | 22 | 1.000 | s_{1}=s_{2} | | |
| fong-spivak-seven-sketches_FO0180 | 23 | 0.413 | \rightarrow, \rightarrow,>, \stackrel{\cong}{\rightarrow} | | |
| fong-spivak-seven-sketches_FO0181 | 23 | 0.832 | \mathrm{id}_{X} | | |
| fong-spivak-seven-sketches_FO0182 | 23 | 0.832 | \operatorname{id}_{X}(x)=x | | |
| fong-spivak-seven-sketches_FO0183 | 23 | 1.000 | f: A \rightarrow B | | |
| fong-spivak-seven-sketches_FO0184 | 23 | 1.000 | f: A \rightarrow \varnothing | | |
| fong-spivak-seven-sketches_FO0185 | 23 | 1.000 | f: A \rightarrow P | | |
| fong-spivak-seven-sketches_FO0186 | 23 | 1.000 | f^{-1}(p) \subseteq A | | |
| fong-spivak-seven-sketches_FO0187 | 24 | 1.000 | S:=\{11,12,13,21,22,23\} | | |
| fong-spivak-seven-sketches_FO0188 | 24 | 1.000 | P=\{a, b, c, d\} | | |
| fong-spivak-seven-sketches_FO0189 | 24 | 1.000 | f: S \rightarrow P | | |
| fong-spivak-seven-sketches_FO0190 | 24 | 1.000 | F: X \rightarrow Y | | |
| fong-spivak-seven-sketches_FO0191 | 24 | 1.000 | G: Y \rightarrow Z | | |
| fong-spivak-seven-sketches_FO0192 | 24 | 1.000 | X \rightarrow Z | | |
| fong-spivak-seven-sketches_FO0193 | 24 | 1.000 | G(F(x)) | | |
| fong-spivak-seven-sketches_FO0194 | 24 | 1.000 | G \circ F | | |
| fong-spivak-seven-sketches_FO0195 | 24 | 1.000 | F ; G | | |
| fong-spivak-seven-sketches_FO0196 | 24 | 1.000 | F(x) \in Y | | |
| fong-spivak-seven-sketches_FO0197 | 24 | 1.000 | G | | |
| fong-spivak-seven-sketches_FO0198 | 24 | 1.000 | \{1\} \rightarrow X | | |
| fong-spivak-seven-sketches_FO0199 | 24 | 0.974 | \{1\} \rightarrow\{1,2,3\} | | |
| fong-spivak-seven-sketches_FO0200 | 24 | 0.974 | \{1,2,3\} | | |
| fong-spivak-seven-sketches_FO0201 | 24 | 1.000 | x:\{1\} \rightarrow X | | |
| fong-spivak-seven-sketches_FO0202 | 24 | 1.000 | F(x)=x ; F | | |
| fong-spivak-seven-sketches_FO0203 | 24 | 1.000 | x \leq x | | |
| fong-spivak-seven-sketches_FO0204 | 24 | 1.000 | y \leq z | | |
| fong-spivak-seven-sketches_FO0205 | 24 | 1.000 | x \leq z | | |
| fong-spivak-seven-sketches_FO0206 | 25 | 0.608 | y \leq x | | |
| fong-spivak-seven-sketches_FO0207 | 25 | 0.608 | x \cong y | | |
| fong-spivak-seven-sketches_FO0208 | 25 | 0.608 | (X, \leq) | | |
| fong-spivak-seven-sketches_FO0209 | 25 | 0.999 | (X,=) | | |
| fong-spivak-seven-sketches_FO0210 | 25 | 0.999 | x \neq y | | |
| fong-spivak-seven-sketches_FO0211 | 25 | 0.924 | X, \leq | | |
| fong-spivak-seven-sketches_FO0212 | 25 | 0.924 | X \times X \subseteq X \times X | | |
| fong-spivak-seven-sketches_FO0213 | 25 | 1.000 | x=y | | |
| fong-spivak-seven-sketches_FO0214 | 26 | 1.000 | G=(V, A, s, t) | | |
| fong-spivak-seven-sketches_FO0215 | 26 | 1.000 | V | | |
| fong-spivak-seven-sketches_FO0216 | 26 | 0.999 | s, t: A \rightarrow V | | |
| fong-spivak-seven-sketches_FO0217 | 26 | 1.000 | s(a)=v | | |
| fong-spivak-seven-sketches_FO0218 | 26 | 1.000 | t(a)=w | | |
| fong-spivak-seven-sketches_FO0219 | 26 | 1.000 | v | | |
| fong-spivak-seven-sketches_FO0220 | 26 | 1.000 | w | | |
| fong-spivak-seven-sketches_FO0221 | 26 | 1.000 | V=\{1,2,3,4\} | | |
| fong-spivak-seven-sketches_FO0222 | 26 | 1.000 | A=\{a, b, c, d, e\} | | |
| fong-spivak-seven-sketches_FO0223 | 26 | 1.000 | s, t: A \rightarrow | | |
| fong-spivak-seven-sketches_FO0226 | 26 | 1.000 | d | | |
| fong-spivak-seven-sketches_FO0227 | 26 | 0.862 | e | | |
| fong-spivak-seven-sketches_FO0228 | 26 | 1.000 | 1 \rightarrow 2 | | |
| fong-spivak-seven-sketches_FO0229 | 26 | 0.833 | (P, \leq) | | |
| fong-spivak-seven-sketches_FO0230 | 26 | 0.842 | v \leq w | | |
| fong-spivak-seven-sketches_FO0231 | 26 | 0.842 | ^{6} | | |
| fong-spivak-seven-sketches_FO0232 | 26 | 0.842 | v \rightarrow w | | |
| fong-spivak-seven-sketches_FO0233 | 26 | 1.000 | v \rightarrow v | | |
| fong-spivak-seven-sketches_FO0234 | 26 | 1.000 | u \rightarrow v | | |
| fong-spivak-seven-sketches_FO0235 | 26 | 1.000 | u \rightarrow w | | |
| fong-spivak-seven-sketches_FO0236 | 27 | 0.959 | \left(p_{1}, p_{2}\right) | | |
| fong-spivak-seven-sketches_FO0237 | 27 | 0.959 | p_{1} \leq p_{2} | | |
| fong-spivak-seven-sketches_FO0238 | 27 | 0.999 | \{\bullet, \circ, *\} | | |
| fong-spivak-seven-sketches_FO0239 | 27 | 0.929 | (x, y) | | |
| fong-spivak-seven-sketches_FO0240 | 27 | 1.000 | \mathbb{N}:=\{0,1,2,3, \ldots\} | | |
| fong-spivak-seven-sketches_FO0241 | 27 | 1.000 | 0 \leq 1 | | |
| fong-spivak-seven-sketches_FO0242 | 27 | 1.000 | 5 \leq 100 | | |
| fong-spivak-seven-sketches_FO0243 | 27 | 1.000 | m \leq n | | |
| fong-spivak-seven-sketches_FO0244 | 27 | 1.000 | n \leq m | | |
| fong-spivak-seven-sketches_FO0245 | 27 | 1.000 | m, n | | |
| fong-spivak-seven-sketches_FO0246 | 27 | 1.000 | a \leq b | | |
| fong-spivak-seven-sketches_FO0247 | 27 | 1.000 | b \leq a | | |
| fong-spivak-seven-sketches_FO0248 | 27 | 1.000 | n=m | | |
| fong-spivak-seven-sketches_FO0249 | 27 | 1.000 | 5 \leq 3 | | |
| fong-spivak-seven-sketches_FO0250 | 27 | 1.000 | 3 \leq 2 | | |
| fong-spivak-seven-sketches_FO0251 | 27 | 1.000 | m | | |
| fong-spivak-seven-sketches_FO0252 | 27 | 1.000 | 2 \leq 4 | | |
| fong-spivak-seven-sketches_FO0253 | 27 | 1.000 | 2 \not \leq 3 | | |
| fong-spivak-seven-sketches_FO0254 | 27 | 0.999 | 1,2, \ldots, 10 | | |
| fong-spivak-seven-sketches_FO0255 | 27 | 0.999 | a \rightarrow b | | |
| fong-spivak-seven-sketches_FO0256 | 28 | 0.967 | -500 \leq-499 \leq 0 \leq \sqrt{2} \leq 100 / 3 | | |
| fong-spivak-seven-sketches_FO0257 | 28 | 1.000 | \mathrm{P}(X) | | |
| fong-spivak-seven-sketches_FO0258 | 28 | 1.000 | X=\{0,1,2\} | | |
| fong-spivak-seven-sketches_FO0259 | 28 | 0.975 | \mathrm{P}\{1,2, \ldots, n\},{ }^{a} | | |
| fong-spivak-seven-sketches_FO0260 | 28 | 1.000 | \mathrm{P} X | | |
| fong-spivak-seven-sketches_FO0261 | 28 | 0.963 | \mathrm{P}(\varnothing), \mathrm{P}\{1\} | | |
| fong-spivak-seven-sketches_FO0262 | 28 | 0.963 | \mathrm{P}\{1,2\} | | |
| fong-spivak-seven-sketches_FO0263 | 29 | 1.000 | \operatorname{Prt}(A) | | |
| fong-spivak-seven-sketches_FO0264 | 29 | 1.000 | q \in Q | | |
| fong-spivak-seven-sketches_FO0265 | 29 | 1.000 | A_{p} \subseteq A_{q} | | |
| fong-spivak-seven-sketches_FO0266 | 29 | 1.000 | g: A \rightarrow Q | | |
| fong-spivak-seven-sketches_FO0267 | 29 | 1.000 | h: P \rightarrow Q | | |
| fong-spivak-seven-sketches_FO0268 | 29 | 1.000 | f ; h=g | | |
| fong-spivak-seven-sketches_FO0269 | 29 | 0.795 | P, \leq | | |
| fong-spivak-seven-sketches_FO0270 | 29 | 0.795 | U | | |
| fong-spivak-seven-sketches_FO0271 | 29 | 1.000 | p \in U | | |
| fong-spivak-seven-sketches_FO0272 | 29 | 1.000 | p \leq q | | |
| fong-spivak-seven-sketches_FO0273 | 29 | 1.000 | q \in U | | |
| fong-spivak-seven-sketches_FO0274 | 29 | 1.000 | p | | |
| fong-spivak-seven-sketches_FO0275 | 29 | 0.912 | \mathrm{U}(P) | | |
| fong-spivak-seven-sketches_FO0276 | 29 | 1.000 | U \leq V | | |
| fong-spivak-seven-sketches_FO0277 | 29 | 0.998 | (\mathbb{B}, \leq) | | |
| fong-spivak-seven-sketches_FO0278 | 29 | 0.457 | \mathrm{U}(\mathbb{B}) | | |
| fong-spivak-seven-sketches_FO0279 | 29 | 0.749 | \subseteq \mathbb{B} | | |
| fong-spivak-seven-sketches_FO0280 | 29 | 0.749 | \notin\{ | | |
| fong-spivak-seven-sketches_FO0281 | 29 | 0.998 | (Q, \leq) | | |
| fong-spivak-seven-sketches_FO0282 | 29 | 1.000 | P \times Q | | |
| fong-spivak-seven-sketches_FO0283 | 29 | 1.000 | (p, q) \leq\left(p^{\prime}, q^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO0284 | 29 | 1.000 | p \leq p^{\prime} | | |
| fong-spivak-seven-sketches_FO0285 | 29 | 1.000 | q \leq q^{\prime} | | |
| fong-spivak-seven-sketches_FO0286 | 30 | 0.889 | (P, \leq \mathrm{op}) | | |
| fong-spivak-seven-sketches_FO0287 | 30 | 0.889 | p \leq{ }^{\mathrm{op}} q | | |
| fong-spivak-seven-sketches_FO0288 | 30 | 0.889 | q \leq p | | |
| fong-spivak-seven-sketches_FO0289 | 30 | 0.988 | A, \leq_{A} | | |
| fong-spivak-seven-sketches_FO0290 | 30 | 0.988 | B, \leq_{B} | | |
| fong-spivak-seven-sketches_FO0291 | 30 | 0.872 | x, y \in A | | |
| fong-spivak-seven-sketches_FO0292 | 30 | 0.872 | x \leq_{A} y | | |
| fong-spivak-seven-sketches_FO0293 | 30 | 0.872 | f(x) \leq_{B} f(y) | | |
| fong-spivak-seven-sketches_FO0294 | 30 | 1.000 | A \rightarrow B | | |
| fong-spivak-seven-sketches_FO0295 | 30 | 1.000 | x \in A | | |
| fong-spivak-seven-sketches_FO0296 | 30 | 1.000 | f(x) \in B | | |
| fong-spivak-seven-sketches_FO0297 | 30 | 1.000 | f(y) | | |
| fong-spivak-seven-sketches_FO0298 | 31 | 1.000 | \mathbb{B} \rightarrow \mathbb{N} | | |
| fong-spivak-seven-sketches_FO0299 | 31 | 1.000 | x \rightarrow y | | |
| fong-spivak-seven-sketches_FO0300 | 31 | 1.000 | F(x) \rightarrow F(y) | | |
| fong-spivak-seven-sketches_FO0301 | 31 | 0.988 | |\cdot|: \mathrm{P}(X) \rightarrow \mathbb{N} | | |
| fong-spivak-seven-sketches_FO0302 | 31 | 1.000 | |S| | | |
| fong-spivak-seven-sketches_FO0303 | 31 | 1.000 | 0 \leq 1 \leq 2 \leq 3 | | |
| fong-spivak-seven-sketches_FO0304 | 32 | 0.997 | |\cdot| | | |
| fong-spivak-seven-sketches_FO0305 | 32 | 0.995 | \mathrm{U}(P) \rightarrow \mathrm{P}(P) | | |
| fong-spivak-seven-sketches_FO0306 | 32 | 0.939 | \mathrm{P}(\mathbb{B}) | | |
| fong-spivak-seven-sketches_FO0307 | 32 | 1.000 | \uparrow p:=\left\{p^{\prime} \in P \mid p \leq p^{\prime}\right\} | | |
| fong-spivak-seven-sketches_FO0308 | 32 | 0.671 | \uparrow: P^{\text {op }} \rightarrow \mathrm{U}(P) | | |
| fong-spivak-seven-sketches_FO0309 | 32 | 1.000 | \uparrow\left(p^{\prime}\right) \subseteq \uparrow(p) | | |
| fong-spivak-seven-sketches_FO0310 | 32 | 1.000 | b \geq a \leq c | | |
| fong-spivak-seven-sketches_FO0311 | 32 | 1.000 | f:\left(P, \leq_{P}\right) \rightarrow\left(Q, \leq_{Q}\right) | | |
| fong-spivak-seven-sketches_FO0312 | 32 | 1.000 | f: P \rightarrow Q | | |
| fong-spivak-seven-sketches_FO0313 | 32 | 1.000 | \left(P, \leq_{P}\right) | | |
| fong-spivak-seven-sketches_FO0314 | 32 | 1.000 | P \rightarrow Q | | |
| fong-spivak-seven-sketches_FO0315 | 32 | 1.000 | \leq_{Q} | | |
| fong-spivak-seven-sketches_FO0316 | 32 | 1.000 | \operatorname{Prt}(X) | | |
| fong-spivak-seven-sketches_FO0317 | 32 | 1.000 | s: X \rightarrow P | | |
| fong-spivak-seven-sketches_FO0318 | 32 | 1.000 | f^{*}: \operatorname{Prt}(Y) \rightarrow \operatorname{Prt}(X) | | |
| fong-spivak-seven-sketches_FO0319 | 32 | 0.994 | s: Y \rightarrow P | | |
| fong-spivak-seven-sketches_FO0320 | 32 | 0.905 | s | | |
| fong-spivak-seven-sketches_FO0321 | 32 | 0.905 | X \rightarrow P .^{7} | | |
| fong-spivak-seven-sketches_FO0322 | 32 | 1.000 | f^{*}(P) | | |
| fong-spivak-seven-sketches_FO0323 | 32 | 1.000 | f^{*}(Q) | | |
| fong-spivak-seven-sketches_FO0324 | 33 | 0.979 | \left(Q, \leq_{Q}\right) | | |
| fong-spivak-seven-sketches_FO0325 | 33 | 0.979 | \left(R, \leq_{R}\right) | | |
| fong-spivak-seven-sketches_FO0326 | 33 | 0.979 | g: Q \rightarrow R | | |
| fong-spivak-seven-sketches_FO0327 | 33 | 0.960 | (f ; g): P \rightarrow R | | |
| fong-spivak-seven-sketches_FO0328 | 33 | 0.629 | \operatorname{id}_{P}: P \rightarrow P | | |
| fong-spivak-seven-sketches_FO0329 | 33 | 0.629 | (P, \leq) \rightarrow\left(P, \leq^{\mathrm{op}}\right) | | |
| fong-spivak-seven-sketches_FO0330 | 33 | 1.000 | p, q \in P | | |
| fong-spivak-seven-sketches_FO0331 | 33 | 0.997 | p_{1}=p_{2} | | |
| fong-spivak-seven-sketches_FO0332 | 33 | 1.000 | g: Q \rightarrow P | | |
| fong-spivak-seven-sketches_FO0333 | 33 | 0.555 | f ; g=\operatorname{id}_{P} | | |
| fong-spivak-seven-sketches_FO0334 | 33 | 0.555 | g \circ f=\operatorname{id}_{Q} | | |
| fong-spivak-seven-sketches_FO0335 | 33 | 1.000 | g | | |
| fong-spivak-seven-sketches_FO0336 | 34 | 1.000 | P, Q | | |
| fong-spivak-seven-sketches_FO0337 | 34 | 1.000 | R | | |
| fong-spivak-seven-sketches_FO0338 | 34 | 1.000 | f(a)=v, f(b)=w, f(c)=x, f(d)=y | | |
| fong-spivak-seven-sketches_FO0339 | 34 | 1.000 | f(e)=z | | |
| fong-spivak-seven-sketches_FO0340 | 34 | 1.000 | x \rightarrow z | | |
| fong-spivak-seven-sketches_FO0341 | 34 | 1.000 | v \rightarrow y | | |
| fong-spivak-seven-sketches_FO0342 | 34 | 0.546 | \operatorname{Prt}(\{*, \bullet, \circ\}) \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO0343 | 34 | 1.000 | P \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO0344 | 34 | 0.656 | f: P \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO0345 | 34 | 0.656 | f^{-1}( | | |
| fong-spivak-seven-sketches_FO0346 | 34 | 0.656 | ) \subseteq P | | |
| fong-spivak-seven-sketches_FO0347 | 34 | 1.000 | p \in f^{-1} | | |
| fong-spivak-seven-sketches_FO0348 | 34 | 1.000 | =f(p) \leq f(q) | | |
| fong-spivak-seven-sketches_FO0349 | 34 | 0.842 | \leq f(q) | | |
| fong-spivak-seven-sketches_FO0350 | 34 | 0.842 | =f(q) | | |
| fong-spivak-seven-sketches_FO0351 | 34 | 0.842 | q \in f^{-1} | | |
| fong-spivak-seven-sketches_FO0352 | 34 | 1.000 | f^{-1} | | |
| fong-spivak-seven-sketches_FO0353 | 34 | 1.000 | f_{U}: P \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO0354 | 34 | 1.000 | f_{U}(p)= | | |
| fong-spivak-seven-sketches_FO0355 | 34 | 1.000 | p \notin U | | |
| fong-spivak-seven-sketches_FO0356 | 34 | 1.000 | f(p)= | | |
| fong-spivak-seven-sketches_FO0357 | 34 | 0.996 | f^{*}: \mathrm{U}(Q) \rightarrow \mathrm{U}(P) | | |
| fong-spivak-seven-sketches_FO0358 | 34 | 1.000 | U \subseteq Q | | |
| fong-spivak-seven-sketches_FO0359 | 34 | 1.000 | f^{-1}(U) \subseteq P | | |
| fong-spivak-seven-sketches_FO0360 | 34 | 1.000 | f^{*} | | |
| fong-spivak-seven-sketches_FO0361 | 34 | 1.000 | u: Q \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO0362 | 34 | 1.000 | (f ; u): P \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO0363 | 35 | 0.637 | (\mathbb{R}, \leq) | | |
| fong-spivak-seven-sketches_FO0364 | 35 | 1.000 | \mathbb{N} \subseteq \mathbb{R} | | |
| fong-spivak-seven-sketches_FO0365 | 35 | 1.000 | \left\{\left.\frac{1}{n+1} \right\rvert\, n \in \mathbb{N}\right\} \subseteq \mathbb{R} | | |
| fong-spivak-seven-sketches_FO0366 | 35 | 0.719 | A \subseteq P | | |
| fong-spivak-seven-sketches_FO0367 | 35 | 1.000 | p \leq a | | |
| fong-spivak-seven-sketches_FO0368 | 35 | 1.000 | q | | |
| fong-spivak-seven-sketches_FO0369 | 35 | 1.000 | q \leq a | | |
| fong-spivak-seven-sketches_FO0370 | 35 | 1.000 | p=\wedge A, p=\wedge_{a \in A} a | | |
| fong-spivak-seven-sketches_FO0371 | 35 | 0.999 | p=\wedge_{A} a | | |
| fong-spivak-seven-sketches_FO0372 | 35 | 0.999 | A=\{a, b\} | | |
| fong-spivak-seven-sketches_FO0373 | 35 | 0.999 | \wedge A | | |
| fong-spivak-seven-sketches_FO0374 | 35 | 1.000 | a \wedge b | | |
| fong-spivak-seven-sketches_FO0375 | 35 | 1.000 | a \leq p | | |
| fong-spivak-seven-sketches_FO0376 | 35 | 1.000 | a \leq q | | |
| fong-spivak-seven-sketches_FO0377 | 35 | 0.990 | p=\vee A | | |
| fong-spivak-seven-sketches_FO0378 | 35 | 0.990 | p=\bigvee_{a \in A} a | | |
| fong-spivak-seven-sketches_FO0379 | 35 | 0.990 | p=a \vee b | | |
| fong-spivak-seven-sketches_FO0380 | 36 | 0.980 | p=\wedge A | | |
| fong-spivak-seven-sketches_FO0381 | 36 | 0.980 | q=\wedge A | | |
| fong-spivak-seven-sketches_FO0382 | 36 | 0.980 | p \neq q! | | |
| fong-spivak-seven-sketches_FO0383 | 36 | 1.000 | p, q | | |
| fong-spivak-seven-sketches_FO0384 | 36 | 1.000 | p \cong q | | |
| fong-spivak-seven-sketches_FO0385 | 36 | 1.000 | x \in P | | |
| fong-spivak-seven-sketches_FO0386 | 36 | 1.000 | p \leq x | | |
| fong-spivak-seven-sketches_FO0387 | 36 | 1.000 | q \leq x | | |
| fong-spivak-seven-sketches_FO0388 | 36 | 1.000 | x \leq p | | |
| fong-spivak-seven-sketches_FO0389 | 36 | 1.000 | x \leq q | | |
| fong-spivak-seven-sketches_FO0390 | 36 | 1.000 | P=\{p, q, r\} | | |
| fong-spivak-seven-sketches_FO0391 | 36 | 1.000 | A=\{p, q\} | | |
| fong-spivak-seven-sketches_FO0392 | 36 | 0.965 | \{a, b\} | | |
| fong-spivak-seven-sketches_FO0393 | 36 | 0.998 | c \leq d | | |
| fong-spivak-seven-sketches_FO0394 | 36 | 0.998 | d \leq c | | |
| fong-spivak-seven-sketches_FO0395 | 36 | 0.998 | c \cong d | | |
| fong-spivak-seven-sketches_FO0396 | 36 | 0.519 | A=\{p\} | | |
| fong-spivak-seven-sketches_FO0397 | 36 | 1.000 | \wedge A \cong p | | |
| fong-spivak-seven-sketches_FO0398 | 36 | 1.000 | \wedge A=p | | |
| fong-spivak-seven-sketches_FO0399 | 36 | 0.999 | \wedge | | |
| fong-spivak-seven-sketches_FO0400 | 37 | 1.000 | p \vee p=p \wedge p=p | | |
| fong-spivak-seven-sketches_FO0401 | 37 | 0.796 | p \vee p | | |
| fong-spivak-seven-sketches_FO0402 | 37 | 0.796 | \vee\{p, p\} | | |
| fong-spivak-seven-sketches_FO0403 | 37 | 0.796 | \{p, p\}=\{p\} | | |
| fong-spivak-seven-sketches_FO0404 | 37 | 0.796 | p \vee p=p | | |
| fong-spivak-seven-sketches_FO0405 | 37 | 0.999 | A, B \subseteq X | | |
| fong-spivak-seven-sketches_FO0406 | 37 | 1.000 | A \wedge B=A \cap B | | |
| fong-spivak-seven-sketches_FO0407 | 37 | 1.000 | A \vee B=A \cup B | | |
| fong-spivak-seven-sketches_FO0408 | 37 | 1.000 | n \mid m | | |
| fong-spivak-seven-sketches_FO0409 | 37 | 0.706 | A \subseteq B \subseteq P | | |
| fong-spivak-seven-sketches_FO0410 | 37 | 1.000 | \wedge B \leq \wedge A | | |
| fong-spivak-seven-sketches_FO0411 | 37 | 0.998 | \bigvee A \leq \bigvee B | | |
| fong-spivak-seven-sketches_FO0412 | 37 | 1.000 | m=\wedge A | | |
| fong-spivak-seven-sketches_FO0413 | 37 | 1.000 | n=\wedge B | | |
| fong-spivak-seven-sketches_FO0414 | 37 | 1.000 | a \in B | | |
| fong-spivak-seven-sketches_FO0415 | 37 | 1.000 | n \leq a | | |
| fong-spivak-seven-sketches_FO0416 | 37 | 0.843 | A^{\prime} | | |
| fong-spivak-seven-sketches_FO0417 | 38 | 1.000 | \Phi: P \rightarrow Q | | |
| fong-spivak-seven-sketches_FO0418 | 38 | 1.000 | f(a \wedge b) \cong | | |
| fong-spivak-seven-sketches_FO0419 | 38 | 0.997 | f(a) \wedge f(b) | | |
| fong-spivak-seven-sketches_FO0420 | 38 | 0.997 | a, b \in P | | |
| fong-spivak-seven-sketches_FO0421 | 38 | 0.997 | f(a \vee b) \cong f(a) \vee f(b) | | |
| fong-spivak-seven-sketches_FO0422 | 38 | 1.000 | f(a) \vee f(b) | | |
| fong-spivak-seven-sketches_FO0423 | 38 | 1.000 | f(a \vee b) | | |
| fong-spivak-seven-sketches_FO0424 | 38 | 1.000 | a \vee b | | |
| fong-spivak-seven-sketches_FO0425 | 38 | 1.000 | f(a \vee b) \neq f(a) \vee f(b) | | |
| fong-spivak-seven-sketches_FO0426 | 38 | 1.000 | f(a), f(b) \in Q | | |
| fong-spivak-seven-sketches_FO0427 | 38 | 1.000 | f(a) \vee f(b) \leq f(a \vee b) | | |
| fong-spivak-seven-sketches_FO0428 | 39 | 0.889 | (3 \times-): \mathbb{Z} \rightarrow \mathbb{R} | | |
| fong-spivak-seven-sketches_FO0429 | 39 | 0.889 | x \in \mathbb{Z} | | |
| fong-spivak-seven-sketches_FO0430 | 39 | 0.889 | 3 x | | |
| fong-spivak-seven-sketches_FO0431 | 39 | 0.711 | 3 x \in \mathbb{Z} \subseteq \mathbb{R} | | |
| fong-spivak-seven-sketches_FO0432 | 39 | 0.711 | 3 \times- | | |
| fong-spivak-seven-sketches_FO0433 | 39 | 1.000 | \lceil z\rceil | | |
| fong-spivak-seven-sketches_FO0434 | 39 | 1.000 | z \in \mathbb{R} | | |
| fong-spivak-seven-sketches_FO0435 | 39 | 1.000 | \lfloor z\rfloor | | |
| fong-spivak-seven-sketches_FO0436 | 39 | 0.582 | \lceil 3.14\rceil=4 | | |
| fong-spivak-seven-sketches_FO0437 | 39 | 0.582 | \lfloor 3.14\rfloor=3 .^{a} | | |
| fong-spivak-seven-sketches_FO0438 | 39 | 0.582 | \mathbb{R} \rightarrow \mathbb{Z} | | |
| fong-spivak-seven-sketches_FO0439 | 39 | 1.000 | \lceil-/ 3\rceil | | |
| fong-spivak-seven-sketches_FO0440 | 39 | 0.788 | (3 \times-) | | |
| fong-spivak-seven-sketches_FO0441 | 39 | 1.000 | \lfloor 3\rfloor=3=\lceil 3\rceil | | |
| fong-spivak-seven-sketches_FO0442 | 39 | 0.989 | 3 \times | | |
| fong-spivak-seven-sketches_FO0443 | 39 | 0.945 | -): \mathbb{Z} \rightarrow \mathbb{R} | | |
| fong-spivak-seven-sketches_FO0444 | 39 | 1.000 | P=Q=\underline{3} | | |
| fong-spivak-seven-sketches_FO0445 | 39 | 1.000 | f, g | | |
| fong-spivak-seven-sketches_FO0446 | 39 | 0.999 | 1 \leq p, q \leq 3 | | |
| fong-spivak-seven-sketches_FO0447 | 39 | 1.000 | f(p) \leq q | | |
| fong-spivak-seven-sketches_FO0448 | 39 | 1.000 | p \leq g(q) | | |
| fong-spivak-seven-sketches_FO0449 | 40 | 1.000 | L: \mathbb{Z} \rightarrow \mathbb{R} | | |
| fong-spivak-seven-sketches_FO0450 | 40 | 1.000 | \operatorname{Prt}(S) | | |
| fong-spivak-seven-sketches_FO0451 | 40 | 1.000 | g: S \rightarrow T | | |
| fong-spivak-seven-sketches_FO0452 | 40 | 0.933 | g_{!}: \operatorname{Prt}(S) \leftrightarrows \operatorname{Prt}(T): g^{*} | | |
| fong-spivak-seven-sketches_FO0453 | 40 | 0.991 | g_{!}: \operatorname{Prt}(S) \rightarrow \operatorname{Prt}(T) | | |
| fong-spivak-seven-sketches_FO0454 | 40 | 1.000 | \sim_{S} | | |
| fong-spivak-seven-sketches_FO0455 | 40 | 1.000 | \sim_{T} | | |
| fong-spivak-seven-sketches_FO0456 | 40 | 0.997 | t_{1}, t_{2} \in T | | |
| fong-spivak-seven-sketches_FO0457 | 40 | 0.997 | t_{1} \sim_{T} t_{2} | | |
| fong-spivak-seven-sketches_FO0458 | 40 | 0.990 | s_{1} \sim_{S} s_{2} | | |
| fong-spivak-seven-sketches_FO0459 | 40 | 0.990 | g\left(s_{1}\right)=t_{1} | | |
| fong-spivak-seven-sketches_FO0460 | 40 | 0.990 | g\left(s_{2}\right)=t_{2} | | |
| fong-spivak-seven-sketches_FO0461 | 41 | 1.000 | t_{2} \sim_{T} t_{3} | | |
| fong-spivak-seven-sketches_FO0462 | 41 | 1.000 | t_{1} \sim_{T}^{?} t_{3} | | |
| fong-spivak-seven-sketches_FO0463 | 41 | 0.522 | g_{!}(\sim S):=\sim T | | |
| fong-spivak-seven-sketches_FO0464 | 41 | 0.813 | g^{*}: \operatorname{Prt}(T) \rightarrow \operatorname{Prt}(S) | | |
| fong-spivak-seven-sketches_FO0465 | 41 | 1.000 | g\left(s_{1}\right) \sim_{T} g\left(s_{2}\right) | | |
| fong-spivak-seven-sketches_FO0466 | 41 | 1.000 | g^{*}\left(\sim_{T}\right):=\sim_{S} | | |
| fong-spivak-seven-sketches_FO0467 | 41 | 1.000 | S=\{1,2,3,4\}, T=\{12,3,4\} | | |
| fong-spivak-seven-sketches_FO0468 | 41 | 1.000 | g(1):=g(2):=12 | | |
| fong-spivak-seven-sketches_FO0469 | 41 | 0.813 | g(3):=3 | | |
| fong-spivak-seven-sketches_FO0470 | 41 | 0.813 | g(4):=4 | | |
| fong-spivak-seven-sketches_FO0471 | 41 | 0.813 | g_{!} | | |
| fong-spivak-seven-sketches_FO0472 | 41 | 0.999 | c: S \rightarrow P | | |
| fong-spivak-seven-sketches_FO0473 | 41 | 0.999 | g_{!}(c) | | |
| fong-spivak-seven-sketches_FO0474 | 41 | 0.999 | S, T | | |
| fong-spivak-seven-sketches_FO0475 | 41 | 1.000 | g^{*} | | |
| fong-spivak-seven-sketches_FO0476 | 41 | 1.000 | T=\{12,3,4\} | | |
| fong-spivak-seven-sketches_FO0477 | 41 | 1.000 | g^{*}(c) | | |
| fong-spivak-seven-sketches_FO0478 | 41 | 1.000 | c: T \rightarrow P | | |
| fong-spivak-seven-sketches_FO0479 | 42 | 1.000 | d: T \rightarrow P^{\prime} | | |
| fong-spivak-seven-sketches_FO0480 | 42 | 1.000 | g_{!}(c) \leq d | | |
| fong-spivak-seven-sketches_FO0481 | 42 | 1.000 | e: T \rightarrow Q | | |
| fong-spivak-seven-sketches_FO0482 | 42 | 1.000 | g_{!}(c) \not \leq e | | |
| fong-spivak-seven-sketches_FO0483 | 42 | 1.000 | g^{*}(d) | | |
| fong-spivak-seven-sketches_FO0484 | 42 | 1.000 | g^{*}(e) | | |
| fong-spivak-seven-sketches_FO0485 | 42 | 0.999 | c \leq g^{*}(d) | | |
| fong-spivak-seven-sketches_FO0486 | 42 | 0.999 | c \not \leq g^{*}(e) | | |
| fong-spivak-seven-sketches_FO0487 | 42 | 1.000 | q:=f(p) | | |
| fong-spivak-seven-sketches_FO0488 | 42 | 1.000 | p \leq g(f(p)) | | |
| fong-spivak-seven-sketches_FO0489 | 42 | 1.000 | f(g(q)) \leq q | | |
| fong-spivak-seven-sketches_FO0490 | 42 | 1.000 | g(f(p)) \leq g(q) | | |
| fong-spivak-seven-sketches_FO0491 | 42 | 1.000 | g^{\prime} | | |
| fong-spivak-seven-sketches_FO0492 | 42 | 1.000 | g(q) \cong g^{\prime}(q) | | |
| fong-spivak-seven-sketches_FO0493 | 43 | 1.000 | A \subseteq Q | | |
| fong-spivak-seven-sketches_FO0494 | 43 | 1.000 | g(A):=\{g(a) \mid a \in A\} | | |
| fong-spivak-seven-sketches_FO0495 | 43 | 1.000 | \wedge A \in Q | | |
| fong-spivak-seven-sketches_FO0496 | 43 | 1.000 | g(A) | | |
| fong-spivak-seven-sketches_FO0497 | 43 | 1.000 | \wedge g(A) | | |
| fong-spivak-seven-sketches_FO0498 | 43 | 1.000 | \bigvee A \in P | | |
| fong-spivak-seven-sketches_FO0499 | 43 | 1.000 | f(A) | | |
| fong-spivak-seven-sketches_FO0500 | 43 | 1.000 | \bigvee f(A) | | |
| fong-spivak-seven-sketches_FO0501 | 43 | 1.000 | m:=\wedge A | | |
| fong-spivak-seven-sketches_FO0502 | 43 | 1.000 | g(m) \leq g(a) | | |
| fong-spivak-seven-sketches_FO0503 | 43 | 1.000 | g(m) | | |
| fong-spivak-seven-sketches_FO0504 | 43 | 1.000 | b \leq g(a) | | |
| fong-spivak-seven-sketches_FO0505 | 43 | 1.000 | b \leq g(m) | | |
| fong-spivak-seven-sketches_FO0506 | 43 | 1.000 | f(b) \leq a | | |
| fong-spivak-seven-sketches_FO0507 | 43 | 1.000 | f(b) | | |
| fong-spivak-seven-sketches_FO0508 | 43 | 1.000 | f(b) \leq m | | |
| fong-spivak-seven-sketches_FO0509 | 43 | 1.000 | f(3.9):=4 | | |
| fong-spivak-seven-sketches_FO0510 | 44 | 1.000 | 1 \vee 2=4 | | |
| fong-spivak-seven-sketches_FO0511 | 44 | 1.000 | g(1) \vee g(2)=1 \vee 2=3.9 \neq 4=g(4) | | |
| fong-spivak-seven-sketches_FO0512 | 44 | 1.000 | \left\{q^{\prime} \in Q \mid p^{\prime} \leq g\left(q^{\prime}\right)\right\} \subseteq\{q \in | | |
| fong-spivak-seven-sketches_FO0513 | 44 | 1.000 | Q \mid p \leq g(q)\} | | |
| fong-spivak-seven-sketches_FO0514 | 44 | 1.000 | f(p) \leq f\left(p^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO0515 | 44 | 1.000 | p_{0} \leq g\left(f\left(p_{0}\right)\right) | | |
| fong-spivak-seven-sketches_FO0516 | 44 | 1.000 | f\left(g\left(q_{0}\right)\right) \leq q_{0} | | |
| fong-spivak-seven-sketches_FO0517 | 44 | 1.000 | p_{0} \in P | | |
| fong-spivak-seven-sketches_FO0518 | 44 | 1.000 | q_{0} \in Q | | |
| fong-spivak-seven-sketches_FO0519 | 44 | 1.000 | p_{0} | | |
| fong-spivak-seven-sketches_FO0520 | 44 | 1.000 | \left\{q_{0}\right\} \subseteq\left\{q \in Q \mid g\left(q_{0}\right) \leq\right. | | |
| fong-spivak-seven-sketches_FO0521 | 44 | 0.999 | g(q)\} | | |
| fong-spivak-seven-sketches_FO0522 | 44 | 0.999 | \wedge\left\{q_{0}\right\}=q_{0} | | |
| fong-spivak-seven-sketches_FO0523 | 44 | 1.000 | f^{*}: \mathrm{P}(Y) \rightarrow \mathrm{P}(X) | | |
| fong-spivak-seven-sketches_FO0524 | 44 | 1.000 | B^{\prime} \subseteq B | | |
| fong-spivak-seven-sketches_FO0525 | 44 | 1.000 | f^{-1}\left(B^{\prime}\right) \subseteq A | | |
| fong-spivak-seven-sketches_FO0526 | 44 | 1.000 | B^{\prime} | | |
| fong-spivak-seven-sketches_FO0527 | 45 | 1.000 | f_{!}(A) | | |
| fong-spivak-seven-sketches_FO0528 | 45 | 1.000 | A^{\prime} \subseteq A | | |
| fong-spivak-seven-sketches_FO0529 | 45 | 1.000 | f_{*} | | |
| fong-spivak-seven-sketches_FO0530 | 45 | 0.971 | f^{*}, f_{!} | | |
| fong-spivak-seven-sketches_FO0531 | 45 | 1.000 | B_{1}, B_{2} \subseteq Y | | |
| fong-spivak-seven-sketches_FO0532 | 45 | 1.000 | f^{*}\left(B_{1}\right) | | |
| fong-spivak-seven-sketches_FO0533 | 45 | 1.000 | f^{*}\left(B_{2}\right) | | |
| fong-spivak-seven-sketches_FO0534 | 45 | 1.000 | A_{1}, A_{2} \subseteq X | | |
| fong-spivak-seven-sketches_FO0535 | 45 | 1.000 | f_{!}\left(A_{1}\right) | | |
| fong-spivak-seven-sketches_FO0536 | 45 | 1.000 | f_{!}\left(A_{2}\right) | | |
| fong-spivak-seven-sketches_FO0537 | 45 | 1.000 | f_{*}\left(A_{1}\right) | | |
| fong-spivak-seven-sketches_FO0538 | 45 | 1.000 | f_{*}\left(A_{2}\right) | | |
| fong-spivak-seven-sketches_FO0539 | 45 | 1.000 | f ; g: P \rightarrow P | | |
| fong-spivak-seven-sketches_FO0540 | 45 | 0.988 | p \leq(f ; g)(p) | | |
| fong-spivak-seven-sketches_FO0541 | 45 | 0.988 | (f ; g ; f ; g)(p) \cong(f ; g)(p) | | |
| fong-spivak-seven-sketches_FO0542 | 45 | 1.000 | \geq | | |
| fong-spivak-seven-sketches_FO0543 | 45 | 1.000 | j: P \rightarrow P | | |
| fong-spivak-seven-sketches_FO0544 | 45 | 0.966 | g \circ f | | |
| fong-spivak-seven-sketches_FO0545 | 45 | 0.966 | (g \circ f)(q) \leq q | | |
| fong-spivak-seven-sketches_FO0546 | 45 | 0.966 | (g \circ f \circ g \circ f)(q) \cong(g \circ f)(q) | | |
| fong-spivak-seven-sketches_FO0547 | 46 | 0.999 | p \leq j(p) | | |
| fong-spivak-seven-sketches_FO0548 | 46 | 1.000 | j(j(p)) \cong j(p) | | |
| fong-spivak-seven-sketches_FO0549 | 46 | 1.000 | 7+2+3 | | |
| fong-spivak-seven-sketches_FO0550 | 46 | 0.986 | j: \exp \rightarrow \exp | | |
| fong-spivak-seven-sketches_FO0551 | 46 | 1.000 | j(x) | | |
| fong-spivak-seven-sketches_FO0552 | 46 | 1.000 | j(y) | | |
| fong-spivak-seven-sketches_FO0553 | 46 | 1.000 | x \leq j(x) | | |
| fong-spivak-seven-sketches_FO0554 | 46 | 1.000 | j | | |
| fong-spivak-seven-sketches_FO0555 | 46 | 1.000 | j(j(x))=j(x) | | |
| fong-spivak-seven-sketches_FO0556 | 46 | 0.771 | { }_{j} | | |
| fong-spivak-seven-sketches_FO0557 | 46 | 1.000 | j(p) | | |
| fong-spivak-seven-sketches_FO0558 | 46 | 0.502 | j: P \rightarrow | | |
| fong-spivak-seven-sketches_FO0559 | 46 | 0.502 | _{j} | | |
| fong-spivak-seven-sketches_FO0560 | 46 | 0.910 | _{j} \rightarrow P | | |
| fong-spivak-seven-sketches_FO0561 | 46 | 0.990 | q \in \operatorname{fix}_{j} | | |
| fong-spivak-seven-sketches_FO0562 | 46 | 0.990 | j(p) \leq q | | |
| fong-spivak-seven-sketches_FO0563 | 46 | 1.000 | j(p) \leq j(q) \cong q | | |
| fong-spivak-seven-sketches_FO0564 | 47 | 0.999 | p \Rightarrow q | | |
| fong-spivak-seven-sketches_FO0565 | 47 | 0.999 | p \Rightarrow p | | |
| fong-spivak-seven-sketches_FO0566 | 47 | 0.999 | q \Rightarrow r | | |
| fong-spivak-seven-sketches_FO0567 | 47 | 0.999 | p \Rightarrow r | | |
| fong-spivak-seven-sketches_FO0568 | 47 | 1.000 | p \Rightarrow j(p) | | |
| fong-spivak-seven-sketches_FO0569 | 47 | 1.000 | j(j(p))=j(p) | | |
| fong-spivak-seven-sketches_FO0570 | 47 | 0.966 | p^{\prime \prime} | | |
| fong-spivak-seven-sketches_FO0571 | 47 | 0.966 | B \Rightarrow p | | |
| fong-spivak-seven-sketches_FO0572 | 47 | 0.966 | B \Rightarrow- | | |
| fong-spivak-seven-sketches_FO0573 | 47 | 1.000 | p^{\prime} | | |
| fong-spivak-seven-sketches_FO0574 | 47 | 0.622 | \boldsymbol{\operatorname { R e l }}(S) | | |
| fong-spivak-seven-sketches_FO0575 | 47 | 0.622 | R \in \boldsymbol{\operatorname { R e l }}(S) | | |
| fong-spivak-seven-sketches_FO0576 | 47 | 1.000 | R \subseteq S \times S | | |
| fong-spivak-seven-sketches_FO0577 | 47 | 1.000 | R \subseteq R^{\prime} | | |
| fong-spivak-seven-sketches_FO0578 | 47 | 1.000 | R\left(s_{1}, s_{2}\right) | | |
| fong-spivak-seven-sketches_FO0579 | 47 | 1.000 | R^{\prime}\left(s_{1}, s_{2}\right) | | |
| fong-spivak-seven-sketches_FO0580 | 47 | 0.999 | \boldsymbol{\operatorname { R e l }}(\{1\}) | | |
| fong-spivak-seven-sketches_FO0581 | 47 | 0.998 | \boldsymbol{\operatorname { R e l }}(\{1,2\}) | | |
| fong-spivak-seven-sketches_FO0582 | 47 | 0.755 | \{1,2\} | | |
| fong-spivak-seven-sketches_FO0583 | 47 | 1.000 | \boldsymbol{\operatorname { P o s }}(S) | | |
| fong-spivak-seven-sketches_FO0584 | 47 | 0.961 | \sqsubseteq | | |
| fong-spivak-seven-sketches_FO0585 | 47 | 0.982 | \leq^{\prime} | | |
| fong-spivak-seven-sketches_FO0586 | 47 | 0.982 | \leq \underline{\leq} \leq^{\prime} | | |
| fong-spivak-seven-sketches_FO0587 | 47 | 0.982 | a \leq^{\prime} b | | |
| fong-spivak-seven-sketches_FO0588 | 47 | 0.982 | a, b \in S | | |
| fong-spivak-seven-sketches_FO0589 | 47 | 1.000 | \boldsymbol{\operatorname { P o s }}(S) \rightarrow \boldsymbol{\operatorname { R e l }}(S) | | |
| fong-spivak-seven-sketches_FO0590 | 47 | 0.977 | \mathrm{Cl}: \boldsymbol{\operatorname { R e l }}(S) \rightarrow \boldsymbol{\operatorname { P o s }}(S) | | |
| fong-spivak-seven-sketches_FO0591 | 47 | 1.000 | s \leq s | | |
| fong-spivak-seven-sketches_FO0592 | 47 | 1.000 | s \leq u | | |
| fong-spivak-seven-sketches_FO0593 | 47 | 1.000 | s \leq t | | |
| fong-spivak-seven-sketches_FO0594 | 47 | 1.000 | t \leq u | | |
| fong-spivak-seven-sketches_FO0595 | 47 | 1.000 | S=\{1,2,3\} | | |
| fong-spivak-seven-sketches_FO0596 | 47 | 1.000 | U(\leq) | | |
| fong-spivak-seven-sketches_FO0597 | 47 | 1.000 | U(\leq):=\left\{\left(s_{1}, s_{2}\right) \mid s_{1} \leq s_{2}\right\} \subseteq S \times S | | |
| fong-spivak-seven-sketches_FO0598 | 48 | 1.000 | Q \subseteq S \times S | | |
| fong-spivak-seven-sketches_FO0599 | 48 | 1.000 | Q^{\prime} \subseteq S \times S | | |
| fong-spivak-seven-sketches_FO0600 | 48 | 1.000 | Q \subseteq U(\leq) | | |
| fong-spivak-seven-sketches_FO0601 | 48 | 1.000 | Q^{\prime} \nsubseteq U(\leq) | | |
| fong-spivak-seven-sketches_FO0602 | 48 | 1.000 | Q, Q^{\prime} | | |
| fong-spivak-seven-sketches_FO0603 | 48 | 0.973 | \mathrm{Cl}(Q) \sqsubseteq \leq | | |
| fong-spivak-seven-sketches_FO0604 | 48 | 0.973 | \operatorname{Pos}(S) | | |
| fong-spivak-seven-sketches_FO0605 | 48 | 0.986 | \mathrm{Cl}\left(Q^{\prime}\right) \nsubseteq \leq | | |
| fong-spivak-seven-sketches_FO0606 | 48 | 1.000 | a \otimes b | | |
| fong-spivak-seven-sketches_FO0609 | 53 | 1.000 | d \in[0, \infty] | | |
| fong-spivak-seven-sketches_FO0610 | 53 | 1.000 | I \in X | | |
| fong-spivak-seven-sketches_FO0611 | 53 | 0.983 | \otimes: X \times X \rightarrow X | | |
| fong-spivak-seven-sketches_FO0612 | 53 | 0.997 | \otimes\left(x_{1}, x_{2}\right)= | | |
| fong-spivak-seven-sketches_FO0613 | 53 | 1.000 | x_{1} \otimes x_{2} | | |
| fong-spivak-seven-sketches_FO0614 | 54 | 1.000 | x_{1}, x_{2}, y_{1}, y_{2} \in X | | |
| fong-spivak-seven-sketches_FO0615 | 54 | 1.000 | x_{1} \leq y_{1} | | |
| fong-spivak-seven-sketches_FO0616 | 54 | 1.000 | x_{2} \leq y_{2} | | |
| fong-spivak-seven-sketches_FO0617 | 54 | 1.000 | x_{1} \otimes x_{2} \leq y_{1} \otimes y_{2} | | |
| fong-spivak-seven-sketches_FO0618 | 54 | 1.000 | I \otimes x=x | | |
| fong-spivak-seven-sketches_FO0619 | 54 | 1.000 | x \otimes I=x | | |
| fong-spivak-seven-sketches_FO0620 | 54 | 1.000 | x, y, z \in X | | |
| fong-spivak-seven-sketches_FO0621 | 54 | 1.000 | (x \otimes y) \otimes z=x \otimes(y \otimes z) | | |
| fong-spivak-seven-sketches_FO0622 | 54 | 1.000 | x \otimes y=y \otimes x | | |
| fong-spivak-seven-sketches_FO0623 | 54 | 0.849 | (X, \leq, I, \otimes) | | |
| fong-spivak-seven-sketches_FO0624 | 54 | 1.000 | I | | |
| fong-spivak-seven-sketches_FO0625 | 54 | 0.933 | \cong | | |
| fong-spivak-seven-sketches_FO0626 | 54 | 0.585 | \{*\} | | |
| fong-spivak-seven-sketches_FO0627 | 54 | 0.999 | -5 \leq \sqrt{2} | | |
| fong-spivak-seven-sketches_FO0628 | 54 | 0.999 | +: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} | | |
| fong-spivak-seven-sketches_FO0629 | 54 | 0.902 | (\mathbb{R}, \leq, 0,+) | | |
| fong-spivak-seven-sketches_FO0630 | 54 | 1.000 | x_{1}+x_{2} \leq y_{1}+y_{2} | | |
| fong-spivak-seven-sketches_FO0631 | 54 | 1.000 | 0+x=x | | |
| fong-spivak-seven-sketches_FO0632 | 54 | 0.989 | x+0=x | | |
| fong-spivak-seven-sketches_FO0633 | 54 | 0.989 | (x+y)+z=x+(y+z) | | |
| fong-spivak-seven-sketches_FO0634 | 54 | 0.989 | x+y=y+x | | |
| fong-spivak-seven-sketches_FO0635 | 54 | 0.554 | M | | |
| fong-spivak-seven-sketches_FO0636 | 54 | 0.554 | M \times M \rightarrow M | | |
| fong-spivak-seven-sketches_FO0637 | 54 | 1.000 | e \in M | | |
| fong-spivak-seven-sketches_FO0638 | 54 | 0.944 | *(m, n) | | |
| fong-spivak-seven-sketches_FO0639 | 54 | 0.944 | m * n | | |
| fong-spivak-seven-sketches_FO0640 | 54 | 1.000 | m, n, p \in M | | |
| fong-spivak-seven-sketches_FO0641 | 54 | 1.000 | m * n=n * m | | |
| fong-spivak-seven-sketches_FO0642 | 54 | 0.912 | \mathbf{D i s c}_{S} | | |
| fong-spivak-seven-sketches_FO0643 | 54 | 0.912 | m=n | | |
| fong-spivak-seven-sketches_FO0644 | 54 | 1.000 | (M, e, *) | | |
| fong-spivak-seven-sketches_FO0645 | 54 | 0.770 | \left(\right. | | |
| fong-spivak-seven-sketches_FO0646 | 54 | 0.770 | \left._{M},=, e, *\right) | | |
| fong-spivak-seven-sketches_FO0647 | 55 | 0.896 | (M, *, e) | | |
| fong-spivak-seven-sketches_FO0648 | 55 | 0.447 | \left._{M},=, *, e\right) | | |
| fong-spivak-seven-sketches_FO0649 | 55 | 1.000 | H | | |
| fong-spivak-seven-sketches_FO0650 | 55 | 1.000 | h \leq h^{\prime} | | |
| fong-spivak-seven-sketches_FO0651 | 55 | 1.000 | h^{\prime} | | |
| fong-spivak-seven-sketches_FO0652 | 55 | 1.000 | \otimes: H \times H \rightarrow H | | |
| fong-spivak-seven-sketches_FO0653 | 55 | 1.000 | h_{1} \otimes h_{2} | | |
| fong-spivak-seven-sketches_FO0654 | 55 | 1.000 | h_{1} | | |
| fong-spivak-seven-sketches_FO0655 | 55 | 1.000 | h_{2} | | |
| fong-spivak-seven-sketches_FO0656 | 55 | 0.355 | \{2 \odot, 5 \diamond, 6 \diamond, 6 \star, 7 \star\}=\{2 \diamond, 3 \diamond, 4 \diamond, 5 \diamond, 6 \circlearrowright\} | | |
| fong-spivak-seven-sketches_FO0657 | 55 | 1.000 | h_{1} \leq i_{1} | | |
| fong-spivak-seven-sketches_FO0658 | 55 | 1.000 | h_{2} \leq i_{2} | | |
| fong-spivak-seven-sketches_FO0659 | 55 | 1.000 | h_{1} \otimes h_{2} \leq i_{1} \otimes i_{2} | | |
| fong-spivak-seven-sketches_FO0660 | 55 | 0.925 | h_{1} \otimes h_{2}=\{10 \star, \mathrm{~J} \star, \mathrm{Q} \star, \mathrm{K} \star, \mathrm{A} \star\} | | |
| fong-spivak-seven-sketches_FO0661 | 55 | 0.999 | i_{1} \otimes i_{2}=\{5 \bullet, 5 \diamond, 6 \diamond, 6 \diamond, Q \diamond\} | | |
| fong-spivak-seven-sketches_FO0662 | 55 | 0.973 | 0,1,2 | | |
| fong-spivak-seven-sketches_FO0663 | 56 | 0.974 | (x \in X) | | |
| fong-spivak-seven-sketches_FO0664 | 56 | 0.974 | (\leq) | | |
| fong-spivak-seven-sketches_FO0665 | 56 | 1.000 | x \otimes y | | |
| fong-spivak-seven-sketches_FO0666 | 57 | 0.997 | x_{1} \otimes x_{2} \leq y_{1} \otimes y_{2} \otimes y_{3} | | |
| fong-spivak-seven-sketches_FO0667 | 57 | 0.990 | 4 \leq 7 | | |
| fong-spivak-seven-sketches_FO0668 | 57 | 1.000 | 2+5 \leq-1+5+3 | | |
| fong-spivak-seven-sketches_FO0669 | 58 | 1.000 | x_{1} \otimes x_{2} \leq | | |
| fong-spivak-seven-sketches_FO0670 | 58 | 1.000 | y_{1} \otimes y_{2} | | |
| fong-spivak-seven-sketches_FO0671 | 59 | 0.985 | \mathcal{X}=(X, \leq I, \otimes) | | |
| fong-spivak-seven-sketches_FO0672 | 59 | 0.381 | \mathcal{X} | | |
| fong-spivak-seven-sketches_FO0673 | 59 | 1.000 | v+w+u | | |
| fong-spivak-seven-sketches_FO0674 | 59 | 1.000 | (v+w)+u | | |
| fong-spivak-seven-sketches_FO0675 | 59 | 1.000 | v+(w+u) | | |
| fong-spivak-seven-sketches_FO0676 | 61 | 1.000 | 2 \mathrm{H}_{2} \mathrm{O}+2 \mathrm{Na} | | |
| fong-spivak-seven-sketches_FO0677 | 61 | 1.000 | 2 \mathrm{NaOH}+\mathrm{H}_{2} | | |
| fong-spivak-seven-sketches_FO0678 | 61 | 0.997 | \in | | |
| fong-spivak-seven-sketches_FO0679 | 61 | 0.543 | 4 \mathrm{H}_{2} \mathrm{O}+6 \mathrm{Ne} \in | | |
| fong-spivak-seven-sketches_FO0680 | 61 | 1.000 | k | | |
| fong-spivak-seven-sketches_FO0681 | 61 | 1.000 | x+y \rightarrow z | | |
| fong-spivak-seven-sketches_FO0682 | 61 | 1.000 | x+y+k \rightarrow z+k | | |
| fong-spivak-seven-sketches_FO0683 | 62 | 1.000 | x \leq I | | |
| fong-spivak-seven-sketches_FO0684 | 62 | 1.000 | x \rightarrow 0 | | |
| fong-spivak-seven-sketches_FO0685 | 63 | 1.000 | 10^{23} | | |
| fong-spivak-seven-sketches_FO0686 | 64 | 1.000 | x \leq x+x | | |
| fong-spivak-seven-sketches_FO0687 | 64 | 0.904 | =0 | | |
| fong-spivak-seven-sketches_FO0688 | 64 | 0.904 | =1 | | |
| fong-spivak-seven-sketches_FO0689 | 64 | 0.299 | =(\mathbb{B}, \leq | | |
| fong-spivak-seven-sketches_FO0690 | 64 | 0.299 | \Lambda) | | |
| fong-spivak-seven-sketches_FO0691 | 64 | 0.986 | \mathcal{V}=(V, \leq, I, \otimes) | | |
| fong-spivak-seven-sketches_FO0692 | 64 | 0.928 | \otimes | | |
| fong-spivak-seven-sketches_FO0693 | 65 | 0.182 | (\mathbb{B}, \leq | | |
| fong-spivak-seven-sketches_FO0694 | 65 | 0.182 | \wedge) | | |
| fong-spivak-seven-sketches_FO0695 | 65 | 0.666 | \mathbb{B}, \leq | | |
| fong-spivak-seven-sketches_FO0696 | 65 | 0.991 | (\mathbb{N}, \leq) | | |
| fong-spivak-seven-sketches_FO0697 | 65 | 0.991 | 5 \leq 16 | | |
| fong-spivak-seven-sketches_FO0698 | 65 | 0.994 | 6+4=10 | | |
| fong-spivak-seven-sketches_FO0699 | 65 | 0.992 | * | | |
| fong-spivak-seven-sketches_FO0700 | 65 | 0.992 | 6 * 4=24 | | |
| fong-spivak-seven-sketches_FO0701 | 65 | 1.000 | m \mid n | | |
| fong-spivak-seven-sketches_FO0702 | 65 | 1.000 | 1 \mid m | | |
| fong-spivak-seven-sketches_FO0703 | 65 | 1.000 | 4 \mid 12 | | |
| fong-spivak-seven-sketches_FO0704 | 65 | 0.995 | (\mathbb{N}, \mid) | | |
| fong-spivak-seven-sketches_FO0705 | 65 | 0.912 | x_{1} \mid y_{1} | | |
| fong-spivak-seven-sketches_FO0706 | 65 | 0.912 | x_{2} \mid y_{2} | | |
| fong-spivak-seven-sketches_FO0707 | 65 | 0.912 | \left(x_{1} * x_{2}\right) \mid\left(y_{1} * y_{2}\right) | | |
| fong-spivak-seven-sketches_FO0708 | 65 | 1.000 | p_{1}, p_{2} \in \mathbb{N} | | |
| fong-spivak-seven-sketches_FO0709 | 65 | 1.000 | x_{1} * p_{1}=y_{1} | | |
| fong-spivak-seven-sketches_FO0710 | 65 | 1.000 | x_{2} * p_{2}=y_{2} | | |
| fong-spivak-seven-sketches_FO0711 | 65 | 1.000 | \left(p_{1} * p_{2}\right) *\left(x_{1} * x_{2}\right)=y_{1} * y_{2} | | |
| fong-spivak-seven-sketches_FO0712 | 66 | 1.000 | \mathrm{P}(S) | | |
| fong-spivak-seven-sketches_FO0713 | 66 | 1.000 | \varnothing \subseteq S | | |
| fong-spivak-seven-sketches_FO0714 | 66 | 1.000 | S \subseteq S | | |
| fong-spivak-seven-sketches_FO0715 | 66 | 1.000 | A \subseteq B | | |
| fong-spivak-seven-sketches_FO0716 | 66 | 1.000 | A \cap B | | |
| fong-spivak-seven-sketches_FO0717 | 66 | 0.902 | { }^{\mathbb{N}} | | |
| fong-spivak-seven-sketches_FO0718 | 66 | 0.901 | \operatorname{Prop}^{\mathbb{N}} | | |
| fong-spivak-seven-sketches_FO0719 | 66 | 0.993 | n=2 | | |
| fong-spivak-seven-sketches_FO0720 | 66 | 0.993 | n \geq 11 | | |
| fong-spivak-seven-sketches_FO0721 | 66 | 0.993 | n+2=5 | | |
| fong-spivak-seven-sketches_FO0722 | 66 | 0.992 | P, Q \in \operatorname{Prop}^{\mathbb{N}} | | |
| fong-spivak-seven-sketches_FO0723 | 66 | 0.992 | P \leq Q | | |
| fong-spivak-seven-sketches_FO0724 | 66 | 1.000 | P(n) | | |
| fong-spivak-seven-sketches_FO0725 | 66 | 1.000 | Q(n) | | |
| fong-spivak-seven-sketches_FO0726 | 66 | 1.000 | [0, \infty] | | |
| fong-spivak-seven-sketches_FO0727 | 66 | 0.999 | 0,1,15.33 \overline{3} | | |
| fong-spivak-seven-sketches_FO0728 | 66 | 0.999 | 2 \pi | | |
| fong-spivak-seven-sketches_FO0729 | 66 | 0.999 | \infty | | |
| fong-spivak-seven-sketches_FO0730 | 66 | 0.725 | ([0, \infty], \geq) | | |
| fong-spivak-seven-sketches_FO0731 | 66 | 0.725 | \infty \geq x | | |
| fong-spivak-seven-sketches_FO0732 | 66 | 0.725 | x \in[0, \infty] | | |
| fong-spivak-seven-sketches_FO0733 | 66 | 0.983 | x+\infty=\infty | | |
| fong-spivak-seven-sketches_FO0734 | 67 | 0.999 | (X, \leq)^{\mathrm{op}}:=(X, \geq) | | |
| fong-spivak-seven-sketches_FO0735 | 67 | 0.955 | \mathcal{X}=(X, \leq) | | |
| fong-spivak-seven-sketches_FO0736 | 67 | 0.955 | \mathcal{X}^{\mathrm{op}}=(X, \geq) | | |
| fong-spivak-seven-sketches_FO0737 | 67 | 0.930 | (X, \leq I, \otimes) | | |
| fong-spivak-seven-sketches_FO0738 | 67 | 0.930 | (X, \geq, I, \otimes) | | |
| fong-spivak-seven-sketches_FO0739 | 67 | 0.838 | x_{1} \geq y_{1} | | |
| fong-spivak-seven-sketches_FO0740 | 67 | 0.838 | x_{2} \geq y_{2} | | |
| fong-spivak-seven-sketches_FO0741 | 67 | 0.838 | X^{\text {op }} | | |
| fong-spivak-seven-sketches_FO0742 | 67 | 0.999 | x_{1} \otimes x_{2} \geq y_{1} \otimes y_{2} | | |
| fong-spivak-seven-sketches_FO0743 | 67 | 0.999 | y_{1} \leq x_{1} | | |
| fong-spivak-seven-sketches_FO0744 | 67 | 0.558 | y_{2} \leq x_{2} | | |
| fong-spivak-seven-sketches_FO0745 | 67 | 0.558 | y_{1} \otimes y_{2} \leq x_{1} \otimes x_{2} | | |
| fong-spivak-seven-sketches_FO0746 | 67 | 0.996 | { }^{\mathrm{op}} | | |
| fong-spivak-seven-sketches_FO0747 | 67 | 1.000 | x \cong x^{\prime} | | |
| fong-spivak-seven-sketches_FO0748 | 67 | 1.000 | x \leq x^{\prime} | | |
| fong-spivak-seven-sketches_FO0749 | 67 | 1.000 | x^{\prime} \leq x | | |
| fong-spivak-seven-sketches_FO0750 | 67 | 1.000 | \mathcal{P}=\left(P, \leq_{P}, I_{P}, \otimes_{P}\right) | | |
| fong-spivak-seven-sketches_FO0751 | 67 | 1.000 | Q=\left(Q, \leq_{Q}, I_{Q}, \otimes_{Q}\right) | | |
| fong-spivak-seven-sketches_FO0752 | 67 | 0.995 | \mathcal{P} | | |
| fong-spivak-seven-sketches_FO0753 | 67 | 1.000 | I_{Q} \leq_{Q} f\left(I_{P}\right) | | |
| fong-spivak-seven-sketches_FO0754 | 67 | 1.000 | f\left(p_{1}\right) \otimes_{Q} f\left(p_{2}\right) \leq_{Q} f\left(p_{1} \otimes_{P} p_{2}\right) | | |
| fong-spivak-seven-sketches_FO0755 | 67 | 1.000 | p_{1}, p_{2} \in P | | |
| fong-spivak-seven-sketches_FO0756 | 68 | 1.000 | I_{Q} \cong f\left(I_{P}\right) | | |
| fong-spivak-seven-sketches_FO0757 | 68 | 1.000 | f\left(p_{1}\right) \otimes_{Q} f\left(p_{2}\right) \cong f\left(p_{1} \otimes_{P} p_{2}\right) | | |
| fong-spivak-seven-sketches_FO0758 | 68 | 0.895 | I_{Q}=f\left(I_{P}\right) | | |
| fong-spivak-seven-sketches_FO0759 | 68 | 0.995 | f\left(p_{1}\right) \otimes_{Q} f\left(p_{2}\right)=f\left(p_{1} \otimes_{P} p_{2}\right) | | |
| fong-spivak-seven-sketches_FO0760 | 68 | 1.000 | i:(\mathbb{N}, \leq, 0,+) \rightarrow(\mathbb{R}, \leq, 0,+) | | |
| fong-spivak-seven-sketches_FO0761 | 68 | 1.000 | i(n)= | | |
| fong-spivak-seven-sketches_FO0762 | 68 | 1.000 | i(m) \leq i(n) | | |
| fong-spivak-seven-sketches_FO0763 | 68 | 1.000 | i(0)=0 | | |
| fong-spivak-seven-sketches_FO0764 | 68 | 1.000 | i(m+n)=i(m)+i(n) | | |
| fong-spivak-seven-sketches_FO0765 | 68 | 0.998 | f:(\mathbb{R}, \leq, 0,+) \rightarrow(\mathbb{N}, \leq, 0,+) | | |
| fong-spivak-seven-sketches_FO0766 | 68 | 1.000 | f(x):=\lfloor x\rfloor | | |
| fong-spivak-seven-sketches_FO0767 | 68 | 1.000 | f(3.14)=3 | | |
| fong-spivak-seven-sketches_FO0768 | 68 | 1.000 | f(0)=0 | | |
| fong-spivak-seven-sketches_FO0769 | 68 | 1.000 | f(x)+f(y) \leq f(x+y) | | |
| fong-spivak-seven-sketches_FO0770 | 68 | 1.000 | f(0.5)+f(0.5) \neq f(0.5+0.5) | | |
| fong-spivak-seven-sketches_FO0771 | 68 | 0.754 | =([0, \infty], \geq, 0,+) | | |
| fong-spivak-seven-sketches_FO0772 | 68 | 0.905 | :=\infty | | |
| fong-spivak-seven-sketches_FO0773 | 68 | 0.998 | :=0 | | |
| fong-spivak-seven-sketches_FO0774 | 68 | 0.683 | g:(\mathbb{B}, \leq | | |
| fong-spivak-seven-sketches_FO0775 | 68 | 0.683 | \wedge) \rightarrow([0, \infty], \geq, 0,+) | | |
| fong-spivak-seven-sketches_FO0776 | 68 | 1.000 | d, u:[0, \infty] \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO0777 | 68 | 1.000 | u | | |
| fong-spivak-seven-sketches_FO0778 | 69 | 0.929 | (\mathbb{N}, \leq, 1, *) | | |
| fong-spivak-seven-sketches_FO0779 | 69 | 0.998 | (\mathbb{N}, \leq, 0,+) \rightarrow(\mathbb{N}, \leq | | |
| fong-spivak-seven-sketches_FO0780 | 69 | 0.833 | , 1, *) | | |
| fong-spivak-seven-sketches_FO0781 | 69 | 0.935 | (\mathbb{Z}, \leq, *, 1) | | |
| fong-spivak-seven-sketches_FO0782 | 69 | 0.949 | \operatorname{Ob}(\mathcal{X}) | | |
| fong-spivak-seven-sketches_FO0783 | 69 | 1.000 | \mathcal{X}(x, y) \in V | | |
| fong-spivak-seven-sketches_FO0784 | 69 | 0.631 | x \in \operatorname{Ob}(X) | | |
| fong-spivak-seven-sketches_FO0785 | 69 | 0.631 | I \leq X(x, x) | | |
| fong-spivak-seven-sketches_FO0786 | 69 | 0.416 | x, y, z \in \operatorname{Ob}(X) | | |
| fong-spivak-seven-sketches_FO0787 | 69 | 0.416 | \mathcal{X}(x, y) \otimes \mathcal{X}(y, z) \leq \mathfrak{X}(x, z) | | |
| fong-spivak-seven-sketches_FO0788 | 69 | 0.999 | \mathcal{X}(x, y) | | |
| fong-spivak-seven-sketches_FO0789 | 70 | 0.986 | \operatorname{Ob}(\mathcal{X})=\{p, q, r, s, t\} | | |
| fong-spivak-seven-sketches_FO0790 | 70 | 1.000 | t \not \leq s | | |
| fong-spivak-seven-sketches_FO0791 | 70 | 1.000 | \mathcal{X}(s, t)= | | |
| fong-spivak-seven-sketches_FO0792 | 70 | 0.989 | X(t, s)= | | |
| fong-spivak-seven-sketches_FO0793 | 70 | 0.598 | x(x, y) | | |
| fong-spivak-seven-sketches_FO0794 | 131 | 0.848 | r | | |
| fong-spivak-seven-sketches_FO0795 | 71 | 0.503 | x, y \in \mathrm{Ob}(X) | | |
| fong-spivak-seven-sketches_FO0796 | 71 | 0.503 | \mathcal{X}(x, y) \in \mathbb{B} | | |
| fong-spivak-seven-sketches_FO0797 | 71 | 0.503 | \mathcal{X}(x, y)= | | |
| fong-spivak-seven-sketches_FO0798 | 71 | 0.958 | X(x, y)= | | |
| fong-spivak-seven-sketches_FO0799 | 71 | 0.507 | X:=\operatorname{Ob}(X) | | |
| fong-spivak-seven-sketches_FO0800 | 71 | 0.952 | \leq \mathcal{X}(x, x) | | |
| fong-spivak-seven-sketches_FO0801 | 71 | 0.798 | \mathcal{X}(x, y) \wedge \mathcal{X}(y, z) \leq \mathcal{X}(x, z) | | |
| fong-spivak-seven-sketches_FO0802 | 71 | 0.998 | b \in | | |
| fong-spivak-seven-sketches_FO0803 | 71 | 0.998 | \leq b | | |
| fong-spivak-seven-sketches_FO0804 | 71 | 0.998 | b= | | |
| fong-spivak-seven-sketches_FO0805 | 71 | 0.998 | \mathcal{X}(x, x)= | | |
| fong-spivak-seven-sketches_FO0806 | 71 | 0.691 | b \in \mathbb{B} | | |
| fong-spivak-seven-sketches_FO0807 | 71 | 0.943 | \mathcal{X}(y, z)= | | |
| fong-spivak-seven-sketches_FO0808 | 71 | 0.943 | \mathcal{X}(x, z)= | | |
| fong-spivak-seven-sketches_FO0809 | 71 | 0.766 | (X, d) | | |
| fong-spivak-seven-sketches_FO0810 | 71 | 1.000 | d: X \times X \rightarrow \mathbb{R}_{\geq 0} | | |
| fong-spivak-seven-sketches_FO0811 | 71 | 1.000 | d(x, y) | | |
| fong-spivak-seven-sketches_FO0812 | 72 | 1.000 | d(x, x)=0 | | |
| fong-spivak-seven-sketches_FO0813 | 72 | 1.000 | d(x, y)=0 | | |
| fong-spivak-seven-sketches_FO0814 | 72 | 1.000 | d(x, y)=d(y, x) | | |
| fong-spivak-seven-sketches_FO0815 | 72 | 1.000 | d(x, y)+d(y, z) \geq d(x, z) | | |
| fong-spivak-seven-sketches_FO0816 | 72 | 0.772 | d: X \times X \rightarrow[0, \infty]=\mathbb{R}_{\geq 0} \cup\{\infty\} | | |
| fong-spivak-seven-sketches_FO0817 | 72 | 0.772 | X, d | | |
| fong-spivak-seven-sketches_FO0818 | 72 | 1.000 | z | | |
| fong-spivak-seven-sketches_FO0819 | 72 | 1.000 | x \rightarrow y \rightarrow z | | |
| fong-spivak-seven-sketches_FO0820 | 72 | 1.000 | 3+5 \geq 7.2 | | |
| fong-spivak-seven-sketches_FO0821 | 72 | 1.000 | 5+7.2 \geq 3 | | |
| fong-spivak-seven-sketches_FO0822 | 72 | 1.000 | 3+7.2 \geq 5 | | |
| fong-spivak-seven-sketches_FO0823 | 72 | 1.000 | 5+3 \geq 7.2,7.2+5 \geq 3 | | |
| fong-spivak-seven-sketches_FO0824 | 72 | 1.000 | 7.2+3 \geq 5 | | |
| fong-spivak-seven-sketches_FO0825 | 72 | 0.764 | d(x, y)=? d(y, x) | | |
| fong-spivak-seven-sketches_FO0826 | 72 | 0.689 | d(\mathrm{US} | | |
| fong-spivak-seven-sketches_FO0827 | 72 | 0.689 | ) | | |
| fong-spivak-seven-sketches_FO0828 | 72 | 0.998 | d( | | |
| fong-spivak-seven-sketches_FO0829 | 72 | 0.998 | )=0 | | |
| fong-spivak-seven-sketches_FO0830 | 72 | 0.852 | U \subseteq X | | |
| fong-spivak-seven-sketches_FO0831 | 73 | 0.547 | ) \neq 0 | | |
| fong-spivak-seven-sketches_FO0832 | 73 | 0.867 | \{r \in \mathbb{R} \mid r<0\} | | |
| fong-spivak-seven-sketches_FO0833 | 73 | 0.996 | (\mathbb{R}, d) | | |
| fong-spivak-seven-sketches_FO0834 | 73 | 0.899 | x, y \in \operatorname{Ob}(\mathcal{X}) | | |
| fong-spivak-seven-sketches_FO0835 | 73 | 0.899 | \mathcal{X}(x, y) \in[0, \infty] | | |
| fong-spivak-seven-sketches_FO0836 | 73 | 1.000 | 0 \geq d(x, x) | | |
| fong-spivak-seven-sketches_FO0837 | 73 | 1.000 | d(x, x) \in[0, \infty] | | |
| fong-spivak-seven-sketches_FO0838 | 73 | 1.000 | d(x, y):=|y-x| | | |
| fong-spivak-seven-sketches_FO0839 | 73 | 1.000 | d(3,7)=4 | | |
| fong-spivak-seven-sketches_FO0840 | 73 | 0.214 | \mathbb{R}_{\geq 0, \geq, 0,+} | | |
| fong-spivak-seven-sketches_FO0841 | 73 | 0.824 | \left(\mathbb{R}_{\geq 0,} \geq, 0,+\right) | | |
| fong-spivak-seven-sketches_FO0842 | 73 | 1.000 | d(U, V):=\max \left(d_{L}(U, V), d_{L}(V, U)\right) | | |
| fong-spivak-seven-sketches_FO0843 | 74 | 1.000 | w \geq 0 | | |
| fong-spivak-seven-sketches_FO0844 | 74 | 0.985 | d_{X} | | |
| fong-spivak-seven-sketches_FO0845 | 74 | 0.985 | d(p, q) | | |
| fong-spivak-seven-sketches_FO0847 | 74 | 0.999 | M_{G} | | |
| fong-spivak-seven-sketches_FO0849 | 75 | 0.998 | d_{Y} | | |
| fong-spivak-seven-sketches_FO0850 | 75 | 0.992 | M_{Y} | | |
| fong-spivak-seven-sketches_FO0851 | 75 | 0.699 | M_{X} | | |
| fong-spivak-seven-sketches_FO0852 | 104 | 1.000 | D | | |
| fong-spivak-seven-sketches_FO0854 | 75 | 0.993 | \mathcal{M}:=(\mathrm{P}(M), \subseteq, M, \cap) | | |
| fong-spivak-seven-sketches_FO0855 | 75 | 0.870 | \mathcal{M} | | |
| fong-spivak-seven-sketches_FO0856 | 75 | 0.966 | a, b \in \operatorname{Ob}(\mathcal{X}) | | |
| fong-spivak-seven-sketches_FO0857 | 75 | 0.595 | M=\{ | | |
| fong-spivak-seven-sketches_FO0858 | 76 | 1.000 | \mathcal{W}:=(\mathbb{N} \cup\{\infty\}, \leq, \infty, \min ) | | |
| fong-spivak-seven-sketches_FO0859 | 76 | 1.000 | \mathbb{N} \cup\{\infty\} | | |
| fong-spivak-seven-sketches_FO0860 | 76 | 1.000 | \mathcal{W} | | |
| fong-spivak-seven-sketches_FO0861 | 76 | 1.000 | f: \mathcal{V} \rightarrow \mathcal{W} | | |
| fong-spivak-seven-sketches_FO0862 | 76 | 1.000 | \mathcal{V} \rightarrow \mathcal{W} | | |
| fong-spivak-seven-sketches_FO0863 | 76 | 0.999 | \mathcal{C} | | |
| fong-spivak-seven-sketches_FO0864 | 76 | 1.000 | \mathcal{C}_{f} | | |
| fong-spivak-seven-sketches_FO0865 | 76 | 0.825 | \operatorname{Ob}\left(\mathcal{C}_{f}\right):=\operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO0866 | 76 | 0.963 | c, d \in \operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO0867 | 76 | 0.963 | \mathcal{C}_{f}(c, d):=f(\mathcal{C}(c, d)) | | |
| fong-spivak-seven-sketches_FO0868 | 76 | 1.000 | c \in \mathcal{C} | | |
| fong-spivak-seven-sketches_FO0869 | 77 | 0.964 | c, d, e \in \mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO0870 | 77 | 1.000 | f:[0, \infty] \rightarrow\{ | | |
| fong-spivak-seven-sketches_FO0871 | 77 | 0.651 | x_{f} \neq x_{g} | | |
| fong-spivak-seven-sketches_FO0872 | 77 | 0.606 | F: \mathcal{X} \rightarrow y | | |
| fong-spivak-seven-sketches_FO0873 | 77 | 0.961 | F: \operatorname{Ob}(\mathcal{X}) \rightarrow \operatorname{Ob}(\mathcal{Y}) | | |
| fong-spivak-seven-sketches_FO0874 | 77 | 0.365 | x_{1}, x_{2} \in \mathrm{Ob}(\mathcal{X}) | | |
| fong-spivak-seven-sketches_FO0875 | 77 | 0.365 | \mathcal{X}\left(x_{1}, x_{2}\right) \leq y\left(F\left(x_{1}\right), F\left(x_{2}\right)\right) | | |
| fong-spivak-seven-sketches_FO0876 | 78 | 0.829 | X\left(x_{1}, x_{2}\right)= | | |
| fong-spivak-seven-sketches_FO0877 | 78 | 0.829 | x_{1} \leq x_{2} | | |
| fong-spivak-seven-sketches_FO0878 | 78 | 1.000 | \left(X, \leq_{X}\right) \rightarrow\left(Y, \leq_{Y}\right) | | |
| fong-spivak-seven-sketches_FO0879 | 78 | 1.000 | x_{1}, x_{2} \in X | | |
| fong-spivak-seven-sketches_FO0880 | 78 | 1.000 | x_{1} \leq_{X} x_{2} | | |
| fong-spivak-seven-sketches_FO0881 | 78 | 1.000 | F\left(x_{1}\right) \leq_{Y} F\left(x_{2}\right) | | |
| fong-spivak-seven-sketches_FO0882 | 78 | 0.996 | \mathcal{V}= | | |
| fong-spivak-seven-sketches_FO0883 | 78 | 0.990 | \left(X, d_{X}\right) | | |
| fong-spivak-seven-sketches_FO0884 | 78 | 0.990 | \left(Y, d_{Y}\right) | | |
| fong-spivak-seven-sketches_FO0885 | 78 | 0.982 | d_{X}\left(x_{1}, x_{2}\right) \geq d_{Y}\left(F\left(x_{1}\right), F\left(x_{2}\right)\right) | | |
| fong-spivak-seven-sketches_FO0886 | 78 | 0.971 | \mathcal{X}^{\mathrm{op}} | | |
| fong-spivak-seven-sketches_FO0887 | 78 | 0.784 | \operatorname{Ob}\left(\mathcal{X}^{\text {op }}\right):=\operatorname{Ob}(\mathcal{X}) | | |
| fong-spivak-seven-sketches_FO0888 | 78 | 0.959 | x, y \in \mathcal{X} | | |
| fong-spivak-seven-sketches_FO0889 | 78 | 0.959 | \mathcal{X}^{\mathrm{op}}(x, y):=\mathcal{X}(y, x) | | |
| fong-spivak-seven-sketches_FO0890 | 78 | 0.334 | \mathrm{t}: \mathfrak{X} \rightarrow X^{\mathrm{op}} | | |
| fong-spivak-seven-sketches_FO0891 | 78 | 0.470 | I \leq X(x, y) | | |
| fong-spivak-seven-sketches_FO0892 | 78 | 0.470 | I \leq X(y, x) | | |
| fong-spivak-seven-sketches_FO0893 | 78 | 0.868 | \mathcal{Y} | | |
| fong-spivak-seven-sketches_FO0894 | 79 | 0.966 | \mathcal{X} \times \mathcal{Y} | | |
| fong-spivak-seven-sketches_FO0895 | 79 | 0.382 | \operatorname{Ob}(\mathcal{X} \times \mathcal{Y}):=\operatorname{Ob}(\mathcal{X}) \times \operatorname{Ob}(y) | | |
| fong-spivak-seven-sketches_FO0896 | 79 | 0.711 | (\mathcal{X} \times \mathcal{Y})\left((x, y),\left(x^{\prime}, y^{\prime}\right)\right):=\mathcal{X}\left(x, x^{\prime}\right) \otimes y\left(y, y^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO0897 | 79 | 0.334 | \left(x^{\prime}, y^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO0898 | 79 | 0.334 | \operatorname{Ob}(\mathcal{X} \times y) | | |
| fong-spivak-seven-sketches_FO0899 | 79 | 0.603 | (x, y) \in \mathrm{Ob}(x \times y) | | |
| fong-spivak-seven-sketches_FO0900 | 79 | 0.603 | I \leq(x \times y)((x, y),(x, y)) | | |
| fong-spivak-seven-sketches_FO0901 | 79 | 1.000 | \left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right) | | |
| fong-spivak-seven-sketches_FO0902 | 79 | 1.000 | \left(x_{3}, y_{3}\right) | | |
| fong-spivak-seven-sketches_FO0903 | 79 | 1.000 | \left(P, \leq_{P}\right) \times\left(Q, \leq_{Q}\right) | | |
| fong-spivak-seven-sketches_FO0904 | 79 | 0.960 | \left(p_{1}, q_{1}\right) \leq\left(p_{2}, q_{2}\right) | | |
| fong-spivak-seven-sketches_FO0905 | 79 | 0.960 | q_{1} \leq q_{2} | | |
| fong-spivak-seven-sketches_FO0906 | 79 | 0.640 | x \times y | | |
| fong-spivak-seven-sketches_FO0907 | 79 | 0.640 | d_{X \times Y}\left((x, y),\left(x^{\prime}, y^{\prime}\right)\right) | | |
| fong-spivak-seven-sketches_FO0908 | 79 | 1.000 | d_{X}\left(x, x^{\prime}\right)+d_{Y}\left(y, y^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO0909 | 80 | 1.000 | \mathbb{R} \times \mathbb{R} | | |
| fong-spivak-seven-sketches_FO0910 | 80 | 1.000 | (5,6) | | |
| fong-spivak-seven-sketches_FO0911 | 80 | 0.876 | (-1,4) | | |
| fong-spivak-seven-sketches_FO0912 | 80 | 0.876 | \sqrt{40} | | |
| fong-spivak-seven-sketches_FO0920 | 80 | 0.482 | \mathfrak{y} | | |
| fong-spivak-seven-sketches_FO0921 | 81 | 1.000 | v, w \in V | | |
| fong-spivak-seven-sketches_FO0922 | 81 | 1.000 | a, v, w \in V | | |
| fong-spivak-seven-sketches_FO0923 | 81 | 1.000 | v \in V | | |
| fong-spivak-seven-sketches_FO0924 | 81 | 1.000 | (-\otimes v): V \rightarrow V | | |
| fong-spivak-seven-sketches_FO0925 | 81 | 0.747 | (v-\circ-): V \rightarrow V | | |
| fong-spivak-seven-sketches_FO0926 | 81 | 1.000 | (-\otimes v) | | |
| fong-spivak-seven-sketches_FO0927 | 81 | 0.959 | ((v-o w) \otimes v) \leq w | | |
| fong-spivak-seven-sketches_FO0928 | 81 | 0.702 | (v-o-) | | |
| fong-spivak-seven-sketches_FO0929 | 81 | 0.950 | x, y \in[0, \infty] | | |
| fong-spivak-seven-sketches_FO0930 | 81 | 0.950 | x \multimap y:=\max (0, y-x) | | |
| fong-spivak-seven-sketches_FO0931 | 81 | 0.950 | a, x, y \in[0, \infty] | | |
| fong-spivak-seven-sketches_FO0932 | 82 | 0.746 | \leq p \multimap q | | |
| fong-spivak-seven-sketches_FO0933 | 82 | 0.746 | \rightarrow | | |
| fong-spivak-seven-sketches_FO0934 | 82 | 0.987 | a= | | |
| fong-spivak-seven-sketches_FO0935 | 82 | 0.987 | p= | | |
| fong-spivak-seven-sketches_FO0936 | 82 | 0.987 | q= | | |
| fong-spivak-seven-sketches_FO0937 | 82 | 0.999 | (a \vee p) \not \leq q | | |
| fong-spivak-seven-sketches_FO0938 | 82 | 0.999 | a \leq(p \multimap q) | | |
| fong-spivak-seven-sketches_FO0939 | 82 | 1.000 | c, d | | |
| fong-spivak-seven-sketches_FO0940 | 82 | 1.000 | c \rightarrow d | | |
| fong-spivak-seven-sketches_FO0941 | 82 | 1.000 | 2 \mathrm{H}_{2} \mathrm{O}+2 \mathrm{Na} \rightarrow 2 \mathrm{NaOH}+\mathrm{H}_{2} | | |
| fong-spivak-seven-sketches_FO0942 | 82 | 0.426 | 2 \mathrm{Na} \rightarrow\left(2 \mathrm{NaOH}+\mathrm{H}_{2}\right) | | |
| fong-spivak-seven-sketches_FO0943 | 82 | 0.935 | \mathcal{V}=(V, \leq, I, \otimes,-\infty) | | |
| fong-spivak-seven-sketches_FO0944 | 82 | 1.000 | -\otimes v:(V, \leq) \rightarrow(V, \leq) | | |
| fong-spivak-seven-sketches_FO0945 | 82 | 0.913 | v \rightarrow-:(V, \leq) \rightarrow(V, \leq) | | |
| fong-spivak-seven-sketches_FO0946 | 82 | 1.000 | A \subseteq V | | |
| fong-spivak-seven-sketches_FO0947 | 82 | 1.000 | \bigvee_{a \in A} a | | |
| fong-spivak-seven-sketches_FO0948 | 82 | 1.000 | \bigvee_{a \in A} v \otimes a | | |
| fong-spivak-seven-sketches_FO0949 | 82 | 1.000 | v \otimes(v \multimap w) \leq w | | |
| fong-spivak-seven-sketches_FO0950 | 82 | 1.000 | v \cong(I \multimap v) | | |
| fong-spivak-seven-sketches_FO0951 | 82 | 1.000 | u, v, w \in V | | |
| fong-spivak-seven-sketches_FO0952 | 82 | 1.000 | (u \multimap v) \otimes(v \multimap w) \leq(u \multimap w) | | |
| fong-spivak-seven-sketches_FO0953 | 83 | 0.748 | (v-\circ-) | | |
| fong-spivak-seven-sketches_FO0954 | 83 | 0.999 | (v \multimap w) \leq(v \multimap w) | | |
| fong-spivak-seven-sketches_FO0955 | 83 | 0.971 | (v \multimap w) \otimes v \leq w | | |
| fong-spivak-seven-sketches_FO0956 | 83 | 0.988 | v \otimes(v \multimap w)=(v \multimap w) \otimes v | | |
| fong-spivak-seven-sketches_FO0957 | 83 | 1.000 | v \otimes I=v \leq v | | |
| fong-spivak-seven-sketches_FO0958 | 83 | 1.000 | v \leq(I \multimap v) | | |
| fong-spivak-seven-sketches_FO0959 | 83 | 0.909 | (I \multimap v)=I \otimes(I \multimap v) \leq v | | |
| fong-spivak-seven-sketches_FO0960 | 83 | 0.966 | u \otimes(u \multimap v) \otimes(v \multimap w) \leq w | | |
| fong-spivak-seven-sketches_FO0961 | 83 | 0.994 | v, w \in \mathrm{Ob}(\mathcal{V}) | | |
| fong-spivak-seven-sketches_FO0962 | 83 | 0.899 | \mathcal{V}(v, w):=(v-o w) \in \mathcal{V} | | |
| fong-spivak-seven-sketches_FO0963 | 83 | 0.858 | I \leq \mathcal{X}(x, x)=(x \multimap x) | | |
| fong-spivak-seven-sketches_FO0964 | 83 | 1.000 | I \otimes x \leq x | | |
| fong-spivak-seven-sketches_FO0965 | 83 | 0.844 | \mathcal{V}=(V, \leq, I, \otimes, \multimap) | | |
| fong-spivak-seven-sketches_FO0966 | 83 | 0.844 | \bigvee A | | |
| fong-spivak-seven-sketches_FO0967 | 83 | 1.000 | 0:=\vee \varnothing | | |
| fong-spivak-seven-sketches_FO0968 | 83 | 1.000 | A \subseteq[0, \infty] | | |
| fong-spivak-seven-sketches_FO0969 | 83 | 0.564 | \vee A | | |
| fong-spivak-seven-sketches_FO0970 | 83 | 0.959 | a \geq \bigvee A | | |
| fong-spivak-seven-sketches_FO0971 | 83 | 0.969 | b \in[0, \infty] | | |
| fong-spivak-seven-sketches_FO0972 | 83 | 0.969 | a \geq b | | |
| fong-spivak-seven-sketches_FO0973 | 83 | 0.969 | \bigvee A \geq b | | |
| fong-spivak-seven-sketches_FO0974 | 84 | 0.728 | A .{ }^{5} | | |
| fong-spivak-seven-sketches_FO0975 | 84 | 0.728 | A=\{2,3\} | | |
| fong-spivak-seven-sketches_FO0976 | 84 | 0.728 | \bigvee A=2 | | |
| fong-spivak-seven-sketches_FO0977 | 84 | 0.573 | B=\{2.5,2.05,2.005, \ldots\} | | |
| fong-spivak-seven-sketches_FO0978 | 84 | 1.000 | A=\varnothing | | |
| fong-spivak-seven-sketches_FO0979 | 84 | 0.994 | \bigvee \varnothing \geq b | | |
| fong-spivak-seven-sketches_FO0980 | 84 | 0.994 | \bigvee \varnothing=\infty | | |
| fong-spivak-seven-sketches_FO0981 | 84 | 0.998 | \vee \varnothing | | |
| fong-spivak-seven-sketches_FO0982 | 84 | 0.408 | \mathcal{V}=\mathbf{B o o l}=(\mathbb{B}, \leq | | |
| fong-spivak-seven-sketches_FO0983 | 84 | 0.959 | \mathcal{V}=\boldsymbol{\operatorname { C o s t }}=([0, \infty], \geq, 0,+) | | |
| fong-spivak-seven-sketches_FO0984 | 84 | 1.000 | x \vee y | | |
| fong-spivak-seven-sketches_FO0985 | 84 | 0.724 | \mathcal{V}=\mathbf{B O O l} | | |
| fong-spivak-seven-sketches_FO0986 | 84 | 0.724 | x, y \in \mathbb{B} | | |
| fong-spivak-seven-sketches_FO0987 | 84 | 0.939 | \mathbf{B o o l}=(\mathbb{B}, \leq | | |
| fong-spivak-seven-sketches_FO0988 | 84 | 0.884 | \mathrm{P}(S), \subseteq, S, \cap | | |
| fong-spivak-seven-sketches_FO0989 | 84 | 1.000 | \mathcal{P}=(P, \leq) | | |
| fong-spivak-seven-sketches_FO0990 | 84 | 1.000 | A \subseteq \mathcal{P} | | |
| fong-spivak-seven-sketches_FO0991 | 84 | 1.000 | M_{A}:=\{p \in P \mid p \leq a | | |
| fong-spivak-seven-sketches_FO0992 | 84 | 1.000 | a \in A\} | | |
| fong-spivak-seven-sketches_FO0993 | 84 | 1.000 | m_{A}:=\bigvee_{p \in M_{A}} p | | |
| fong-spivak-seven-sketches_FO0994 | 84 | 1.000 | m_{A} | | |
| fong-spivak-seven-sketches_FO0995 | 84 | 1.000 | m_{A} \leq a | | |
| fong-spivak-seven-sketches_FO0996 | 84 | 1.000 | p \in M_{A} | | |
| fong-spivak-seven-sketches_FO0997 | 84 | 1.000 | m^{\prime} \in P | | |
| fong-spivak-seven-sketches_FO0998 | 84 | 1.000 | m^{\prime} \leq a | | |
| fong-spivak-seven-sketches_FO0999 | 84 | 1.000 | m^{\prime} \leq m | | |
| fong-spivak-seven-sketches_FO1000 | 84 | 1.000 | m^{\prime} | | |
| fong-spivak-seven-sketches_FO1001 | 84 | 1.000 | M_{A} | | |
| fong-spivak-seven-sketches_FO1002 | 84 | 0.666 | \{3.1,3.01,3.001, \ldots\} | | |
| fong-spivak-seven-sketches_FO1003 | 85 | 1.000 | V \subseteq X | | |
| fong-spivak-seven-sketches_FO1004 | 85 | 0.302 | u \in U | | |
| fong-spivak-seven-sketches_FO1005 | 85 | 0.302 | \forall_{u \in U .} \exists_{v \in V .} u \leq v | | |
| fong-spivak-seven-sketches_FO1006 | 85 | 0.999 | \mathcal{V}=\mathrm{P}(M) | | |
| fong-spivak-seven-sketches_FO1007 | 85 | 1.000 | d(U, V) \in \mathrm{P}(M) | | |
| fong-spivak-seven-sketches_FO1008 | 85 | 1.000 | -\otimes v: V \rightarrow V | | |
| fong-spivak-seven-sketches_FO1009 | 85 | 0.995 | v \multimap- | | |
| fong-spivak-seven-sketches_FO1010 | 85 | 1.000 | v \multimap w | | |
| fong-spivak-seven-sketches_FO1011 | 85 | 0.703 | \sum | | |
| fong-spivak-seven-sketches_FO1012 | 85 | 1.000 | \mathcal{V}=(V, \leq, \otimes, I) | | |
| fong-spivak-seven-sketches_FO1013 | 85 | 1.000 | M: X \times Y \rightarrow V | | |
| fong-spivak-seven-sketches_FO1014 | 85 | 0.999 | M(x, y) | | |
| fong-spivak-seven-sketches_FO1015 | 86 | 1.000 | N: Y \times Z \rightarrow V | | |
| fong-spivak-seven-sketches_FO1016 | 86 | 0.997 | (M * N): X \times Z \rightarrow V | | |
| fong-spivak-seven-sketches_FO1017 | 86 | 0.595 | \mathcal{V}=\mathbf{B o o l} | | |
| fong-spivak-seven-sketches_FO1018 | 86 | 0.595 | M * N | | |
| fong-spivak-seven-sketches_FO1019 | 86 | 0.996 | X=\{1,2,3\}, Y=\{1,2\} | | |
| fong-spivak-seven-sketches_FO1020 | 86 | 0.996 | Z=\{1,2,3\} | | |
| fong-spivak-seven-sketches_FO1021 | 86 | 0.996 | M: X \times Y \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO1022 | 86 | 0.996 | N: Y \times Z \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO1023 | 86 | 1.000 | I_{X}: X \times X \rightarrow V | | |
| fong-spivak-seven-sketches_FO1024 | 86 | 0.964 | 2 \times 2 | | |
| fong-spivak-seven-sketches_FO1025 | 86 | 0.171 | ([0, \infty], \geq, 0,+) | | |
| fong-spivak-seven-sketches_FO1026 | 86 | 1.000 | I_{X} * M=M | | |
| fong-spivak-seven-sketches_FO1027 | 86 | 1.000 | M: W \times X \rightarrow V, N: X \times Y \rightarrow V | | |
| fong-spivak-seven-sketches_FO1028 | 86 | 1.000 | P: Y \times Z \rightarrow V | | |
| fong-spivak-seven-sketches_FO1029 | 86 | 1.000 | (M * N) * P=M *(N * P) | | |
| fong-spivak-seven-sketches_FO1030 | 87 | 0.999 | M_{Y}^{2} | | |
| fong-spivak-seven-sketches_FO1031 | 87 | 1.000 | M_{Y}^{3} | | |
| fong-spivak-seven-sketches_FO1032 | 87 | 1.000 | M_{Y}^{2}=M_{Y}^{3} | | |
| fong-spivak-seven-sketches_FO1033 | 87 | 1.000 | M_{Y}^{n} | | |
| fong-spivak-seven-sketches_FO1034 | 87 | 0.911 | M_{X}^{2}, M_{X}^{3} | | |
| fong-spivak-seven-sketches_FO1035 | 87 | 0.911 | M_{X}^{4} | | |
| fong-spivak-seven-sketches_FO1036 | 89 | 1.000 | \$ 2.5 | | |
| fong-spivak-seven-sketches_FO1037 | 89 | 0.798 | 8 \% | | |
| fong-spivak-seven-sketches_FO1038 | 91 | 0.984 | \mathcal{C} \rightarrow | | |
| fong-spivak-seven-sketches_FO1039 | 91 | 1.000 | \mathcal{C} \rightarrow \mathcal{D} | | |
| fong-spivak-seven-sketches_FO1040 | 93 | 1.000 | { }^{2} \mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO1041 | 93 | 0.963 | \mathcal{C}(c, d),{ }^{3} | | |
| fong-spivak-seven-sketches_FO1042 | 93 | 0.964 | c \in \mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO1043 | 93 | 0.964 | \mathrm{id}_{c} \in \mathcal{C}(c, c) | | |
| fong-spivak-seven-sketches_FO1044 | 93 | 0.939 | f \in \mathcal{C}(c, d) | | |
| fong-spivak-seven-sketches_FO1045 | 93 | 0.939 | g \in \mathcal{C}(d, e) | | |
| fong-spivak-seven-sketches_FO1046 | 93 | 0.998 | f ; g \in \mathcal{C}(c, e) | | |
| fong-spivak-seven-sketches_FO1047 | 93 | 0.948 | f: c \rightarrow d | | |
| fong-spivak-seven-sketches_FO1048 | 93 | 0.620 | \operatorname{id}_{c} ; f=f | | |
| fong-spivak-seven-sketches_FO1049 | 93 | 0.620 | f ; \operatorname{id}_{d}=f | | |
| fong-spivak-seven-sketches_FO1050 | 93 | 1.000 | f: c_{0} \rightarrow c_{1}, g: c_{1} \rightarrow c_{2} | | |
| fong-spivak-seven-sketches_FO1051 | 93 | 1.000 | h: c_{2} \rightarrow c_{3} | | |
| fong-spivak-seven-sketches_FO1052 | 93 | 1.000 | (f ; g) ; h=f ;(g ; h) | | |
| fong-spivak-seven-sketches_FO1053 | 93 | 0.947 | f ; g ; h | | |
| fong-spivak-seven-sketches_FO1054 | 93 | 0.998 | \mathcal{C}(c, d) | | |
| fong-spivak-seven-sketches_FO1055 | 93 | 0.998 | \operatorname{Hom}_{\mathcal{C}}(c, d) | | |
| fong-spivak-seven-sketches_FO1056 | 94 | 1.000 | p: v \rightarrow w | | |
| fong-spivak-seven-sketches_FO1057 | 94 | 0.999 | \mathrm{id}_{v} | | |
| fong-spivak-seven-sketches_FO1058 | 94 | 1.000 | q: w \rightarrow x | | |
| fong-spivak-seven-sketches_FO1059 | 94 | 1.000 | p ; q | | |
| fong-spivak-seven-sketches_FO1060 | 94 | 1.000 | v \rightarrow x | | |
| fong-spivak-seven-sketches_FO1061 | 94 | 0.980 | (V, \leq) | | |
| fong-spivak-seven-sketches_FO1062 | 94 | 0.999 | \operatorname{Free}(G) | | |
| fong-spivak-seven-sketches_FO1063 | 94 | 0.754 | v_{1} | | |
| fong-spivak-seven-sketches_FO1064 | 94 | 0.754 | v_{2} | | |
| fong-spivak-seven-sketches_FO1065 | 94 | 0.754 | \mathrm{id}_{v_{1}}: v_{1} \rightarrow v_{1}, f_{1}: v_{1} \rightarrow v_{2} | | |
| fong-spivak-seven-sketches_FO1066 | 94 | 0.607 | \mathrm{id}_{v_{2}}: v_{2} \rightarrow v_{2} | | |
| fong-spivak-seven-sketches_FO1067 | 94 | 0.607 | \mathrm{id}_{v_{1}} | | |
| fong-spivak-seven-sketches_FO1068 | 94 | 0.607 | v_{1}, f_{1} | | |
| fong-spivak-seven-sketches_FO1069 | 94 | 0.997 | f_{1} | | |
| fong-spivak-seven-sketches_FO1070 | 94 | 0.997 | \mathrm{id}_{v_{2}} | | |
| fong-spivak-seven-sketches_FO1071 | 94 | 0.562 | f_{1}=f_{1} | | |
| fong-spivak-seven-sketches_FO1072 | 94 | 0.524 | \mathrm{id}_{v_{2}}{ }_{9} \mathrm{id}_{v_{2}}=\mathrm{id}_{v_{2}} | | |
| fong-spivak-seven-sketches_FO1073 | 95 | 0.621 | \mathbf{0} | | |
| fong-spivak-seven-sketches_FO1074 | 95 | 1.000 | \mathbf{n} | | |
| fong-spivak-seven-sketches_FO1075 | 95 | 0.968 | \{z, s,(s ; s),(s ; s ; s), \ldots\} | | |
| fong-spivak-seven-sketches_FO1076 | 95 | 0.967 | \mathrm{id}_{z} | | |
| fong-spivak-seven-sketches_FO1077 | 95 | 1.000 | \mathbb{N}=\{0,1,2,3, \ldots\} | | |
| fong-spivak-seven-sketches_FO1078 | 95 | 0.966 | *: M \times M \rightarrow M | | |
| fong-spivak-seven-sketches_FO1079 | 95 | 1.000 | m * 1=m=1 * m | | |
| fong-spivak-seven-sketches_FO1080 | 95 | 1.000 | (m * n) * p=m *(n * p) | | |
| fong-spivak-seven-sketches_FO1081 | 95 | 0.851 | M:=\mathcal{C}(\bullet, \bullet) | | |
| fong-spivak-seven-sketches_FO1082 | 95 | 0.851 | e=\mathrm{i} \mathrm{d}_{\bullet} | | |
| fong-spivak-seven-sketches_FO1083 | 95 | 0.922 | *: \mathcal{C}(\bullet, \bullet) \times \mathcal{C}(\bullet, \bullet) \rightarrow \mathcal{C}(\bullet, \bullet) | | |
| fong-spivak-seven-sketches_FO1084 | 95 | 0.922 | *=9 | | |
| fong-spivak-seven-sketches_FO1085 | 95 | 1.000 | m, n \in \mathbb{N} | | |
| fong-spivak-seven-sketches_FO1086 | 96 | 0.920 | f \circ h | | |
| fong-spivak-seven-sketches_FO1087 | 96 | 0.920 | g \circ i | | |
| fong-spivak-seven-sketches_FO1088 | 96 | 0.547 | g_{9} i | | |
| fong-spivak-seven-sketches_FO1089 | 96 | 0.795 | f ; h | | |
| fong-spivak-seven-sketches_FO1090 | 96 | 0.795 | \operatorname{id}_{A} | | |
| fong-spivak-seven-sketches_FO1091 | 97 | 0.499 | s ; s=z | | |
| fong-spivak-seven-sketches_FO1092 | 97 | 0.499 | s ; s ; s=z ; s=s | | |
| fong-spivak-seven-sketches_FO1093 | 97 | 0.493 | ; s=z | | |
| fong-spivak-seven-sketches_FO1094 | 97 | 0.493 | =z | | |
| fong-spivak-seven-sketches_FO1095 | 97 | 0.493 | \{z, s\} | | |
| fong-spivak-seven-sketches_FO1096 | 97 | 0.556 | z=z | | |
| fong-spivak-seven-sketches_FO1097 | 97 | 0.556 | z ; s=s ; z=s | | |
| fong-spivak-seven-sketches_FO1098 | 97 | 1.000 | \mathbb{Z} / 2 \mathbb{Z} | | |
| fong-spivak-seven-sketches_FO1099 | 97 | 1.000 | \mathcal{D} | | |
| fong-spivak-seven-sketches_FO1100 | 97 | 1.000 | \operatorname{Ob}(\mathcal{P})=P | | |
| fong-spivak-seven-sketches_FO1101 | 97 | 1.000 | p \rightarrow q | | |
| fong-spivak-seven-sketches_FO1102 | 97 | 1.000 | p \not \leq q | | |
| fong-spivak-seven-sketches_FO1103 | 98 | 1.000 | (C, \leq) | | |
| fong-spivak-seven-sketches_FO1104 | 98 | 0.985 | C:=\mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO1105 | 98 | 0.985 | c_{1} \leq c_{2} | | |
| fong-spivak-seven-sketches_FO1106 | 98 | 0.985 | \mathcal{C}\left(c_{1}, c_{2}\right) \neq \varnothing | | |
| fong-spivak-seven-sketches_FO1107 | 98 | 1.000 | c_{1} \rightarrow c_{2} | | |
| fong-spivak-seven-sketches_FO1108 | 99 | 0.810 | \mathrm{Ob}( | | |
| fong-spivak-seven-sketches_FO1109 | 99 | 0.998 | \boldsymbol{\operatorname { S e t }}(S, T)=\{f: S \rightarrow T \mid f | | |
| fong-spivak-seven-sketches_FO1110 | 99 | 0.958 | \mathrm{id}_{S}: S \rightarrow S | | |
| fong-spivak-seven-sketches_FO1111 | 99 | 0.908 | \operatorname{id}_{S}(s):=s | | |
| fong-spivak-seven-sketches_FO1112 | 99 | 1.000 | f: S \rightarrow T | | |
| fong-spivak-seven-sketches_FO1113 | 99 | 1.000 | g: T \rightarrow U | | |
| fong-spivak-seven-sketches_FO1114 | 99 | 1.000 | f ; g: S \rightarrow U | | |
| fong-spivak-seven-sketches_FO1115 | 99 | 1.000 | (f ; g)(s):=g(f(s)) | | |
| fong-spivak-seven-sketches_FO1116 | 99 | 1.000 | \underline{2}=\{1,2\} | | |
| fong-spivak-seven-sketches_FO1117 | 99 | 1.000 | \underline{3}=\{1,2,3\} | | |
| fong-spivak-seven-sketches_FO1118 | 99 | 0.952 | \boldsymbol{\operatorname { S e t }}(\mathbf{2}, \mathbf{3}) | | |
| fong-spivak-seven-sketches_FO1119 | 100 | 0.984 | \mathcal{C}^{\mathrm{op}} | | |
| fong-spivak-seven-sketches_FO1120 | 100 | 0.268 | \operatorname{Ob}\left(\mathcal{C}{ }^{\text {op }}\right):=\operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO1121 | 100 | 0.984 | \mathcal{C}^{\mathrm{op}}(c, d):=\mathcal{C}(d, c) | | |
| fong-spivak-seven-sketches_FO1122 | 100 | 0.980 | \{\mathbf{m}, \square\} | | |
| fong-spivak-seven-sketches_FO1123 | 100 | 0.980 | \{0,1\} | | |
| fong-spivak-seven-sketches_FO1124 | 100 | 0.999 | a \cong b | | |
| fong-spivak-seven-sketches_FO1125 | 100 | 1.000 | c \leq a | | |
| fong-spivak-seven-sketches_FO1126 | 100 | 1.000 | c \leq b | | |
| fong-spivak-seven-sketches_FO1127 | 100 | 1.000 | c \geq a | | |
| fong-spivak-seven-sketches_FO1128 | 100 | 1.000 | c \geq b | | |
| fong-spivak-seven-sketches_FO1129 | 100 | 0.931 | g: B \rightarrow A | | |
| fong-spivak-seven-sketches_FO1130 | 100 | 0.931 | f ; g=\operatorname{id}_{A} | | |
| fong-spivak-seven-sketches_FO1131 | 100 | 0.931 | g ; f=\operatorname{id}_{B} | | |
| fong-spivak-seven-sketches_FO1132 | 100 | 1.000 | g=f^{-1} | | |
| fong-spivak-seven-sketches_FO1133 | 100 | 1.000 | f=g^{-1} | | |
| fong-spivak-seven-sketches_FO1134 | 100 | 1.000 | A:=\{a, b, c\} | | |
| fong-spivak-seven-sketches_FO1135 | 100 | 1.000 | f: A \rightarrow \underline{3} | | |
| fong-spivak-seven-sketches_FO1136 | 100 | 1.000 | f(a)=2, f(b)=1, f(c)=3 | | |
| fong-spivak-seven-sketches_FO1137 | 100 | 0.954 | A \in | | |
| fong-spivak-seven-sketches_FO1138 | 100 | 1.000 | A \cong \underline{n} | | |
| fong-spivak-seven-sketches_FO1139 | 100 | 1.000 | f^{-1}: \underline{3} \rightarrow A | | |
| fong-spivak-seven-sketches_FO1140 | 100 | 0.998 | A \rightarrow \underline{3} | | |
| fong-spivak-seven-sketches_FO1141 | 100 | 1.000 | \mathrm{id}_{c} | | |
| fong-spivak-seven-sketches_FO1142 | 101 | 0.880 | \boldsymbol{\operatorname { F r e e }}(G) | | |
| fong-spivak-seven-sketches_FO1143 | 101 | 1.000 | f: \underline{2} \rightarrow \underline{3} | | |
| fong-spivak-seven-sketches_FO1144 | 101 | 1.000 | g: \underline{3} \rightarrow \underline{2} | | |
| fong-spivak-seven-sketches_FO1145 | 101 | 0.330 | f ; g=\operatorname{id}_{\underline{2}} | | |
| fong-spivak-seven-sketches_FO1146 | 101 | 0.330 | g{ }_{9} f \neq \operatorname{id}_{\underline{3}} | | |
| fong-spivak-seven-sketches_FO1147 | 103 | 0.998 | F: \mathcal{C} \rightarrow \mathcal{D} | | |
| fong-spivak-seven-sketches_FO1148 | 103 | 0.989 | F(c) \in \mathrm{Ob}(\mathcal{D}) | | |
| fong-spivak-seven-sketches_FO1149 | 103 | 0.203 | f: c_{1} \rightarrow c_{2} | | |
| fong-spivak-seven-sketches_FO1150 | 103 | 0.203 | \mathcal{C}_{\text {, one }} | | |
| fong-spivak-seven-sketches_FO1151 | 103 | 0.203 | F(f): F\left(c_{1}\right) \rightarrow | | |
| fong-spivak-seven-sketches_FO1152 | 103 | 1.000 | F\left(c_{2}\right) | | |
| fong-spivak-seven-sketches_FO1153 | 103 | 0.628 | F\left(\mathrm{id}_{c}\right)=\mathrm{id}_{F(c)} | | |
| fong-spivak-seven-sketches_FO1154 | 103 | 0.870 | c_{1}, c_{2}, c_{3} \in \operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO1155 | 103 | 0.870 | f \in \mathcal{C}\left(c_{1}, c_{2}\right), g \in | | |
| fong-spivak-seven-sketches_FO1156 | 103 | 1.000 | \mathcal{C}\left(c_{2}, c_{3}\right) | | |
| fong-spivak-seven-sketches_FO1157 | 103 | 1.000 | F(f ; g)=F(f) ; F(g) | | |
| fong-spivak-seven-sketches_FO1158 | 103 | 0.975 | F: 2 \rightarrow 3 | | |
| fong-spivak-seven-sketches_FO1159 | 103 | 1.000 | n_{0} | | |
| fong-spivak-seven-sketches_FO1160 | 103 | 1.000 | F\left(m_{0}\right)=n_{0} | | |
| fong-spivak-seven-sketches_FO1161 | 103 | 1.000 | F\left(f_{1}\right)=g_{1} | | |
| fong-spivak-seven-sketches_FO1162 | 103 | 1.000 | F\left(m_{1}\right)=n_{1} | | |
| fong-spivak-seven-sketches_FO1163 | 103 | 1.000 | F\left(m_{0}\right)=n_{0}, F\left(m_{1}\right)=n_{2} | | |
| fong-spivak-seven-sketches_FO1164 | 103 | 0.996 | F\left(f_{1}\right)=g_{1} ; g_{2} | | |
| fong-spivak-seven-sketches_FO1165 | 103 | 0.772 | 2 \rightarrow 3 | | |
| fong-spivak-seven-sketches_FO1166 | 104 | 1.000 | \mathcal{F} | | |
| fong-spivak-seven-sketches_FO1167 | 104 | 1.000 | F: \mathcal{F} \rightarrow \mathcal{C} | | |
| fong-spivak-seven-sketches_FO1168 | 104 | 1.000 | A, B^{\prime} | | |
| fong-spivak-seven-sketches_FO1169 | 104 | 1.000 | B, C^{\prime} | | |
| fong-spivak-seven-sketches_FO1170 | 104 | 1.000 | D^{\prime} | | |
| fong-spivak-seven-sketches_FO1171 | 104 | 0.994 | \mathcal{C}=\bullet \rightarrow \bullet | | |
| fong-spivak-seven-sketches_FO1172 | 104 | 0.994 | \mathcal{D}=\bullet \rightrightarrows \bullet | | |
| fong-spivak-seven-sketches_FO1173 | 104 | 1.000 | F, G: \mathcal{C} \rightarrow \mathcal{D} | | |
| fong-spivak-seven-sketches_FO1174 | 104 | 0.979 | A^{\prime}, B | | |
| fong-spivak-seven-sketches_FO1175 | 104 | 0.979 | B^{\prime}, C | | |
| fong-spivak-seven-sketches_FO1176 | 104 | 0.979 | C^{\prime} | | |
| fong-spivak-seven-sketches_FO1177 | 104 | 0.352 | f^{\circ} ; h=g{ }^{\circ} i | | |
| fong-spivak-seven-sketches_FO1178 | 104 | 1.000 | F(f ; h)=F(g ; i) | | |
| fong-spivak-seven-sketches_FO1179 | 104 | 0.932 | f^{\prime} ; h^{\prime}=g^{\prime} ; i^{\prime} | | |
| fong-spivak-seven-sketches_FO1180 | 104 | 0.999 | \mathcal{N} | | |
| fong-spivak-seven-sketches_FO1181 | 104 | 1.000 | \operatorname{Ob}(\mathcal{N})=\mathbb{N} | | |
| fong-spivak-seven-sketches_FO1182 | 104 | 1.000 | m \rightarrow n | | |
| fong-spivak-seven-sketches_FO1183 | 104 | 1.000 | F: \mathcal{N} \rightarrow \mathcal{N} | | |
| fong-spivak-seven-sketches_FO1184 | 104 | 1.000 | F(n) \in \mathbb{N} | | |
| fong-spivak-seven-sketches_FO1185 | 104 | 1.000 | F(m) \rightarrow F(n) | | |
| fong-spivak-seven-sketches_FO1186 | 104 | 1.000 | F(m) \leq F(n) | | |
| fong-spivak-seven-sketches_FO1187 | 105 | 1.000 | \mathbb{N} \rightarrow \mathbb{N} | | |
| fong-spivak-seven-sketches_FO1188 | 105 | 0.873 | \mathrm{id}_{\mathcal{C}}: \mathcal{C} \rightarrow \mathcal{C} | | |
| fong-spivak-seven-sketches_FO1189 | 105 | 1.000 | \operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO1190 | 105 | 1.000 | \operatorname{Ob}(\mathcal{D}) | | |
| fong-spivak-seven-sketches_FO1191 | 105 | 0.987 | c_{1}, c_{2} \in \mathcal{C} | | |
| fong-spivak-seven-sketches_FO1192 | 105 | 0.987 | \mathcal{C}\left(c_{1}, c_{2}\right) | | |
| fong-spivak-seven-sketches_FO1193 | 105 | 0.987 | \mathcal{D}\left(F\left(c_{1}\right), F\left(c_{2}\right)\right) | | |
| fong-spivak-seven-sketches_FO1194 | 105 | 0.982 | G: \mathcal{D} \rightarrow \mathcal{E} | | |
| fong-spivak-seven-sketches_FO1195 | 105 | 1.000 | G): \mathcal{C} \rightarrow \mathcal{E} | | |
| fong-spivak-seven-sketches_FO1196 | 105 | 0.991 | F: \mathcal{C} \rightarrow | | |
| fong-spivak-seven-sketches_FO1197 | 105 | 0.508 | I: \mathcal{C} \rightarrow | | |
| fong-spivak-seven-sketches_FO1198 | 105 | 1.000 | c=\stackrel{\text { string }}{\circ} | | |
| fong-spivak-seven-sketches_FO1199 | 105 | 0.954 | ^{5} | | |
| fong-spivak-seven-sketches_FO1200 | 105 | 0.976 | \mathrm{id}_{1} | | |
| fong-spivak-seven-sketches_FO1201 | 105 | 0.976 | F: \mathbf{1} \rightarrow | | |
| fong-spivak-seven-sketches_FO1202 | 105 | 1.000 | F(1) | | |
| fong-spivak-seven-sketches_FO1203 | 105 | 1.000 | F_{S}: \mathbf{1} \rightarrow | | |
| fong-spivak-seven-sketches_FO1204 | 105 | 1.000 | F_{S}(1)=S | | |
| fong-spivak-seven-sketches_FO1205 | 105 | 1.000 | r \in I(c) | | |
| fong-spivak-seven-sketches_FO1206 | 106 | 0.999 | Z:=F(z) | | |
| fong-spivak-seven-sketches_FO1207 | 106 | 1.000 | S:=F(s): Z \rightarrow Z | | |
| fong-spivak-seven-sketches_FO1208 | 106 | 1.000 | S ; S=S | | |
| fong-spivak-seven-sketches_FO1209 | 106 | 0.837 | Z=\mathbb{N} | | |
| fong-spivak-seven-sketches_FO1210 | 106 | 0.917 | 13+(2 * 4) | | |
| fong-spivak-seven-sketches_FO1211 | 106 | 0.990 | Z=\mathbb{N}_{\geq 2} | | |
| fong-spivak-seven-sketches_FO1213 | 107 | 1.000 | \alpha: F \Rightarrow G | | |
| fong-spivak-seven-sketches_FO1214 | 107 | 0.884 | \alpha_{c}: F(c) \rightarrow G(c) | | |
| fong-spivak-seven-sketches_FO1215 | 107 | 1.000 | \alpha | | |
| fong-spivak-seven-sketches_FO1216 | 107 | 1.000 | \alpha: F \rightarrow G | | |
| fong-spivak-seven-sketches_FO1217 | 107 | 0.996 | \alpha_{c} | | |
| fong-spivak-seven-sketches_FO1218 | 107 | 0.988 | D: \mathcal{J} \rightarrow \mathcal{C} | | |
| fong-spivak-seven-sketches_FO1219 | 107 | 0.988 | \mathcal{J} | | |
| fong-spivak-seven-sketches_FO1220 | 107 | 1.000 | D(f)=D\left(f^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO1221 | 107 | 0.738 | f, f^{\prime}: a \rightarrow b | | |
| fong-spivak-seven-sketches_FO1222 | 107 | 0.738 | \mathcal{J}^{7} | | |
| fong-spivak-seven-sketches_FO1223 | 107 | 1.000 | F(c) \rightrightarrows G(d) | | |
| fong-spivak-seven-sketches_FO1224 | 107 | 0.528 | F(f) \stackrel{\circ}{9} \alpha_{d}=\alpha_{c}{ }^{\circ} G(f) | | |
| fong-spivak-seven-sketches_FO1225 | 108 | 1.000 | \alpha_{1}: v \rightarrow x | | |
| fong-spivak-seven-sketches_FO1226 | 108 | 1.000 | \alpha_{2}: w \rightarrow y | | |
| fong-spivak-seven-sketches_FO1227 | 108 | 1.000 | \alpha_{1}=c | | |
| fong-spivak-seven-sketches_FO1228 | 108 | 1.000 | \alpha_{2}=g | | |
| fong-spivak-seven-sketches_FO1229 | 108 | 0.150 | \mathcal{C} \xrightarrow[G]{F} \stackrel{F}{\underset{G}{\leftrightarrows}} D | | |
| fong-spivak-seven-sketches_FO1230 | 108 | 0.959 | A, B: \mathbf{1} \rightarrow | | |
| fong-spivak-seven-sketches_FO1231 | 108 | 1.000 | \mathcal{D}^{\mathcal{C}} | | |
| fong-spivak-seven-sketches_FO1232 | 108 | 0.678 | \mathcal{D}^{\mathcal{P}}(F, G) | | |
| fong-spivak-seven-sketches_FO1233 | 108 | 0.962 | \mathcal{D}^{\mathrm{e}} | | |
| fong-spivak-seven-sketches_FO1234 | 108 | 0.875 | F \in \mathcal{D}^{\mathcal{C}} | | |
| fong-spivak-seven-sketches_FO1235 | 108 | 0.774 | \mathbf{S e t}^{\mathbf{1}} | | |
| fong-spivak-seven-sketches_FO1236 | 108 | 0.923 | \left(F_{0}, F_{1}, F_{2}, \ldots\right) | | |
| fong-spivak-seven-sketches_FO1237 | 108 | 0.923 | F_{0} \leq F_{1} \leq F_{2} \leq \cdots | | |
| fong-spivak-seven-sketches_FO1238 | 109 | 1.000 | \alpha_{n}: F_{n} \rightarrow G_{n} | | |
| fong-spivak-seven-sketches_FO1239 | 109 | 1.000 | F_{n} \leq G_{n} | | |
| fong-spivak-seven-sketches_FO1240 | 109 | 0.987 | F \Rightarrow G | | |
| fong-spivak-seven-sketches_FO1241 | 109 | 0.987 | \mathcal{N}^{\mathcal{N}} | | |
| fong-spivak-seven-sketches_FO1242 | 109 | 1.000 | F, G: \mathcal{C} \rightarrow \mathcal{P} | | |
| fong-spivak-seven-sketches_FO1243 | 109 | 1.000 | F \rightarrow G | | |
| fong-spivak-seven-sketches_FO1244 | 109 | 0.999 | F, G: \mathcal{P} \rightarrow \mathcal{C} | | |
| fong-spivak-seven-sketches_FO1245 | 109 | 0.865 | F ; G \cong \operatorname{id}_{\text {Pro }} | | |
| fong-spivak-seven-sketches_FO1246 | 109 | 0.865 | G ; F \cong \operatorname{id}_{\text {Bool-Cat }} | | |
| fong-spivak-seven-sketches_FO1247 | 109 | 1.000 | I, J: \mathcal{C} \rightarrow | | |
| fong-spivak-seven-sketches_FO1248 | 109 | 1.000 | \alpha: I \rightarrow | | |
| fong-spivak-seven-sketches_FO1249 | 109 | 0.490 | := | | |
| fong-spivak-seven-sketches_FO1250 | 109 | 0.490 | ^{\mathcal{C}} | | |
| fong-spivak-seven-sketches_FO1251 | 110 | 0.986 | I( | | |
| fong-spivak-seven-sketches_FO1252 | 110 | 0.986 | )=\{a, b, c, d, e\} | | |
| fong-spivak-seven-sketches_FO1253 | 110 | 0.986 | )=\{1,2,3,4\} | | |
| fong-spivak-seven-sketches_FO1254 | 110 | 0.983 | G, H: \mathbf{G r} \rightarrow | | |
| fong-spivak-seven-sketches_FO1255 | 110 | 0.989 | G, H \in \mathbf{G r} | | |
| fong-spivak-seven-sketches_FO1256 | 110 | 0.999 | G \rightarrow H | | |
| fong-spivak-seven-sketches_FO1257 | 110 | 0.999 | \alpha: G \rightarrow H | | |
| fong-spivak-seven-sketches_FO1258 | 110 | 1.000 | \alpha_{\text {Vertex }} | | |
| fong-spivak-seven-sketches_FO1259 | 110 | 1.000 | \alpha_{\text {Arrow }} | | |
| fong-spivak-seven-sketches_FO1262 | 111 | 1.000 | G, H | | |
| fong-spivak-seven-sketches_FO1263 | 111 | 0.995 | \alpha_{\text {Arrow }}(a)=d | | |
| fong-spivak-seven-sketches_FO1264 | 112 | 0.598 | F: \mathbf{G r} \rightarrow \mathbf{D D S} | | |
| fong-spivak-seven-sketches_FO1265 | 113 | 0.384 | F{ }_{9}^{\circ} I: \mathbf{G r} \rightarrow | | |
| fong-spivak-seven-sketches_FO1266 | 113 | 0.384 | F_{9}^{\circ} I | | |
| fong-spivak-seven-sketches_FO1267 | 114 | 0.731 | I: \mathcal{D} \rightarrow | | |
| fong-spivak-seven-sketches_FO1268 | 114 | 0.731 | F: I: \mathcal{C} \rightarrow | | |
| fong-spivak-seven-sketches_FO1269 | 114 | 1.000 | \alpha: I \Rightarrow J | | |
| fong-spivak-seven-sketches_FO1270 | 114 | 1.000 | \alpha_{F}: F_{9}^{\circ} I \Rightarrow | | |
| fong-spivak-seven-sketches_FO1271 | 114 | 0.314 | F_{9}^{\circ} J | | |
| fong-spivak-seven-sketches_FO1272 | 114 | 0.314 | \left(F{ }_{9}^{\circ} I\right)(c) \rightarrow\left(F{ }_{\circ}^{\circ} J\right)(c) | | |
| fong-spivak-seven-sketches_FO1273 | 114 | 0.314 | c \in \operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO1274 | 114 | 0.314 | \left(\alpha_{F}\right)_{c}:=\alpha_{F c} | | |
| fong-spivak-seven-sketches_FO1275 | 114 | 0.338 | \Delta_{F}: \mathcal{D} | | |
| fong-spivak-seven-sketches_FO1276 | 114 | 1.000 | \Delta | | |
| fong-spivak-seven-sketches_FO1277 | 114 | 0.998 | P(a, b) | | |
| fong-spivak-seven-sketches_FO1278 | 114 | 0.997 | Q(f(p), q) \cong P(p, g(q)) | | |
| fong-spivak-seven-sketches_FO1279 | 114 | 1.000 | L: \mathcal{C} \rightarrow \mathcal{D} | | |
| fong-spivak-seven-sketches_FO1280 | 114 | 1.000 | R: \mathcal{D} \rightarrow \mathcal{C} | | |
| fong-spivak-seven-sketches_FO1281 | 114 | 1.000 | L | | |
| fong-spivak-seven-sketches_FO1282 | 114 | 0.844 | d \in \mathcal{D} | | |
| fong-spivak-seven-sketches_FO1283 | 114 | 0.615 | d .^{8} | | |
| fong-spivak-seven-sketches_FO1284 | 114 | 0.998 | f: c \rightarrow R(d) | | |
| fong-spivak-seven-sketches_FO1285 | 114 | 0.998 | g:=\alpha_{c, d}(f) | | |
| fong-spivak-seven-sketches_FO1286 | 114 | 1.000 | g: L(c) \rightarrow d | | |
| fong-spivak-seven-sketches_FO1287 | 115 | 1.000 | L \dashv R | | |
| fong-spivak-seven-sketches_FO1288 | 115 | 0.801 | B \in \mathrm{Ob}(\mathbf{S e t}) | | |
| fong-spivak-seven-sketches_FO1289 | 115 | 1.000 | A, C | | |
| fong-spivak-seven-sketches_FO1290 | 115 | 0.912 | B \rightarrow C | | |
| fong-spivak-seven-sketches_FO1291 | 115 | 0.912 | C^{B} | | |
| fong-spivak-seven-sketches_FO1292 | 115 | 1.000 | 10^{3} | | |
| fong-spivak-seven-sketches_FO1293 | 115 | 1.000 | \left|C^{B}\right|=|C|^{|B|} | | |
| fong-spivak-seven-sketches_FO1294 | 115 | 1.000 | (A \times B) \rightarrow C | | |
| fong-spivak-seven-sketches_FO1295 | 115 | 1.000 | A \rightarrow C^{B} | | |
| fong-spivak-seven-sketches_FO1296 | 115 | 0.996 | b \multimap c | | |
| fong-spivak-seven-sketches_FO1297 | 115 | 0.983 | -\times B | | |
| fong-spivak-seven-sketches_FO1298 | 115 | 1.000 | (-)^{B} | | |
| fong-spivak-seven-sketches_FO1299 | 115 | 1.000 | X \times B | | |
| fong-spivak-seven-sketches_FO1300 | 115 | 1.000 | X^{B} | | |
| fong-spivak-seven-sketches_FO1301 | 115 | 0.953 | \mathcal{C}^{\mathrm{op}} \times \mathcal{D} \rightarrow | | |
| fong-spivak-seven-sketches_FO1302 | 115 | 0.953 | f: c^{\prime} \rightarrow c | | |
| fong-spivak-seven-sketches_FO1303 | 115 | 1.000 | g: d \rightarrow d^{\prime} | | |
| fong-spivak-seven-sketches_FO1304 | 116 | 1.000 | -\times B: X \times B \rightarrow Y \times B | | |
| fong-spivak-seven-sketches_FO1305 | 116 | 1.000 | (-)^{B}: X^{B} \rightarrow Y^{B} | | |
| fong-spivak-seven-sketches_FO1306 | 116 | 1.000 | +: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N} | | |
| fong-spivak-seven-sketches_FO1307 | 116 | 1.000 | (a, b) \mapsto a+b | | |
| fong-spivak-seven-sketches_FO1308 | 116 | 1.000 | p: \mathbb{N} \rightarrow \mathbb{N}^{\mathbb{N}} | | |
| fong-spivak-seven-sketches_FO1309 | 116 | 1.000 | p(3) | | |
| fong-spivak-seven-sketches_FO1310 | 116 | 1.000 | \Delta_{F} | | |
| fong-spivak-seven-sketches_FO1311 | 117 | 1.000 | \Sigma_{F} | | |
| fong-spivak-seven-sketches_FO1312 | 117 | 1.000 | \Pi_{F} | | |
| fong-spivak-seven-sketches_FO1313 | 117 | 0.910 | J: \mathbf{G r} \rightarrow | | |
| fong-spivak-seven-sketches_FO1314 | 117 | 1.000 | \Sigma_{F}(J) | | |
| fong-spivak-seven-sketches_FO1315 | 117 | 1.000 | \Pi_{F}(J) | | |
| fong-spivak-seven-sketches_FO1316 | 117 | 1.000 | (e, f) | | |
| fong-spivak-seven-sketches_FO1317 | 118 | 1.000 | \Pi_{F}(I) | | |
| fong-spivak-seven-sketches_FO1318 | 118 | 1.000 | \Delta_{F}, \Sigma_{F} | | |
| fong-spivak-seven-sketches_FO1319 | 118 | 0.931 | \mathbf{1} | | |
| fong-spivak-seven-sketches_FO1320 | 118 | 0.911 | \mathcal{C} \rightarrow \mathbf{1} | | |
| fong-spivak-seven-sketches_FO1321 | 118 | 0.915 | \mathcal{C} \boldsymbol{\rightarrow} \mathbf{1} | | |
| fong-spivak-seven-sketches_FO1322 | 118 | 0.133 | \Sigma_{!} | | |
| fong-spivak-seven-sketches_FO1323 | 118 | 0.133 | \Pi_{!}: \mathcal{} | | |
| fong-spivak-seven-sketches_FO1324 | 118 | 0.988 | \Sigma_{!}(I) | | |
| fong-spivak-seven-sketches_FO1325 | 118 | 0.988 | \Pi_{!}(I) | | |
| fong-spivak-seven-sketches_FO1326 | 118 | 0.985 | I: \mathcal{G} \rightarrow | | |
| fong-spivak-seven-sketches_FO1327 | 119 | 1.000 | \mathcal{G} | | |
| fong-spivak-seven-sketches_FO1328 | 119 | 0.769 | !: \mathcal{G} \rightarrow \mathbf{1} | | |
| fong-spivak-seven-sketches_FO1329 | 119 | 1.000 | \Pi_{!} | | |
| fong-spivak-seven-sketches_FO1330 | 119 | 0.788 | \mathbf{1}, \Sigma | | |
| fong-spivak-seven-sketches_FO1331 | 120 | 0.999 | C \rightarrow Z | | |
| fong-spivak-seven-sketches_FO1332 | 120 | 0.539 | \{\bullet\} | | |
| fong-spivak-seven-sketches_FO1333 | 120 | 0.821 | C \rightarrow\{\bullet\} | | |
| fong-spivak-seven-sketches_FO1334 | 120 | 1.000 | c \in C | | |
| fong-spivak-seven-sketches_FO1335 | 120 | 1.000 | \bullet \in\{\bullet\} | | |
| fong-spivak-seven-sketches_FO1336 | 120 | 0.918 | z \in P | | |
| fong-spivak-seven-sketches_FO1337 | 120 | 1.000 | c \in P | | |
| fong-spivak-seven-sketches_FO1338 | 120 | 1.000 | c \leq z | | |
| fong-spivak-seven-sketches_FO1339 | 120 | 0.924 | Z^{\prime} | | |
| fong-spivak-seven-sketches_FO1340 | 120 | 1.000 | a: Z \rightarrow Z^{\prime} | | |
| fong-spivak-seven-sketches_FO1341 | 120 | 1.000 | b: Z^{\prime} \rightarrow Z | | |
| fong-spivak-seven-sketches_FO1342 | 120 | 1.000 | a ; b: Z \rightarrow Z | | |
| fong-spivak-seven-sketches_FO1343 | 120 | 0.707 | Z \rightarrow Z | | |
| fong-spivak-seven-sketches_FO1344 | 120 | 0.707 | \mathrm{id}_{Z} | | |
| fong-spivak-seven-sketches_FO1345 | 120 | 0.707 | a ; b=\mathrm{id}_{Z} | | |
| fong-spivak-seven-sketches_FO1346 | 120 | 0.900 | b ; a=\mathrm{id}_{\mathrm{Z}^{\prime}} | | |
| fong-spivak-seven-sketches_FO1347 | 120 | 1.000 | X, Y | | |
| fong-spivak-seven-sketches_FO1348 | 120 | 1.000 | p_{X}: X \times Y \rightarrow X | | |
| fong-spivak-seven-sketches_FO1349 | 120 | 1.000 | p_{Y}: X \times Y \rightarrow Y | | |
| fong-spivak-seven-sketches_FO1350 | 120 | 1.000 | f: C \rightarrow X | | |
| fong-spivak-seven-sketches_FO1351 | 120 | 1.000 | g: C \rightarrow Y | | |
| fong-spivak-seven-sketches_FO1352 | 121 | 1.000 | C \rightarrow X \times Y | | |
| fong-spivak-seven-sketches_FO1353 | 121 | 1.000 | \langle f, g\rangle | | |
| fong-spivak-seven-sketches_FO1354 | 121 | 1.000 | p_{X}(x, y):=x | | |
| fong-spivak-seven-sketches_FO1355 | 121 | 1.000 | p_{Y}(x, y):=y | | |
| fong-spivak-seven-sketches_FO1356 | 121 | 1.000 | \langle f, g\rangle(c):=(f(c), g(c)) | | |
| fong-spivak-seven-sketches_FO1357 | 121 | 1.000 | \underline{6} \times \underline{4} | | |
| fong-spivak-seven-sketches_FO1358 | 121 | 1.000 | \underline{6} | | |
| fong-spivak-seven-sketches_FO1359 | 121 | 1.000 | \underline{4} | | |
| fong-spivak-seven-sketches_FO1360 | 121 | 0.943 | x, y \in P | | |
| fong-spivak-seven-sketches_FO1361 | 121 | 1.000 | x \wedge y | | |
| fong-spivak-seven-sketches_FO1362 | 122 | 0.997 | \mathcal{C} \times \mathcal{D} | | |
| fong-spivak-seven-sketches_FO1363 | 122 | 1.000 | (c, d) | | |
| fong-spivak-seven-sketches_FO1364 | 122 | 0.887 | (c, d) \rightarrow\left(c^{\prime}, d^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO1365 | 122 | 0.887 | (f, g) | | |
| fong-spivak-seven-sketches_FO1366 | 122 | 0.887 | f: c \rightarrow c^{\prime} | | |
| fong-spivak-seven-sketches_FO1367 | 122 | 0.944 | (f, g),\left(f^{\prime}, g^{\prime}\right)=\left(f ; f^{\prime}, g ; g^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO1368 | 122 | 1.000 | \mathbf{1} \times \mathbf{2} | | |
| fong-spivak-seven-sketches_FO1369 | 122 | 0.941 | \mathcal{P} \times Q | | |
| fong-spivak-seven-sketches_FO1370 | 122 | 1.000 | X \stackrel{f}{\leftarrow} C \xrightarrow{g} Y | | |
| fong-spivak-seven-sketches_FO1371 | 122 | 0.623 | (X, Y) | | |
| fong-spivak-seven-sketches_FO1372 | 122 | 0.437 | X \stackrel{p_{X}}{\hookleftarrow} X \times Y \xrightarrow{p_{Y}} Y | | |
| fong-spivak-seven-sketches_FO1373 | 122 | 0.437 | \boldsymbol{\operatorname { C o n e }}(X, Y) | | |
| fong-spivak-seven-sketches_FO1374 | 122 | 0.857 | X \stackrel{f^{\prime}}{\leftarrow} C^{\prime} \xrightarrow{g^{\prime}} Y | | |
| fong-spivak-seven-sketches_FO1375 | 122 | 0.857 | a: C \rightarrow C^{\prime} | | |
| fong-spivak-seven-sketches_FO1376 | 122 | 0.714 | X \stackrel{p_{X}}{\longleftrightarrow} X \times Y \xrightarrow{p_{Y}} Y | | |
| fong-spivak-seven-sketches_FO1377 | 123 | 0.659 | \mathfrak{J} | | |
| fong-spivak-seven-sketches_FO1378 | 123 | 0.996 | \left(C, c_{*}\right) | | |
| fong-spivak-seven-sketches_FO1379 | 123 | 1.000 | C \in \mathcal{C} | | |
| fong-spivak-seven-sketches_FO1380 | 123 | 0.915 | j \in \mathcal{J} | | |
| fong-spivak-seven-sketches_FO1381 | 123 | 0.915 | c_{j}: C \rightarrow D(j) | | |
| fong-spivak-seven-sketches_FO1382 | 123 | 0.672 | f: j \rightarrow k | | |
| fong-spivak-seven-sketches_FO1383 | 123 | 0.672 | c_{k}=c_{j} ; D(f) | | |
| fong-spivak-seven-sketches_FO1384 | 123 | 0.997 | \left(C, c_{*}\right) \rightarrow\left(C^{\prime}, c_{*}^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO1385 | 123 | 0.271 | c_{j}=a | | |
| fong-spivak-seven-sketches_FO1386 | 123 | 0.271 | c_{j}^{\prime} | | |
| fong-spivak-seven-sketches_FO1387 | 123 | 0.271 | \operatorname{Cone}(D) | | |
| fong-spivak-seven-sketches_FO1388 | 123 | 0.742 | \lim (D) | | |
| fong-spivak-seven-sketches_FO1389 | 123 | 0.997 | \lim (D)=\left(C, c_{*}\right) | | |
| fong-spivak-seven-sketches_FO1390 | 123 | 0.997 | c_{j} | | |
| fong-spivak-seven-sketches_FO1391 | 123 | 0.995 | j^{\text {th }} | | |
| fong-spivak-seven-sketches_FO1392 | 123 | 0.998 | \mathcal{J}=\varnothing | | |
| fong-spivak-seven-sketches_FO1393 | 123 | 0.198 | v, w | | |
| fong-spivak-seven-sketches_FO1394 | 123 | 0.198 | \mathcal{J}= | | |
| fong-spivak-seven-sketches_FO1395 | 123 | 0.198 | \cdot | | |
| fong-spivak-seven-sketches_FO1396 | 124 | 0.879 | V, A, s, t | | |
| fong-spivak-seven-sketches_FO1397 | 124 | 0.994 | D: \mathcal{J} \rightarrow | | |
| fong-spivak-seven-sketches_FO1398 | 124 | 0.994 | V=\left\{v_{1}, \ldots, v_{n}\right\} | | |
| fong-spivak-seven-sketches_FO1399 | 124 | 1.000 | p_{i}:\left(\lim _{\mathcal{J}} D\right) \rightarrow D\left(v_{i}\right) | | |
| fong-spivak-seven-sketches_FO1400 | 124 | 1.000 | p_{i}\left(d_{1}, \ldots, d_{n}\right):=d_{i} | | |
| fong-spivak-seven-sketches_FO1401 | 124 | 0.999 | J | | |
| fong-spivak-seven-sketches_FO1402 | 124 | 0.999 | n=0 | | |
| fong-spivak-seven-sketches_FO1403 | 124 | 1.000 | D: \mathbf{1} \rightarrow | | |
| fong-spivak-seven-sketches_FO1404 | 124 | 0.784 | \left(d_{1}, \ldots, d_{n}\right) | | |
| fong-spivak-seven-sketches_FO1405 | 124 | 0.784 | D(a)\left(d_{i}\right)=d_{j} | | |
| fong-spivak-seven-sketches_FO1406 | 124 | 0.784 | a: i \rightarrow j | | |
| fong-spivak-seven-sketches_FO1407 | 124 | 0.986 | \stackrel{x}{\bullet} \xrightarrow{f} \stackrel{a}{\bullet} \stackrel{g}{\bullet} | | |
| fong-spivak-seven-sketches_FO1408 | 124 | 0.999 | X \xrightarrow{f} A \stackrel{g}{\leftarrow} Y | | |
| fong-spivak-seven-sketches_FO1409 | 125 | 1.000 | c_{a} | | |
| fong-spivak-seven-sketches_FO1410 | 125 | 0.579 | X \times_{A} Y | | |
| fong-spivak-seven-sketches_FO1411 | 125 | 1.000 | X=\underline{6}, Y=\underline{4} | | |
| fong-spivak-seven-sketches_FO1412 | 125 | 1.000 | f: \underline{6} \rightarrow A | | |
| fong-spivak-seven-sketches_FO1413 | 125 | 1.000 | g: \underline{4} \rightarrow A | | |
| fong-spivak-seven-sketches_FO1414 | 125 | 1.000 | g(2)=g(4)= | | |
| fong-spivak-seven-sketches_FO1415 | 125 | 1.000 | f(5)= | | |
| fong-spivak-seven-sketches_FO1416 | 125 | 1.000 | (i, j) \in \underline{6} \times \underline{4} | | |
| fong-spivak-seven-sketches_FO1417 | 125 | 1.000 | f(i) | | |
| fong-spivak-seven-sketches_FO1418 | 125 | 1.000 | g(j) | | |
| fong-spivak-seven-sketches_FO1419 | 125 | 0.516 | F^{\text {op }}: \mathcal{C}{ }^{\text {op }} \rightarrow \mathcal{D}^{\text {op }} | | |
| fong-spivak-seven-sketches_FO1420 | 126 | 0.997 | F^{\text {op }} | | |
| fong-spivak-seven-sketches_FO1421 | 126 | 0.981 | D^{\mathrm{op}}: \mathcal{J}^{\mathrm{op}} \rightarrow \mathcal{C}^{\mathrm{op}} | | |
| fong-spivak-seven-sketches_FO1422 | 126 | 1.000 | \Delta, \Sigma, \Pi | | |
| fong-spivak-seven-sketches_FO1423 | 130 | 1.000 | \$ 10 \leq \$ 20,5 \mathrm{~W} \leq 6 \mathrm{~W} | | |
| fong-spivak-seven-sketches_FO1424 | 131 | 1.000 | (p, r) \in P \times R | | |
| fong-spivak-seven-sketches_FO1425 | 131 | 1.000 | \Phi: P \times R \rightarrow | | |
| fong-spivak-seven-sketches_FO1426 | 131 | 1.000 | \Phi(p, r)= | | |
| fong-spivak-seven-sketches_FO1427 | 131 | 1.000 | p^{\prime} \leq p | | |
| fong-spivak-seven-sketches_FO1428 | 131 | 1.000 | \Phi\left(p^{\prime}, r\right)= | | |
| fong-spivak-seven-sketches_FO1429 | 131 | 1.000 | r \leq r^{\prime} | | |
| fong-spivak-seven-sketches_FO1430 | 131 | 1.000 | \Phi\left(p, r^{\prime}\right)= | | |
| fong-spivak-seven-sketches_FO1431 | 131 | 1.000 | r^{\prime} \geq r | | |
| fong-spivak-seven-sketches_FO1432 | 131 | 0.685 | P^{\mathrm{op}} \times R \rightarrow | | |
| fong-spivak-seven-sketches_FO1433 | 131 | 0.516 | 5 \mathrm{~W} \leq 10 \mathrm{~W} | | |
| fong-spivak-seven-sketches_FO1434 | 132 | 0.500 | X=(X, \leq) | | |
| fong-spivak-seven-sketches_FO1435 | 132 | 0.982 | X^{\mathrm{op}}=(X, \geq) | | |
| fong-spivak-seven-sketches_FO1436 | 132 | 0.982 | x \geq y | | |
| fong-spivak-seven-sketches_FO1437 | 132 | 0.521 | X=\left(X, \leq_{X}\right) | | |
| fong-spivak-seven-sketches_FO1438 | 132 | 0.521 | y=\left(Y, \leq_{Y}\right) | | |
| fong-spivak-seven-sketches_FO1439 | 132 | 0.521 | \mathfrak{X} | | |
| fong-spivak-seven-sketches_FO1440 | 132 | 0.775 | \Phi: \mathcal{X} \rightarrow y | | |
| fong-spivak-seven-sketches_FO1441 | 132 | 1.000 | \Phi(x, y)= | | |
| fong-spivak-seven-sketches_FO1442 | 132 | 1.000 | x^{\prime} \leq_{X} x | | |
| fong-spivak-seven-sketches_FO1443 | 132 | 1.000 | y \leq_{Y} y^{\prime} | | |
| fong-spivak-seven-sketches_FO1444 | 132 | 1.000 | \Phi(x, y) \leq_{\text {Bool }} \Phi\left(x^{\prime}, y^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO1445 | 132 | 1.000 | x^{\prime} | | |
| fong-spivak-seven-sketches_FO1446 | 132 | 0.605 | \mathcal{X}^{\mathrm{op}} \times \mathcal{Y} | | |
| fong-spivak-seven-sketches_FO1447 | 132 | 0.994 | \Lambda: \mathcal{X} \rightarrow y | | |
| fong-spivak-seven-sketches_FO1448 | 132 | 0.994 | \Lambda(x, y)= | | |
| fong-spivak-seven-sketches_FO1449 | 132 | 0.372 | X^{\mathrm{op}} \times y | | |
| fong-spivak-seven-sketches_FO1450 | 132 | 0.637 | \Rightarrow: \mathbb{B} \times \mathbb{B} \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO1451 | 132 | 1.000 | b, c, d \in \mathbb{B} | | |
| fong-spivak-seven-sketches_FO1453 | 133 | 0.671 | \Phi: \operatorname{Ob}(\mathcal{X}) \times \operatorname{Ob}(\mathcal{Y}) \rightarrow V | | |
| fong-spivak-seven-sketches_FO1454 | 133 | 0.671 | x, x^{\prime} \in \mathcal{X} | | |
| fong-spivak-seven-sketches_FO1455 | 133 | 0.671 | y, y^{\prime} \in y | | |
| fong-spivak-seven-sketches_FO1456 | 134 | 0.821 | \Phi: X \rightarrow Y | | |
| fong-spivak-seven-sketches_FO1457 | 134 | 1.000 | W \leq N | | |
| fong-spivak-seven-sketches_FO1458 | 134 | 1.000 | N | | |
| fong-spivak-seven-sketches_FO1459 | 134 | 1.000 | E | | |
| fong-spivak-seven-sketches_FO1460 | 134 | 1.000 | \Phi(N, e)= | | |
| fong-spivak-seven-sketches_FO1461 | 134 | 1.000 | \Phi(E, a)= | | |
| fong-spivak-seven-sketches_FO1462 | 134 | 1.000 | W | | |
| fong-spivak-seven-sketches_FO1463 | 134 | 1.000 | \Phi(W, d)= | | |
| fong-spivak-seven-sketches_FO1464 | 134 | 1.000 | W \leq N \leq c \leq a | | |
| fong-spivak-seven-sketches_FO1465 | 134 | 1.000 | \Phi(m, n) \in \mathbb{B} | | |
| fong-spivak-seven-sketches_FO1467 | 135 | 1.000 | d_{X} * M_{\Phi} * d_{Y} | | |
| fong-spivak-seven-sketches_FO1468 | 135 | 0.835 | \Phi=M_{X}^{3} * M_{\Phi} * M_{Y}^{2} | | |
| fong-spivak-seven-sketches_FO1469 | 135 | 1.000 | M_{X}^{3} * M_{\Phi} * M_{Y}^{2} | | |
| fong-spivak-seven-sketches_FO1470 | 136 | 0.873 | ([0, \infty], \leq) | | |
| fong-spivak-seven-sketches_FO1471 | 136 | --- | 100, more_than_\ | --- | |
| fong-spivak-seven-sketches_FO1472 | 136 | 0.965 | \Phi:(T \times E) \mapsto \$ | | |
| fong-spivak-seven-sketches_FO1473 | 136 | 0.695 | T, E | | |
| fong-spivak-seven-sketches_FO1474 | 136 | 0.695 | \$ | | |
| fong-spivak-seven-sketches_FO1475 | 137 | 0.940 | \$ 500 \mathrm{~K} | | |
| fong-spivak-seven-sketches_FO1476 | 137 | --- | 500K given \ | --- | |
| fong-spivak-seven-sketches_FO1477 | 137 | 0.928 | \$ 500 | | |
| fong-spivak-seven-sketches_FO1478 | 137 | --- | 1M indicates also that "I can provide a mean and boring movie given \ | --- | |
| fong-spivak-seven-sketches_FO1479 | 137 | 0.952 | \mathcal{P}, Q | | |
| fong-spivak-seven-sketches_FO1480 | 137 | 0.952 | \mathcal{R} | | |
| fong-spivak-seven-sketches_FO1481 | 138 | 0.497 | \Phi: \mathcal{P} \mapsto Q | | |
| fong-spivak-seven-sketches_FO1482 | 138 | 0.497 | \Psi: \mathcal{Q} \rightarrow \mathcal{R} | | |
| fong-spivak-seven-sketches_FO1483 | 138 | 1.000 | \Psi | | |
| fong-spivak-seven-sketches_FO1484 | 138 | 0.223 | \Phi_{9}^{9} \Psi | | |
| fong-spivak-seven-sketches_FO1485 | 138 | 0.984 | p \in \mathcal{P} | | |
| fong-spivak-seven-sketches_FO1486 | 138 | 0.984 | r \in \mathcal{R} | | |
| fong-spivak-seven-sketches_FO1487 | 138 | 0.984 | \mathcal{Q} | | |
| fong-spivak-seven-sketches_FO1489 | 139 | 0.604 | \mathcal{X}, \mathcal{Y} | | |
| fong-spivak-seven-sketches_FO1490 | 139 | 0.604 | \mathcal{Z} | | |
| fong-spivak-seven-sketches_FO1491 | 139 | 0.604 | \Phi: \mathfrak{X} \rightarrow y | | |
| fong-spivak-seven-sketches_FO1492 | 139 | 0.666 | \Psi: y \rightarrow z | | |
| fong-spivak-seven-sketches_FO1493 | 139 | 0.666 | \Phi ; \Psi: \mathcal{X} \rightarrow z | | |
| fong-spivak-seven-sketches_FO1494 | 139 | 0.405 | \Phi: X \rightarrow y | | |
| fong-spivak-seven-sketches_FO1496 | 139 | 0.564 | _{\mathcal{V}} | | |
| fong-spivak-seven-sketches_FO1497 | 139 | 0.426 | \mathcal{X} \rightarrow y | | |
| fong-spivak-seven-sketches_FO1498 | 139 | 0.505 | _{\text {Bool }} | | |
| fong-spivak-seven-sketches_FO1499 | 140 | 0.744 | \mathrm{U}_{x}: \mathcal{X} \rightarrow \mathcal{X} | | |
| fong-spivak-seven-sketches_FO1500 | 140 | 0.988 | \mathcal{X}(x, y) \in \mathcal{V} | | |
| fong-spivak-seven-sketches_FO1501 | 140 | 0.988 | \mathrm{U}_{x} | | |
| fong-spivak-seven-sketches_FO1502 | 140 | 0.666 | U_{X}: \mathcal{X} \rightarrow \mathcal{X} | | |
| fong-spivak-seven-sketches_FO1503 | 140 | 0.982 | \Phi: \mathcal{P} \rightarrow Q | | |
| fong-spivak-seven-sketches_FO1504 | 140 | 0.982 | U_{\mathcal{P}} | | |
| fong-spivak-seven-sketches_FO1505 | 140 | 0.637 | \mathrm{U}_{\mathrm{Q}} | | |
| fong-spivak-seven-sketches_FO1506 | 140 | 0.296 | \mathrm{U}_{\mathcal{P}} ; \Phi=\Phi | | |
| fong-spivak-seven-sketches_FO1507 | 140 | 0.296 | \Phi=\Phi | | |
| fong-spivak-seven-sketches_FO1508 | 140 | 0.673 | \Phi \leq \mathrm{U}_{\mathcal{P}} ; \Phi | | |
| fong-spivak-seven-sketches_FO1509 | 140 | 0.996 | \mathrm{U}_{\mathcal{P}} ; \Phi \leq \Phi | | |
| fong-spivak-seven-sketches_FO1510 | 140 | 1.000 | \bigvee_{p_{1} \in P}\left(\mathcal{P}\left(p, p_{1}\right) \otimes \Phi\left(p_{1}, q\right)\right) \leq \Phi(p, q) | | |
| fong-spivak-seven-sketches_FO1511 | 140 | 1.000 | \mathcal{P}\left(p, p_{1}\right) \otimes \Phi\left(p_{1}, q\right) \leq \Phi(p, q) | | |
| fong-spivak-seven-sketches_FO1512 | 140 | 1.000 | p_{1} \in \mathcal{P} | | |
| fong-spivak-seven-sketches_FO1513 | 140 | 0.696 | (=, \leq, \leq,=) | | |
| fong-spivak-seven-sketches_FO1514 | 141 | 0.964 | \leq, \leq | | |
| fong-spivak-seven-sketches_FO1515 | 141 | 0.567 | \Phi: \mathcal{P} \rightarrow Q, \Psi: \mathcal{Q} \rightarrow \mathcal{R} | | |
| fong-spivak-seven-sketches_FO1516 | 141 | 0.567 | \Upsilon: \mathcal{R} \rightarrow \mathcal{S} | | |
| fong-spivak-seven-sketches_FO1517 | 142 | 0.426 | { }_{\mathrm{C}} | | |
| fong-spivak-seven-sketches_FO1518 | 142 | 0.977 | F: \mathcal{P} \rightarrow Q | | |
| fong-spivak-seven-sketches_FO1519 | 142 | 0.977 | \widehat{F}: \mathcal{P} \rightarrow Q | | |
| fong-spivak-seven-sketches_FO1520 | 142 | 0.488 | \check{F}: Q \mapsto \mathcal{P} | | |
| fong-spivak-seven-sketches_FO1521 | 142 | 0.995 | F(p) \in Q | | |
| fong-spivak-seven-sketches_FO1522 | 142 | 1.000 | \widehat{F} | | |
| fong-spivak-seven-sketches_FO1523 | 142 | 0.979 | Q \rightarrow \mathcal{P} | | |
| fong-spivak-seven-sketches_FO1524 | 142 | 1.000 | \check{F} | | |
| fong-spivak-seven-sketches_FO1525 | 142 | 0.710 | \mathrm{id}: \mathcal{P} \rightarrow \mathcal{P} | | |
| fong-spivak-seven-sketches_FO1526 | 142 | 0.699 | \widehat{\mathrm{id}}=\check{\mathrm{id}}: \mathcal{P} \mapsto \mathcal{P} | | |
| fong-spivak-seven-sketches_FO1527 | 142 | 1.000 | \mathrm{U}_{\mathcal{P}} | | |
| fong-spivak-seven-sketches_FO1528 | 143 | 0.991 | \widehat{\mathrm{id}} | | |
| fong-spivak-seven-sketches_FO1529 | 143 | 0.991 | \mathcal{P} \rightarrow \mathcal{P} | | |
| fong-spivak-seven-sketches_FO1530 | 143 | 0.637 | +: \mathbb{R} \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} | | |
| fong-spivak-seven-sketches_FO1531 | 143 | 0.979 | a+b+c \in \mathbb{R} | | |
| fong-spivak-seven-sketches_FO1532 | 143 | 0.979 | (a, b, c) \leq\left(a^{\prime}, b^{\prime}, c^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO1533 | 143 | 1.000 | a \leq a^{\prime} | | |
| fong-spivak-seven-sketches_FO1534 | 143 | 1.000 | b \leq b^{\prime} | | |
| fong-spivak-seven-sketches_FO1535 | 143 | 1.000 | c \leq c^{\prime} | | |
| fong-spivak-seven-sketches_FO1536 | 143 | 1.000 | a+b+c \leq a^{\prime}+b^{\prime}+c^{\prime} | | |
| fong-spivak-seven-sketches_FO1537 | 143 | 0.955 | \widehat{+}:(\mathbb{R} \times \mathbb{R} \times \mathbb{R}) \rightarrow \mathbb{R} | | |
| fong-spivak-seven-sketches_FO1538 | 143 | 0.955 | (a, b, c, d) | | |
| fong-spivak-seven-sketches_FO1539 | 143 | 1.000 | a+b+c \leq d | | |
| fong-spivak-seven-sketches_FO1540 | 143 | 1.000 | \check{+} | | |
| fong-spivak-seven-sketches_FO1541 | 143 | 0.994 | G: Q \rightarrow \mathcal{P} | | |
| fong-spivak-seven-sketches_FO1542 | 143 | 0.990 | \widehat{F}=\widetilde{G} | | |
| fong-spivak-seven-sketches_FO1543 | 143 | 1.000 | \widehat{\mathrm{id}}=\widetilde{\mathrm{id}} | | |
| fong-spivak-seven-sketches_FO1544 | 143 | 0.998 | \boldsymbol{\operatorname { C o l }}(\Phi) | | |
| fong-spivak-seven-sketches_FO1545 | 143 | 0.976 | \operatorname{Ob}(\operatorname{Col}(\Phi)):=\operatorname{Ob}(\mathcal{X}) \sqcup \operatorname{Ob}(\mathcal{Y}) | | |
| fong-spivak-seven-sketches_FO1546 | 143 | 0.567 | a, b \in \operatorname{Ob}(\operatorname{Col}(\Phi)) | | |
| fong-spivak-seven-sketches_FO1547 | 143 | 0.567 | \operatorname{Col}(\Phi)(a, b) \in \mathcal{V} | | |
| fong-spivak-seven-sketches_FO1548 | 143 | 0.994 | i_{x}: \mathcal{X} \rightarrow \boldsymbol{\operatorname { C o l }}(\Phi) | | |
| fong-spivak-seven-sketches_FO1549 | 143 | 0.994 | i_{y}: y \rightarrow \boldsymbol{\operatorname { C o l }}(\Phi) | | |
| fong-spivak-seven-sketches_FO1550 | 144 | 0.600 | \Phi: x \rightarrow y | | |
| fong-spivak-seven-sketches_FO1551 | 145 | 1.000 | 5+3=8 | | |
| fong-spivak-seven-sketches_FO1552 | 145 | 0.709 | \overline{5}=\{ | | |
| fong-spivak-seven-sketches_FO1553 | 145 | 0.456 | \overline{3}=\{ | | |
| fong-spivak-seven-sketches_FO1554 | 145 | 0.456 | \overline{8}=\{ | | |
| fong-spivak-seven-sketches_FO1555 | 145 | 0.895 | \sqcup | | |
| fong-spivak-seven-sketches_FO1556 | 145 | 0.998 | \overline{5} \sqcup \overline{3} \cong \overline{8} | | |
| fong-spivak-seven-sketches_FO1557 | 145 | 1.000 | \overline{5} \sqcup \overline{3} | | |
| fong-spivak-seven-sketches_FO1558 | 145 | 1.000 | \overline{8} | | |
| fong-spivak-seven-sketches_FO1559 | 146 | 1.000 | \mathcal{P}(a, b)= | | |
| fong-spivak-seven-sketches_FO1560 | 146 | 1.000 | x_{0} \leq x_{1} | | |
| fong-spivak-seven-sketches_FO1561 | 146 | 1.000 | x_{0} \leq x_{3} | | |
| fong-spivak-seven-sketches_FO1562 | 146 | 1.000 | x_{0} \leq x_{1}, x_{1} \leq x_{2} | | |
| fong-spivak-seven-sketches_FO1563 | 146 | 1.000 | x_{2} \leq x_{3} | | |
| fong-spivak-seven-sketches_FO1564 | 146 | 1.000 | g: B \rightarrow C | | |
| fong-spivak-seven-sketches_FO1565 | 146 | 1.000 | h: C \rightarrow D | | |
| fong-spivak-seven-sketches_FO1566 | 146 | 0.918 | f ;(g ; h) | | |
| fong-spivak-seven-sketches_FO1567 | 146 | 0.918 | (f ; g) ; h | | |
| fong-spivak-seven-sketches_FO1568 | 146 | 0.743 | f ;(g ; h)=(f ; g) ; h | | |
| fong-spivak-seven-sketches_FO1569 | 147 | 0.647 | \mathrm{id}_{x} | | |
| fong-spivak-seven-sketches_FO1570 | 147 | 0.926 | f=f=f | | |
| fong-spivak-seven-sketches_FO1571 | 147 | 0.926 | \mathrm{id}_{y} | | |
| fong-spivak-seven-sketches_FO1572 | 147 | 1.000 | f: x \rightarrow y | | |
| fong-spivak-seven-sketches_FO1573 | 148 | 0.943 | \{1\}, \times | | |
| fong-spivak-seven-sketches_FO1574 | 148 | 1.000 | S \times T=\{(s, t) \mid s \in S, t \in T\} | | |
| fong-spivak-seven-sketches_FO1575 | 148 | 0.999 | (p \otimes q) \otimes r=p \otimes(q \otimes r) | | |
| fong-spivak-seven-sketches_FO1576 | 148 | 1.000 | T \times U | | |
| fong-spivak-seven-sketches_FO1577 | 148 | 1.000 | S \times T | | |
| fong-spivak-seven-sketches_FO1578 | 148 | 1.000 | \alpha_{s, t, u} | | |
| fong-spivak-seven-sketches_FO1579 | 148 | 0.731 | I \in \operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO1580 | 149 | 0.990 | \otimes: \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C} | | |
| fong-spivak-seven-sketches_FO1581 | 149 | 0.967 | \lambda_{c}: I \otimes c \cong c | | |
| fong-spivak-seven-sketches_FO1582 | 149 | 0.835 | \rho_{c}: c \otimes I \cong c | | |
| fong-spivak-seven-sketches_FO1583 | 149 | 0.981 | \alpha_{c, d, e}:(c \otimes d) \otimes e \cong c \otimes(d \otimes e) | | |
| fong-spivak-seven-sketches_FO1584 | 149 | 0.964 | \sigma_{\mathcal{C}, d}: c \otimes d \cong d \otimes c | | |
| fong-spivak-seven-sketches_FO1585 | 149 | 0.964 | c, d \in \mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO1586 | 149 | 0.964 | \sigma \circ \sigma=\mathrm{id} | | |
| fong-spivak-seven-sketches_FO1587 | 149 | 0.999 | (\mathcal{P}, I, \otimes) | | |
| fong-spivak-seven-sketches_FO1588 | 149 | 0.999 | p, q \in \mathcal{P} | | |
| fong-spivak-seven-sketches_FO1589 | 149 | 1.000 | \mathcal{P}(p, q) | | |
| fong-spivak-seven-sketches_FO1590 | 149 | 1.000 | I:=\{1\} | | |
| fong-spivak-seven-sketches_FO1591 | 149 | 0.513 | \otimes:=\times | | |
| fong-spivak-seven-sketches_FO1592 | 149 | 0.513 | \times | | |
| fong-spivak-seven-sketches_FO1593 | 149 | 0.537 | (S, T) \in \operatorname{Ob}( | | |
| fong-spivak-seven-sketches_FO1594 | 149 | 0.537 | (S \times T) \in | | |
| fong-spivak-seven-sketches_FO1595 | 149 | 0.988 | \{(s, t) \mid s \in S, t \in T\} | | |
| fong-spivak-seven-sketches_FO1596 | 149 | 1.000 | f: S \rightarrow S^{\prime} | | |
| fong-spivak-seven-sketches_FO1597 | 149 | 1.000 | g: T \rightarrow T^{\prime} | | |
| fong-spivak-seven-sketches_FO1598 | 149 | 1.000 | (f \times g):(S \times T) \rightarrow\left(S^{\prime} \times T^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO1599 | 149 | 1.000 | (f \times g)(s, t):=(f(s), g(t)) | | |
| fong-spivak-seven-sketches_FO1600 | 149 | 0.441 | \mathrm{identities}^{:} \mathrm{id}_{S} \times \mathrm{id}_{T}=\mathrm{id}_{S \times T} | | |
| fong-spivak-seven-sketches_FO1601 | 149 | 1.000 | S \xrightarrow{f} S^{\prime} \xrightarrow{f^{\prime}} S^{\prime \prime} | | |
| fong-spivak-seven-sketches_FO1602 | 149 | 1.000 | T \xrightarrow{g} | | |
| fong-spivak-seven-sketches_FO1603 | 149 | 1.000 | T^{\prime} \xrightarrow{g^{\prime}} T^{\prime \prime} | | |
| fong-spivak-seven-sketches_FO1604 | 150 | 1.000 | \{1\} \times S \cong S | | |
| fong-spivak-seven-sketches_FO1605 | 150 | 0.742 | \{(1, s) \mid s \in S\} \rightarrow S | | |
| fong-spivak-seven-sketches_FO1606 | 150 | 0.742 | 1, s | | |
| fong-spivak-seven-sketches_FO1607 | 150 | 0.984 | 1, \times | | |
| fong-spivak-seven-sketches_FO1608 | 150 | 0.755 | A=B=C=D=F=G=\mathbb{Z} | | |
| fong-spivak-seven-sketches_FO1609 | 150 | 0.755 | E=\mathbb{B}= | | |
| fong-spivak-seven-sketches_FO1610 | 150 | 0.976 | f_{C}(a)=|a|, f_{D}(a)=a * 5, g_{E}(d, b)= | | |
| fong-spivak-seven-sketches_FO1611 | 150 | 0.976 | d \leq b | | |
| fong-spivak-seven-sketches_FO1612 | 150 | 0.976 | g_{F}(d, b)=d-b | | |
| fong-spivak-seven-sketches_FO1613 | 150 | 1.000 | h(c, e)= | | |
| fong-spivak-seven-sketches_FO1614 | 150 | 1.000 | 1-c | | |
| fong-spivak-seven-sketches_FO1615 | 150 | 1.000 | g_{E}(5,3) | | |
| fong-spivak-seven-sketches_FO1616 | 150 | 1.000 | g_{F}(5,3) | | |
| fong-spivak-seven-sketches_FO1617 | 150 | 1.000 | g_{E}(3,5) | | |
| fong-spivak-seven-sketches_FO1618 | 150 | 1.000 | g_{F}(3,5) | | |
| fong-spivak-seven-sketches_FO1619 | 150 | 0.992 | h(5 | | |
| fong-spivak-seven-sketches_FO1620 | 150 | 0.758 | h(-5 | | |
| fong-spivak-seven-sketches_FO1621 | 150 | 1.000 | A \times B \rightarrow G \times F | | |
| fong-spivak-seven-sketches_FO1622 | 150 | 1.000 | q_{G}(-2,3) | | |
| fong-spivak-seven-sketches_FO1623 | 150 | 1.000 | q_{F}(-2,3) | | |
| fong-spivak-seven-sketches_FO1624 | 150 | 1.000 | q_{G}(2,3) | | |
| fong-spivak-seven-sketches_FO1625 | 150 | 1.000 | q_{F}(2,3) | | |
| fong-spivak-seven-sketches_FO1626 | 150 | 0.542 | X(x, y) \in \mathcal{V} | | |
| fong-spivak-seven-sketches_FO1627 | 150 | 0.356 | x \in \mathrm{Ob}(X) | | |
| fong-spivak-seven-sketches_FO1628 | 150 | 0.356 | \mathrm{id}_{x}: I \rightarrow X(x, x) | | |
| fong-spivak-seven-sketches_FO1629 | 151 | 0.608 | x, y, z \in \mathrm{Ob}(X) | | |
| fong-spivak-seven-sketches_FO1630 | 151 | 0.608 | 9: \mathcal{X}(x, y) \otimes \mathcal{X}(y, z) \rightarrow \mathfrak{X}(x, z) | | |
| fong-spivak-seven-sketches_FO1631 | 151 | 1.000 | \mathcal{V}=( | | |
| fong-spivak-seven-sketches_FO1632 | 151 | 1.000 | ,\{1\}, \times) | | |
| fong-spivak-seven-sketches_FO1633 | 151 | 0.961 | \operatorname{id}_{x}: I \rightarrow \mathcal{X}(x, x) | | |
| fong-spivak-seven-sketches_FO1634 | 151 | 0.961 | I \leq \mathcal{X}(x, x) | | |
| fong-spivak-seven-sketches_FO1635 | 153 | 0.644 | (\mathcal{C}, I, \otimes) | | |
| fong-spivak-seven-sketches_FO1636 | 153 | 0.999 | c^{*} \in \operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO1637 | 153 | 1.000 | \eta_{c}: I \rightarrow c^{*} \otimes c | | |
| fong-spivak-seven-sketches_FO1638 | 153 | 1.000 | \epsilon_{c}: c \otimes c^{*} \rightarrow I | | |
| fong-spivak-seven-sketches_FO1639 | 153 | 0.516 | c^{*} | | |
| fong-spivak-seven-sketches_FO1640 | 153 | 0.997 | \xrightarrow{c} | | |
| fong-spivak-seven-sketches_FO1641 | 153 | 0.997 | \stackrel{c^{*}}{\leftarrow} | | |
| fong-spivak-seven-sketches_FO1642 | 154 | 0.877 | \epsilon_{\text {sound }} | | |
| fong-spivak-seven-sketches_FO1643 | 154 | 0.877 | \eta_{\text {sound* }} | | |
| fong-spivak-seven-sketches_FO1644 | 154 | 0.879 | \mathcal{C}(b \otimes c, d) \cong \mathcal{C}(b, c \multimap d) | | |
| fong-spivak-seven-sketches_FO1645 | 154 | 1.000 | c^{\prime} | | |
| fong-spivak-seven-sketches_FO1646 | 154 | 1.000 | c^{*} \cong c^{\prime} | | |
| fong-spivak-seven-sketches_FO1647 | 154 | 1.000 | c \cong c^{* *} | | |
| fong-spivak-seven-sketches_FO1648 | 154 | 0.947 | c^{*} \otimes d | | |
| fong-spivak-seven-sketches_FO1649 | 154 | 0.999 | \mathcal{C}(b \otimes c, d) \cong \mathcal{C}(b, c \rightarrow d) | | |
| fong-spivak-seven-sketches_FO1650 | 154 | 0.899 | \mathrm{id}_{b} \otimes \eta_{c} | | |
| fong-spivak-seven-sketches_FO1651 | 154 | 1.000 | \alpha: A \rightarrow B | | |
| fong-spivak-seven-sketches_FO1652 | 154 | 1.000 | \beta: B \rightarrow C | | |
| fong-spivak-seven-sketches_FO1653 | 155 | 0.999 | \beta \circ \alpha | | |
| fong-spivak-seven-sketches_FO1654 | 155 | 1.000 | \beta | | |
| fong-spivak-seven-sketches_FO1655 | 155 | 0.873 | (\varnothing, \sqcup) | | |
| fong-spivak-seven-sketches_FO1656 | 155 | 1.000 | A \sqcup A:=\{(a, 1),(a, 2) \mid a \in A\} | | |
| fong-spivak-seven-sketches_FO1657 | 155 | 1.000 | (a, 1) | | |
| fong-spivak-seven-sketches_FO1658 | 155 | 1.000 | (a, 2) | | |
| fong-spivak-seven-sketches_FO1659 | 155 | 1.000 | \eta_{A}: \varnothing \rightarrow A \sqcup A | | |
| fong-spivak-seven-sketches_FO1660 | 155 | 1.000 | \epsilon_{A}: A \sqcup A \rightarrow \varnothing | | |
| fong-spivak-seven-sketches_FO1661 | 155 | 0.997 | \varnothing \rightarrow \underline{3} \sqcup \underline{3} | | |
| fong-spivak-seven-sketches_FO1662 | 155 | 1.000 | \underline{3} \sqcup \underline{3} \rightarrow \varnothing | | |
| fong-spivak-seven-sketches_FO1663 | 155 | 0.993 | (\mathcal{V}, I, \otimes) | | |
| fong-spivak-seven-sketches_FO1664 | 155 | 0.562 | \operatorname{Prof}_{\mathcal{V}} | | |
| fong-spivak-seven-sketches_FO1665 | 155 | 0.863 | \alpha ; \beta | | |
| fong-spivak-seven-sketches_FO1666 | 155 | 0.863 | A \sqcup C | | |
| fong-spivak-seven-sketches_FO1667 | 155 | 0.863 | A \sqcup B \sqcup C | | |
| fong-spivak-seven-sketches_FO1668 | 155 | 1.000 | \alpha \subseteq(A \sqcup B) \times(A \sqcup B) | | |
| fong-spivak-seven-sketches_FO1669 | 155 | 1.000 | \beta \subseteq(B \sqcup C) \times(B \sqcup C) | | |
| fong-spivak-seven-sketches_FO1670 | 155 | 1.000 | \iota_{A \sqcup B}: A \sqcup B \rightarrow A \sqcup B \sqcup C, \iota_{B \sqcup C}: B \sqcup C \rightarrow A \sqcup B \sqcup C | | |
| fong-spivak-seven-sketches_FO1671 | 155 | 1.000 | \iota_{A \sqcup C}: A \sqcup C \rightarrow | | |
| fong-spivak-seven-sketches_FO1672 | 156 | 1.000 | { }_{\mathcal{V}} | | |
| fong-spivak-seven-sketches_FO1673 | 156 | 0.383 | \Phi: X_{1} \mapsto X_{2} | | |
| fong-spivak-seven-sketches_FO1674 | 156 | 0.383 | \Psi: y_{1} \rightarrow y_{2} | | |
| fong-spivak-seven-sketches_FO1675 | 156 | 0.262 | (\Phi \times \Psi): \mathfrak{X}_{1} \times y_{1} \rightarrow x_{2} \times y_{2} | | |
| fong-spivak-seven-sketches_FO1676 | 156 | 0.857 | \mathbf{P r o f}_{\text {Bool }} | | |
| fong-spivak-seven-sketches_FO1677 | 156 | 1.000 | \Phi \times \Psi | | |
| fong-spivak-seven-sketches_FO1678 | 156 | 0.593 | (1,1)=I | | |
| fong-spivak-seven-sketches_FO1679 | 156 | 0.467 | X \times \mathbf{1} \rightarrow X | | |
| fong-spivak-seven-sketches_FO1680 | 156 | 0.467 | \mathbf{1} \times X \rightarrow X | | |
| fong-spivak-seven-sketches_FO1681 | 157 | 0.894 | \mathcal{X}^{\text {op }} | | |
| fong-spivak-seven-sketches_FO1682 | 157 | 0.225 | \eta_{x}: \mathbf{1} \rightarrow \mathcal{X}^{\mathrm{op}} \times \mathcal{X} | | |
| fong-spivak-seven-sketches_FO1683 | 157 | 0.981 | \eta_{x}\left(1, x, x^{\prime}\right):=\mathcal{X}\left(x, x^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO1684 | 157 | 0.981 | \epsilon_{x}:(\mathcal{X} \times | | |
| fong-spivak-seven-sketches_FO1685 | 157 | 0.710 | \left.\mathcal{X}^{\mathrm{op}}\right) \rightarrow \mathbf{1} | | |
| fong-spivak-seven-sketches_FO1686 | 157 | 0.710 | \epsilon_{\mathcal{X}}\left(x, x^{\prime}, 1\right):=\mathcal{X}\left(x, x^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO1687 | 160 | 1.000 | u=15 x | | |
| fong-spivak-seven-sketches_FO1688 | 160 | 1.000 | v=3 x+21 y | | |
| fong-spivak-seven-sketches_FO1689 | 160 | 1.000 | x+7 y | | |
| fong-spivak-seven-sketches_FO1690 | 161 | 0.608 | (\mathcal{C}, 0,+) | | |
| fong-spivak-seven-sketches_FO1691 | 161 | 0.950 | \operatorname{Ob}(\mathcal{C})=\mathbb{N} | | |
| fong-spivak-seven-sketches_FO1692 | 161 | 0.950 | 0 \in \mathbb{N} | | |
| fong-spivak-seven-sketches_FO1693 | 161 | 0.994 | \mathcal{C}(m, n) | | |
| fong-spivak-seven-sketches_FO1694 | 161 | 0.655 | \operatorname{id}_{n}: n \rightarrow n | | |
| fong-spivak-seven-sketches_FO1695 | 161 | 0.999 | \sigma_{m, n}: m+n \rightarrow n+m | | |
| fong-spivak-seven-sketches_FO1696 | 161 | 1.000 | f: m \rightarrow n | | |
| fong-spivak-seven-sketches_FO1697 | 161 | 1.000 | g: n \rightarrow p | | |
| fong-spivak-seven-sketches_FO1698 | 161 | 1.000 | (f ; g): m \rightarrow p | | |
| fong-spivak-seven-sketches_FO1699 | 161 | 1.000 | f: m \rightarrow m^{\prime} | | |
| fong-spivak-seven-sketches_FO1700 | 161 | 1.000 | g: n \rightarrow n^{\prime} | | |
| fong-spivak-seven-sketches_FO1701 | 161 | 0.963 | (f+g): m+n \rightarrow m^{\prime}+n^{\prime} | | |
| fong-spivak-seven-sketches_FO1702 | 162 | 1.000 | \underline{m}=\{1, \ldots m\} | | |
| fong-spivak-seven-sketches_FO1703 | 162 | 1.000 | \underline{n}=\{1, \ldots, n\} | | |
| fong-spivak-seven-sketches_FO1704 | 162 | 1.000 | f+g: m+n \longrightarrow m^{\prime}+n^{\prime} | | |
| fong-spivak-seven-sketches_FO1705 | 162 | 1.000 | f: 3 \rightarrow 2 | | |
| fong-spivak-seven-sketches_FO1706 | 162 | 1.000 | g: 2 \rightarrow 4 | | |
| fong-spivak-seven-sketches_FO1707 | 162 | 1.000 | f+g | | |
| fong-spivak-seven-sketches_FO1708 | 162 | 0.998 | \sigma_{m, n} | | |
| fong-spivak-seven-sketches_FO1709 | 162 | 1.000 | \underline{m} \rightarrow \underline{n} | | |
| fong-spivak-seven-sketches_FO1710 | 162 | 0.987 | m \neq n | | |
| fong-spivak-seven-sketches_FO1711 | 162 | 0.987 | \mathbf{B i j}(m, n) | | |
| fong-spivak-seven-sketches_FO1712 | 162 | 0.987 | \mathbf{B i j} | | |
| fong-spivak-seven-sketches_FO1713 | 162 | 1.000 | \underline{m} \sqcup \underline{n} | | |
| fong-spivak-seven-sketches_FO1714 | 162 | 0.952 | R \subseteq \underline{m} \times \underline{n} | | |
| fong-spivak-seven-sketches_FO1715 | 162 | 1.000 | S \subseteq \underline{n} \times \underline{p} | | |
| fong-spivak-seven-sketches_FO1716 | 163 | 0.934 | F(n)=n | | |
| fong-spivak-seven-sketches_FO1717 | 163 | 0.934 | n \in \operatorname{Ob}(\mathcal{C})=\operatorname{Ob}(\mathcal{D})=\mathbb{N} | | |
| fong-spivak-seven-sketches_FO1718 | 163 | 0.977 | f: m_{1} \rightarrow m_{2} | | |
| fong-spivak-seven-sketches_FO1719 | 163 | 0.977 | g: n_{1} \rightarrow n_{2} | | |
| fong-spivak-seven-sketches_FO1720 | 163 | 0.977 | F(f)+F(g)=F(f+g) | | |
| fong-spivak-seven-sketches_FO1721 | 163 | 0.966 | i | | |
| fong-spivak-seven-sketches_FO1722 | 163 | 0.503 | F: | | |
| fong-spivak-seven-sketches_FO1723 | 163 | 0.503 | \mathbf{R e l}_{\text {Fin }} | | |
| fong-spivak-seven-sketches_FO1724 | 163 | 0.503 | f: \underline{m} \rightarrow \underline{n} | | |
| fong-spivak-seven-sketches_FO1725 | 163 | 1.000 | F(f):=\{(i, j) \mid f(i)=j\} \subseteq \underline{m} \times \underline{n} | | |
| fong-spivak-seven-sketches_FO1726 | 163 | 0.616 | \iota | | |
| fong-spivak-seven-sketches_FO1727 | 163 | 0.835 | V \rightarrow \mathbb{N} | | |
| fong-spivak-seven-sketches_FO1728 | 163 | 0.835 | \operatorname{in}(v) | | |
| fong-spivak-seven-sketches_FO1729 | 163 | 0.835 | \operatorname{out}(v) | | |
| fong-spivak-seven-sketches_FO1730 | 163 | 0.793 | \iota: \underline{m} \sqcup O \xrightarrow{\cong} I \sqcup \underline{n} | | |
| fong-spivak-seven-sketches_FO1731 | 163 | 0.793 | I=\{(v, i) \mid v \in V, 1 \leq i \leq \operatorname{in}(v)\} | | |
| fong-spivak-seven-sketches_FO1732 | 163 | 1.000 | O=\{(v, i) \mid v \in V, 1 \leq i \leq \operatorname{out}(v)\} | | |
| fong-spivak-seven-sketches_FO1733 | 163 | 0.933 | e_{v, j}^{u, i}: u \rightarrow v | | |
| fong-spivak-seven-sketches_FO1734 | 163 | 0.933 | i, j \in \mathbb{N} | | |
| fong-spivak-seven-sketches_FO1735 | 163 | 1.000 | \iota(u, i)=(v, j) | | |
| fong-spivak-seven-sketches_FO1736 | 163 | 1.000 | R_{1}+R_{2} | | |
| fong-spivak-seven-sketches_FO1737 | 163 | 1.000 | R_{1} \subseteq \underline{m_{1}} \times \underline{n_{1}} | | |
| fong-spivak-seven-sketches_FO1738 | 163 | 1.000 | R_{2} \subseteq \underline{m_{2}} \times \underline{n_{2}} | | |
| fong-spivak-seven-sketches_FO1739 | 163 | 1.000 | R_{1} \sqcup R_{2} \subseteq | | |
| fong-spivak-seven-sketches_FO1740 | 163 | 0.420 | \left(\underline{m_{1}} \times \underline{n_{1}}\right) \sqcup\left(\underline{m_{2}} \times \underline{n_{2}}\right) \subseteq\left(\underline{m_{1}} \sqcup \underline{m_{2}}\right) \times\left(\underline{n_{1} \sqcup} \underline{n_{2}}\right) | | |
| fong-spivak-seven-sketches_FO1741 | 164 | 0.999 | m=2 | | |
| fong-spivak-seven-sketches_FO1742 | 164 | 0.999 | n=3 | | |
| fong-spivak-seven-sketches_FO1743 | 164 | 1.000 | V=\{a, b, c\} | | |
| fong-spivak-seven-sketches_FO1744 | 164 | 0.994 | \operatorname{in}(a)=1 | | |
| fong-spivak-seven-sketches_FO1745 | 164 | 0.994 | \operatorname{out}(a)=3 | | |
| fong-spivak-seven-sketches_FO1746 | 164 | 0.996 | (m, n) | | |
| fong-spivak-seven-sketches_FO1747 | 164 | 0.179 | (V | | |
| fong-spivak-seven-sketches_FO1748 | 164 | 0.179 | \iota) | | |
| fong-spivak-seven-sketches_FO1749 | 164 | 0.179 | (n, p) | | |
| fong-spivak-seven-sketches_FO1750 | 164 | 0.179 | V^{\prime} | | |
| fong-spivak-seven-sketches_FO1751 | 164 | 0.179 | \iota^{\prime} | | |
| fong-spivak-seven-sketches_FO1752 | 164 | 0.147 | (m, p) | | |
| fong-spivak-seven-sketches_FO1753 | 164 | 0.147 | \left(V \sqcup V^{\prime},[\right. | | |
| fong-spivak-seven-sketches_FO1754 | 164 | 0.147 | ],[ | | |
| fong-spivak-seven-sketches_FO1755 | 164 | 0.147 | \left.], \iota^{\prime \prime}\right) | | |
| fong-spivak-seven-sketches_FO1756 | 164 | 0.777 | V \sqcup V^{\prime} \rightarrow \mathbb{N} | | |
| fong-spivak-seven-sketches_FO1757 | 164 | 0.191 | i n^{\prime} | | |
| fong-spivak-seven-sketches_FO1758 | 164 | 0.191 | \iota^{\prime \prime}: \underline{m} \sqcup O \sqcup O^{\prime} \rightarrow I \sqcup I^{\prime} \sqcup \underline{p} | | |
| fong-spivak-seven-sketches_FO1759 | 165 | 0.911 | \operatorname{Ob}(\mathbf{P G})=\mathbb{N} | | |
| fong-spivak-seven-sketches_FO1760 | 165 | 0.965 | \mathbf{P G}(m, n)=\{(V, i n, o u t, \iota) \mid | | |
| fong-spivak-seven-sketches_FO1761 | 165 | 0.996 | (n, n) | | |
| fong-spivak-seven-sketches_FO1762 | 165 | 0.928 | \left(\varnothing,!,!, \mathrm{id}_{\underline{n}}\right) | | |
| fong-spivak-seven-sketches_FO1763 | 165 | 0.928 | \varnothing \rightarrow \mathbb{N} | | |
| fong-spivak-seven-sketches_FO1764 | 165 | 0.423 | G:=(V | | |
| fong-spivak-seven-sketches_FO1765 | 165 | 0.423 | G^{\prime}:=\left(V^{\prime}, i n^{\prime}, o u t^{\prime}, \iota^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO1766 | 165 | 0.863 | (\varnothing,!,!,!) | | |
| fong-spivak-seven-sketches_FO1767 | 165 | 1.000 | R \subseteq P \times P | | |
| fong-spivak-seven-sketches_FO1768 | 165 | 0.798 | (V, A, s, t) | | |
| fong-spivak-seven-sketches_FO1769 | 165 | 0.798 | \{(s(a), t(a)) \mid a \in A\} \subseteq V \times V | | |
| fong-spivak-seven-sketches_FO1770 | 166 | 0.821 | P, \leq_{P} | | |
| fong-spivak-seven-sketches_FO1771 | 166 | 1.000 | R(x, y) | | |
| fong-spivak-seven-sketches_FO1772 | 166 | 1.000 | f(x) \leq_{Q} f(y) | | |
| fong-spivak-seven-sketches_FO1773 | 166 | 0.989 | P, Q, R | | |
| fong-spivak-seven-sketches_FO1774 | 166 | 1.000 | a, b \in Q | | |
| fong-spivak-seven-sketches_FO1775 | 166 | 1.000 | (g(a), g(b)) \in R | | |
| fong-spivak-seven-sketches_FO1776 | 166 | 1.000 | g:\left(Q, \leq_{Q}\right) \rightarrow\left(P, \leq_{P}\right) | | |
| fong-spivak-seven-sketches_FO1777 | 167 | 0.998 | \operatorname{Mor}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO1778 | 167 | 0.996 | \operatorname{Mor}(\mathcal{C}) \rightarrow | | |
| fong-spivak-seven-sketches_FO1779 | 167 | 0.615 | \mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO1780 | 167 | 1.000 | \mathcal{G} \rightarrow \mathcal{C} | | |
| fong-spivak-seven-sketches_FO1781 | 167 | 0.999 | f: V \rightarrow \operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO1782 | 167 | 0.999 | g: A \rightarrow \operatorname{Mor}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO1783 | 167 | 1.000 | \operatorname{dom}(g(a))=f(s(a)) | | |
| fong-spivak-seven-sketches_FO1784 | 167 | 1.000 | \operatorname{cod}(g(a))=f(t(a)) | | |
| fong-spivak-seven-sketches_FO1785 | 167 | 1.000 | \boldsymbol{\operatorname { L o o p }}(A) | | |
| fong-spivak-seven-sketches_FO1786 | 167 | 1.000 | A=\{a\} | | |
| fong-spivak-seven-sketches_FO1787 | 167 | 1.000 | \{a\} | | |
| fong-spivak-seven-sketches_FO1788 | 167 | 0.999 | G, s, t | | |
| fong-spivak-seven-sketches_FO1789 | 167 | 0.999 | s, t: G \rightarrow \mathbb{N} | | |
| fong-spivak-seven-sketches_FO1790 | 167 | 1.000 | g \in G | | |
| fong-spivak-seven-sketches_FO1791 | 167 | 1.000 | s(g), t(g) \in \mathbb{N} | | |
| fong-spivak-seven-sketches_FO1792 | 167 | 0.860 | s, t | | |
| fong-spivak-seven-sketches_FO1793 | 167 | 0.577 | \Gamma=(V | | |
| fong-spivak-seven-sketches_FO1794 | 167 | 0.577 | \ell: V \rightarrow G | | |
| fong-spivak-seven-sketches_FO1795 | 167 | 0.999 | s(\ell(v))=\operatorname{in}(v) | | |
| fong-spivak-seven-sketches_FO1796 | 167 | 0.999 | t(\ell(v))=\operatorname{out}(v) | | |
| fong-spivak-seven-sketches_FO1797 | 167 | 1.000 | (m, n) \in \mathbb{N} \times \mathbb{N} | | |
| fong-spivak-seven-sketches_FO1798 | 168 | 0.852 | \ell^{\prime} | | |
| fong-spivak-seven-sketches_FO1799 | 168 | 1.000 | \ell | | |
| fong-spivak-seven-sketches_FO1800 | 168 | 1.000 | G=\{f: 1 \rightarrow 1, g: 2 \rightarrow 2, h: 2 \rightarrow 1\} | | |
| fong-spivak-seven-sketches_FO1801 | 168 | 0.987 | 3 \rightarrow 2 | | |
| fong-spivak-seven-sketches_FO1802 | 168 | 1.000 | V \rightarrow \varnothing | | |
| fong-spivak-seven-sketches_FO1803 | 168 | 1.000 | V=\varnothing | | |
| fong-spivak-seven-sketches_FO1804 | 168 | 1.000 | (n, m) | | |
| fong-spivak-seven-sketches_FO1805 | 168 | 0.997 | (\varnothing,!,!, \sigma) | | |
| fong-spivak-seven-sketches_FO1806 | 168 | 0.997 | \sigma: n \rightarrow m | | |
| fong-spivak-seven-sketches_FO1807 | 168 | 0.985 | (m, n) \in \mathbb{N}^{2} | | |
| fong-spivak-seven-sketches_FO1808 | 168 | 0.886 | \operatorname{prop} \mathcal{C} | | |
| fong-spivak-seven-sketches_FO1809 | 168 | 0.886 | \operatorname{Free}(G) \rightarrow \mathcal{C} | | |
| fong-spivak-seven-sketches_FO1810 | 168 | 1.000 | G \rightarrow \mathcal{C} | | |
| fong-spivak-seven-sketches_FO1811 | 168 | 1.000 | s(g) \rightarrow t(g) | | |
| fong-spivak-seven-sketches_FO1812 | 169 | 0.995 | e: m \rightarrow n | | |
| fong-spivak-seven-sketches_FO1813 | 169 | 0.976 | \mathrm{id}_{0}: 0 \rightarrow 0 | | |
| fong-spivak-seven-sketches_FO1814 | 169 | 0.976 | \mathrm{id}_{1}: 1 \rightarrow 1 | | |
| fong-spivak-seven-sketches_FO1815 | 169 | 1.000 | \sigma: 2 \rightarrow 2 | | |
| fong-spivak-seven-sketches_FO1816 | 169 | 1.000 | g: s(g) \rightarrow t(g) | | |
| fong-spivak-seven-sketches_FO1817 | 169 | 1.000 | \alpha: m \rightarrow n | | |
| fong-spivak-seven-sketches_FO1818 | 169 | 1.000 | \beta: p \rightarrow q | | |
| fong-spivak-seven-sketches_FO1819 | 169 | 1.000 | \alpha+\beta: m+p \rightarrow n+q | | |
| fong-spivak-seven-sketches_FO1820 | 169 | 1.000 | \beta: n \rightarrow p | | |
| fong-spivak-seven-sketches_FO1821 | 169 | 1.000 | \alpha ; \beta: m \rightarrow p | | |
| fong-spivak-seven-sketches_FO1822 | 169 | 1.000 | \operatorname{Expr}(G) | | |
| fong-spivak-seven-sketches_FO1823 | 169 | 1.000 | f: 1 \rightarrow 1 | | |
| fong-spivak-seven-sketches_FO1824 | 169 | 0.963 | f: \mathrm{id}_{1}: 1 \rightarrow 1 | | |
| fong-spivak-seven-sketches_FO1825 | 169 | 0.997 | h+\mathrm{id}_{1}: 3 \rightarrow 2 | | |
| fong-spivak-seven-sketches_FO1826 | 169 | 0.551 | \left(h+\mathrm{id}_{1}\right) ; \sigma ; g ; \sigma: 3 \rightarrow 2 | | |
| fong-spivak-seven-sketches_FO1827 | 169 | 0.981 | f ; \mathrm{id}_{1} | | |
| fong-spivak-seven-sketches_FO1828 | 169 | 0.708 | f ; \mathrm{id}_{1}=f | | |
| fong-spivak-seven-sketches_FO1829 | 169 | 0.923 | \sigma | | |
| fong-spivak-seven-sketches_FO1830 | 169 | 0.923 | \mathrm{X}: 2 \rightarrow 2 | | |
| fong-spivak-seven-sketches_FO1831 | 170 | 0.975 | \left(g+\mathrm{id}_{1}\right) \stackrel{\circ}{9}\left(\mathrm{id}_{1}+\sigma\right) \stackrel{\circ}{9}\left(\mathrm{id}_{1}+g\right) \stackrel{\circ}{9}\left(h+\mathrm{id}_{1}\right) | | |
| fong-spivak-seven-sketches_FO1832 | 170 | 1.000 | G=\{f: 1 \rightarrow 1, g: 2 \rightarrow | | |
| fong-spivak-seven-sketches_FO1833 | 170 | 0.285 | 2, h: 2 \rightarrow 1\} | | |
| fong-spivak-seven-sketches_FO1834 | 170 | 0.285 | \left(f+\mathrm{id}_{1}+\mathrm{id}_{1}\right) \stackrel{\circ}{9}\left(\sigma+\mathrm{id}_{1}\right) \stackrel{\circ}{9}\left(\mathrm{id}_{1}+h\right) \stackrel{\circ}{9} \sigma \stackrel{\circ}{g} | | |
| fong-spivak-seven-sketches_FO1835 | 170 | 0.817 | G, s, t, E | | |
| fong-spivak-seven-sketches_FO1836 | 170 | 1.000 | E \subseteq \operatorname{Expr}(G) \times \operatorname{Expr}(G) | | |
| fong-spivak-seven-sketches_FO1837 | 170 | 1.000 | e_{1} | | |
| fong-spivak-seven-sketches_FO1838 | 170 | 1.000 | e_{2} | | |
| fong-spivak-seven-sketches_FO1839 | 170 | 1.000 | \left(e_{1}, e_{2}\right) \in E | | |
| fong-spivak-seven-sketches_FO1840 | 170 | 0.936 | (G, s, t, E) | | |
| fong-spivak-seven-sketches_FO1841 | 170 | 1.000 | e_{1}=e_{2} | | |
| fong-spivak-seven-sketches_FO1842 | 171 | 1.000 | g \in G, f(g) | | |
| fong-spivak-seven-sketches_FO1843 | 171 | 1.000 | f\left(e_{1}\right)=f\left(e_{2}\right) | | |
| fong-spivak-seven-sketches_FO1844 | 171 | 1.000 | f(e) | | |
| fong-spivak-seven-sketches_FO1845 | 171 | 0.686 | (G, s, t) | | |
| fong-spivak-seven-sketches_FO1846 | 171 | 0.793 | G, s, t, \varnothing | | |
| fong-spivak-seven-sketches_FO1847 | 171 | 0.997 | R, 0,+, 1, * | | |
| fong-spivak-seven-sketches_FO1848 | 171 | 0.997 | 0,1 \in R | | |
| fong-spivak-seven-sketches_FO1849 | 171 | 0.775 | R \times R \rightarrow R | | |
| fong-spivak-seven-sketches_FO1850 | 171 | 0.924 | R,+, 0 | | |
| fong-spivak-seven-sketches_FO1851 | 171 | 0.911 | R, *, 1 | | |
| fong-spivak-seven-sketches_FO1852 | 171 | 1.000 | a *(b+c)=a * b+a * c | | |
| fong-spivak-seven-sketches_FO1853 | 171 | 1.000 | (a+b) * c=a * c+b * c | | |
| fong-spivak-seven-sketches_FO1854 | 171 | 1.000 | a, b, c \in R | | |
| fong-spivak-seven-sketches_FO1855 | 171 | 1.000 | a * 0=0=0 * a | | |
| fong-spivak-seven-sketches_FO1856 | 171 | 1.000 | a \in R | | |
| fong-spivak-seven-sketches_FO1857 | 171 | 0.800 | (\mathbb{N}, 0,+, 1, *) | | |
| fong-spivak-seven-sketches_FO1858 | 171 | 0.952 | (R, *, 1) | | |
| fong-spivak-seven-sketches_FO1859 | 172 | 0.737 | V, 0, \vee, I, \otimes | | |
| fong-spivak-seven-sketches_FO1860 | 172 | 0.737 | 0= | | |
| fong-spivak-seven-sketches_FO1861 | 172 | 0.971 | \bigvee \varnothing | | |
| fong-spivak-seven-sketches_FO1862 | 172 | 1.000 | \operatorname{Mat}_{n}(R) | | |
| fong-spivak-seven-sketches_FO1863 | 172 | 1.000 | (n \times n) | | |
| fong-spivak-seven-sketches_FO1864 | 172 | 1.000 | M \in \operatorname{Mat}_{n}(R) | | |
| fong-spivak-seven-sketches_FO1865 | 172 | 1.000 | M: \underline{n} \times \underline{n} \rightarrow | | |
| fong-spivak-seven-sketches_FO1866 | 172 | 1.000 | M+N | | |
| fong-spivak-seven-sketches_FO1867 | 172 | 1.000 | (M+N)(i, j):=M(i, j)+N(i, j) | | |
| fong-spivak-seven-sketches_FO1868 | 172 | 0.977 | (M * N)(i, j):=\sum_{k \in \underline{n}} M(i, k) * N(k, j) | | |
| fong-spivak-seven-sketches_FO1869 | 172 | 1.000 | 0(i, j):=0 | | |
| fong-spivak-seven-sketches_FO1870 | 172 | 1.000 | i, j \in \underline{n} | | |
| fong-spivak-seven-sketches_FO1871 | 172 | 1.000 | 1 \in \operatorname{Mat}_{n}(R) | | |
| fong-spivak-seven-sketches_FO1872 | 172 | 1.000 | \operatorname{Mat}_{n}(\mathbb{N}) | | |
| fong-spivak-seven-sketches_FO1873 | 172 | 0.993 | \mathbb{R}, 0,+, 1, * | | |
| fong-spivak-seven-sketches_FO1874 | 172 | 1.000 | M=R | | |
| fong-spivak-seven-sketches_FO1875 | 173 | 1.000 | R=\mathbb{N} | | |
| fong-spivak-seven-sketches_FO1876 | 173 | 1.000 | a=15 x | | |
| fong-spivak-seven-sketches_FO1877 | 173 | 1.000 | b=3 x+21 y | | |
| fong-spivak-seven-sketches_FO1878 | 174 | 0.999 | 15 x | | |
| fong-spivak-seven-sketches_FO1879 | 174 | 1.000 | 3 x=x+2 x | | |
| fong-spivak-seven-sketches_FO1880 | 174 | 1.000 | 21 y | | |
| fong-spivak-seven-sketches_FO1881 | 174 | 1.000 | (x, y, z) | | |
| fong-spivak-seven-sketches_FO1882 | 175 | 1.000 | s, t: G_{R} \rightarrow \mathbb{N} | | |
| fong-spivak-seven-sketches_FO1883 | 175 | 0.686 | \boldsymbol{\operatorname { F r e e }}\left(G_{R}\right) | | |
| fong-spivak-seven-sketches_FO1884 | 175 | 0.686 | G_{R} | | |
| fong-spivak-seven-sketches_FO1885 | 175 | 0.686 | \mathbf{S F G}_{R}:=\boldsymbol{\operatorname { F r e e }}\left(G_{R}\right) | | |
| fong-spivak-seven-sketches_FO1886 | 175 | 0.896 | \operatorname{Free}\left(G_{R}\right) | | |
| fong-spivak-seven-sketches_FO1887 | 176 | 0.978 | (m \times n) | | |
| fong-spivak-seven-sketches_FO1888 | 176 | 0.978 | M:(\underline{m} \times \underline{n}) \rightarrow R | | |
| fong-spivak-seven-sketches_FO1889 | 176 | 0.815 | (n \times p) | | |
| fong-spivak-seven-sketches_FO1890 | 176 | 0.815 | (m \times p) | | |
| fong-spivak-seven-sketches_FO1891 | 176 | 0.815 | M ; N | | |
| fong-spivak-seven-sketches_FO1892 | 176 | 0.958 | a \in \underline{m} | | |
| fong-spivak-seven-sketches_FO1893 | 176 | 0.958 | c \in \underline{p} | | |
| fong-spivak-seven-sketches_FO1894 | 176 | 0.871 | \sum_{b \in \underline{n}} | | |
| fong-spivak-seven-sketches_FO1895 | 176 | 1.000 | A v | | |
| fong-spivak-seven-sketches_FO1896 | 176 | 1.000 | v ; A | | |
| fong-spivak-seven-sketches_FO1897 | 176 | 1.000 | A: m \rightarrow n | | |
| fong-spivak-seven-sketches_FO1898 | 176 | 1.000 | b: p \rightarrow q | | |
| fong-spivak-seven-sketches_FO1899 | 176 | 1.000 | A+B: m+p \rightarrow n+q | | |
| fong-spivak-seven-sketches_FO1900 | 176 | 1.000 | m \times q | | |
| fong-spivak-seven-sketches_FO1901 | 176 | 1.000 | n \times p | | |
| fong-spivak-seven-sketches_FO1902 | 176 | 1.000 | A+B | | |
| fong-spivak-seven-sketches_FO1903 | 177 | 1.000 | 1 \times 1 | | |
| fong-spivak-seven-sketches_FO1904 | 177 | 1.000 | (a): 1 \rightarrow 1 | | |
| fong-spivak-seven-sketches_FO1905 | 177 | 1.000 | x \in R | | |
| fong-spivak-seven-sketches_FO1906 | 177 | 1.000 | a * x | | |
| fong-spivak-seven-sketches_FO1907 | 177 | 0.897 | \supset | | |
| fong-spivak-seven-sketches_FO1908 | 177 | 0.897 | 2 \times 1 | | |
| fong-spivak-seven-sketches_FO1909 | 177 | 0.897 | \binom{1}{1} | | |
| fong-spivak-seven-sketches_FO1910 | 177 | 1.000 | x+y | | |
| fong-spivak-seven-sketches_FO1911 | 221 | 0.984 | 1 \rightarrow 1 | | |
| fong-spivak-seven-sketches_FO1912 | 313 | 0.395 | 2 \rightarrow 1 | | |
| fong-spivak-seven-sketches_FO1915 | 177 | 1.000 | 0 \times n | | |
| fong-spivak-seven-sketches_FO1916 | 177 | 1.000 | n \times 0 | | |
| fong-spivak-seven-sketches_FO1917 | 177 | 0.749 | 0 \times 3 | | |
| fong-spivak-seven-sketches_FO1918 | 177 | 0.749 | 3 \times n | | |
| fong-spivak-seven-sketches_FO1919 | 177 | 1.000 | 2 \times n | | |
| fong-spivak-seven-sketches_FO1920 | 178 | 0.999 | S: \mathbf{S F G}_{R} \rightarrow \mathbf{M a t}(R) | | |
| fong-spivak-seven-sketches_FO1921 | 178 | 1.000 | S(g) | | |
| fong-spivak-seven-sketches_FO1922 | 178 | 1.000 | m \times n | | |
| fong-spivak-seven-sketches_FO1923 | 178 | 1.000 | (i, j) | | |
| fong-spivak-seven-sketches_FO1924 | 178 | 1.000 | \mathrm{id}_{0}: 0 \rightarrow 0, \mathrm{id}_{1}: 1 \rightarrow 1, \sigma: 2 \rightarrow 2 | | |
| fong-spivak-seven-sketches_FO1925 | 178 | 1.000 | (\alpha+\beta):(m+p) \rightarrow(n+q) | | |
| fong-spivak-seven-sketches_FO1926 | 178 | 0.999 | \mathrm{id}_{0}, \mathrm{id}_{1} | | |
| fong-spivak-seven-sketches_FO1927 | 178 | 0.682 | 0 \rightarrow 0 | | |
| fong-spivak-seven-sketches_FO1928 | 178 | 0.682 | \mathrm{id}_{1}, \sigma | | |
| fong-spivak-seven-sketches_FO1929 | 178 | 0.712 | (a) | | |
| fong-spivak-seven-sketches_FO1930 | 178 | 0.848 | n=p | | |
| fong-spivak-seven-sketches_FO1931 | 178 | 0.848 | \alpha+\beta | | |
| fong-spivak-seven-sketches_FO1932 | 179 | 1.000 | \beta: n \rightarrow q | | |
| fong-spivak-seven-sketches_FO1933 | 179 | 0.994 | j \in \underline{n} | | |
| fong-spivak-seven-sketches_FO1934 | 179 | 0.779 | \alpha / | | |
| fong-spivak-seven-sketches_FO1935 | 179 | 0.606 | S(\alpha) ; S(\beta) | | |
| fong-spivak-seven-sketches_FO1936 | 179 | 0.358 | \operatorname{Mat}(R) | | |
| fong-spivak-seven-sketches_FO1937 | 179 | 0.684 | m+i | | |
| fong-spivak-seven-sketches_FO1938 | 179 | 0.999 | (n+j) | | |
| fong-spivak-seven-sketches_FO1939 | 179 | 1.000 | S(\alpha)+S(\beta)=S(\alpha+\beta) | | |
| fong-spivak-seven-sketches_FO1940 | 181 | 0.989 | M \in \operatorname{Mat}(R) | | |
| fong-spivak-seven-sketches_FO1941 | 181 | 1.000 | g \in \mathbf{S F G}_{R} | | |
| fong-spivak-seven-sketches_FO1942 | 181 | 1.000 | S(g)=M | | |
| fong-spivak-seven-sketches_FO1943 | 181 | 0.890 | M \in \mathbf{M a t}(R) | | |
| fong-spivak-seven-sketches_FO1944 | 181 | 1.000 | S(g)(i, j) | | |
| fong-spivak-seven-sketches_FO1945 | 181 | 1.000 | M(i, j) | | |
| fong-spivak-seven-sketches_FO1946 | 181 | 0.939 | \left(\begin{array}{cc}a & b \\ c & d\end{array}\right) | | |
| fong-spivak-seven-sketches_FO1947 | 181 | 1.000 | \left(\begin{array}{l}0 \\ 1 \\ 2\end{array}\right) | | |
| fong-spivak-seven-sketches_FO1948 | 181 | 1.000 | \left(\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right) | | |
| fong-spivak-seven-sketches_FO1949 | 181 | 1.000 | \left(\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right) | | |
| fong-spivak-seven-sketches_FO1950 | 182 | 1.000 | \mathbf{S F G}_{R} | | |
| fong-spivak-seven-sketches_FO1951 | 182 | 1.000 | a, b \in R | | |
| fong-spivak-seven-sketches_FO1952 | 183 | 0.997 | \binom{0}{6} | | |
| fong-spivak-seven-sketches_FO1953 | 183 | 0.549 | R=(\mathbb{N}, 0,+, 1, *) | | |
| fong-spivak-seven-sketches_FO1954 | 183 | 1.000 | R=\mathbb{N} / 3 \mathbb{N} | | |
| fong-spivak-seven-sketches_FO1955 | 184 | 0.931 | M, \mu, \eta | | |
| fong-spivak-seven-sketches_FO1956 | 184 | 0.931 | \mathcal{C}, I, \otimes | | |
| fong-spivak-seven-sketches_FO1957 | 184 | 1.000 | \mu: M \otimes M \rightarrow M | | |
| fong-spivak-seven-sketches_FO1958 | 184 | 1.000 | \eta: I \rightarrow M | | |
| fong-spivak-seven-sketches_FO1959 | 184 | 0.939 | (\mu \otimes \mathrm{id}) ; \mu=(\mathrm{id} \otimes \mu) ; \mu | | |
| fong-spivak-seven-sketches_FO1960 | 184 | 1.000 | (\eta \otimes \mathrm{id}) ; \mu=\mathrm{id}=(\mathrm{id} \otimes \eta) ; \mu | | |
| fong-spivak-seven-sketches_FO1961 | 184 | 0.959 | \sigma_{M, M} ; \mu=\mu | | |
| fong-spivak-seven-sketches_FO1962 | 184 | 1.000 | \sigma_{M, M} | | |
| fong-spivak-seven-sketches_FO1963 | 184 | 1.000 | \mu: M \times M \rightarrow M | | |
| fong-spivak-seven-sketches_FO1964 | 184 | 0.935 | \eta(1) \in M | | |
| fong-spivak-seven-sketches_FO1965 | 184 | 1.000 | (a * b) * c=a *(b * c) | | |
| fong-spivak-seven-sketches_FO1966 | 184 | 1.000 | a * e=a=e * a | | |
| fong-spivak-seven-sketches_FO1967 | 184 | 0.784 | \mu: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} | | |
| fong-spivak-seven-sketches_FO1968 | 184 | 0.784 | \mu(a, b)=a * b | | |
| fong-spivak-seven-sketches_FO1969 | 184 | 0.784 | \eta \in \mathbb{R} | | |
| fong-spivak-seven-sketches_FO1970 | 184 | 1.000 | \eta=1 | | |
| fong-spivak-seven-sketches_FO1971 | 184 | 1.000 | (\mathbb{R}, *, 1) | | |
| fong-spivak-seven-sketches_FO1972 | 184 | 0.995 | \mu(a, b)=a+b | | |
| fong-spivak-seven-sketches_FO1973 | 184 | 0.985 | \eta=0 | | |
| fong-spivak-seven-sketches_FO1974 | 184 | 0.985 | (\mathbb{R},+, 0) | | |
| fong-spivak-seven-sketches_FO1975 | 184 | 0.457 | \mu=>- | | |
| fong-spivak-seven-sketches_FO1976 | 184 | 0.457 | \eta=\sigma | | |
| fong-spivak-seven-sketches_FO1977 | 184 | 0.315 | (R) | | |
| fong-spivak-seven-sketches_FO1978 | 185 | 0.665 | U: \operatorname{Mat}(R) \rightarrow | | |
| fong-spivak-seven-sketches_FO1979 | 185 | 0.999 | R^{n} | | |
| fong-spivak-seven-sketches_FO1980 | 185 | 0.999 | M: m \rightarrow n | | |
| fong-spivak-seven-sketches_FO1981 | 185 | 1.000 | R^{m} \rightarrow R^{n} | | |
| fong-spivak-seven-sketches_FO1982 | 185 | 0.881 | \{1\} | | |
| fong-spivak-seven-sketches_FO1983 | 185 | 0.828 | (M, \mu, \eta) | | |
| fong-spivak-seven-sketches_FO1984 | 185 | 0.828 | (U(M), U(\mu), U(\eta)) | | |
| fong-spivak-seven-sketches_FO1985 | 185 | 0.960 | \operatorname{prop} \operatorname{Mat}(R) | | |
| fong-spivak-seven-sketches_FO1986 | 185 | 0.960 | R=\mathbb{R} | | |
| fong-spivak-seven-sketches_FO1987 | 185 | 0.212 | \mathbf{M a t}(R) | | |
| fong-spivak-seven-sketches_FO1988 | 185 | 0.157 | \operatorname{Mat}(R)^{\mathrm{op}} | | |
| fong-spivak-seven-sketches_FO1989 | 185 | 0.245 | \varnothing, \mathbb{N} | | |
| fong-spivak-seven-sketches_FO1990 | 185 | 1.000 | \mu: 2 \rightarrow 1 | | |
| fong-spivak-seven-sketches_FO1991 | 185 | 1.000 | \eta: 0 \rightarrow 1 | | |
| fong-spivak-seven-sketches_FO1992 | 185 | 1.000 | \mathcal{M} \rightarrow \mathcal{C} | | |
| fong-spivak-seven-sketches_FO1993 | 186 | 0.998 | g: m \rightarrow n | | |
| fong-spivak-seven-sketches_FO1994 | 186 | 0.998 | x \in R^{m} | | |
| fong-spivak-seven-sketches_FO1995 | 186 | 1.000 | y \in R^{n} | | |
| fong-spivak-seven-sketches_FO1996 | 186 | 1.000 | \mathrm{B}(g) \subseteq R^{m} \times R^{n} | | |
| fong-spivak-seven-sketches_FO1997 | 186 | 1.000 | (x,(x, x)) | | |
| fong-spivak-seven-sketches_FO1998 | 186 | 0.954 | g^{\text {op }} | | |
| fong-spivak-seven-sketches_FO1999 | 186 | 0.954 | g^{\text {op }}: n \rightarrow m | | |
| fong-spivak-seven-sketches_FO2000 | 186 | 0.954 | \mathrm{B}\left(g^{\text {op }}\right) \subseteq R^{n} \times R^{m} | | |
| fong-spivak-seven-sketches_FO2001 | 186 | 0.996 | (b, a) | | |
| fong-spivak-seven-sketches_FO2002 | 186 | 0.854 | \mathrm{B}(-\mathrm{C}) | | |
| fong-spivak-seven-sketches_FO2003 | 186 | 0.854 | -\mathrm{C}: 1 \rightarrow 2 | | |
| fong-spivak-seven-sketches_FO2004 | 186 | 0.833 | \mathrm{B}(>-) | | |
| fong-spivak-seven-sketches_FO2005 | 186 | 0.833 | >: 2 \rightarrow 1 | | |
| fong-spivak-seven-sketches_FO2006 | 187 | 1.000 | B \subseteq R^{m} \times R^{n} | | |
| fong-spivak-seven-sketches_FO2007 | 187 | 1.000 | N: n \rightarrow p | | |
| fong-spivak-seven-sketches_FO2008 | 187 | 1.000 | B_{1}:=\left\{(x, M x) \mid x \in R^{m}\right\} | | |
| fong-spivak-seven-sketches_FO2009 | 187 | 1.000 | B_{2}:=\left\{(y, N y) \mid y \in R^{n}\right\} | | |
| fong-spivak-seven-sketches_FO2010 | 187 | 0.999 | \left\{(x, x ; M ; N) \mid x \in R^{m}\right\} | | |
| fong-spivak-seven-sketches_FO2011 | 187 | 0.881 | B_{1} \subseteq R^{m} \times R^{n} | | |
| fong-spivak-seven-sketches_FO2012 | 187 | 0.881 | B_{2} \subseteq R^{n} \times R^{p} | | |
| fong-spivak-seven-sketches_FO2013 | 187 | 0.881 | B_{1} ; B_{2} \subseteq R^{m} \times R^{p} | | |
| fong-spivak-seven-sketches_FO2014 | 187 | 0.953 | \mathbf{R e l}_{R} | | |
| fong-spivak-seven-sketches_FO2015 | 187 | 1.000 | C \subseteq R^{p} \times R^{q} | | |
| fong-spivak-seven-sketches_FO2016 | 187 | 1.000 | B+C | | |
| fong-spivak-seven-sketches_FO2017 | 187 | 0.755 | g \in G_{R} | | |
| fong-spivak-seven-sketches_FO2018 | 187 | 0.755 | G_{R}^{\text {op }}:=\left\{g^{\text {op }} \mid g \in G_{R}\right\} | | |
| fong-spivak-seven-sketches_FO2019 | 187 | 0.998 | \mathbf{S F G}_{R}^{+} | | |
| fong-spivak-seven-sketches_FO2020 | 188 | 1.000 | g: m \rightarrow n, h: \ell \rightarrow n | | |
| fong-spivak-seven-sketches_FO2021 | 188 | 1.000 | h^{\mathrm{op}}: n \rightarrow \ell | | |
| fong-spivak-seven-sketches_FO2022 | 188 | 0.467 | g_{9}^{\circ}\left(h^{\mathrm{op}}\right) | | |
| fong-spivak-seven-sketches_FO2023 | 188 | 0.274 | g{ }^{\circ}\left(h^{\mathrm{op}}\right) \subseteq R^{m} \times R^{\ell} | | |
| fong-spivak-seven-sketches_FO2024 | 188 | 0.807 | g: m \rightarrow n, h: m \rightarrow p | | |
| fong-spivak-seven-sketches_FO2025 | 188 | 0.807 | \left(g^{\mathrm{op}}\right): n \rightarrow m | | |
| fong-spivak-seven-sketches_FO2026 | 188 | 0.856 | g^{\mathrm{op}} ; h | | |
| fong-spivak-seven-sketches_FO2027 | 188 | 0.963 | S(g) \in \operatorname{Mat}(R) | | |
| fong-spivak-seven-sketches_FO2028 | 188 | 0.994 | b_{1}, b_{2} \in \mathrm{~B}(g) | | |
| fong-spivak-seven-sketches_FO2029 | 188 | 0.994 | b_{1}+b_{2} | | |
| fong-spivak-seven-sketches_FO2030 | 188 | 0.994 | r * b_{1} | | |
| fong-spivak-seven-sketches_FO2031 | 188 | 0.994 | r \in R | | |
| fong-spivak-seven-sketches_FO2032 | 189 | 0.848 | \mathbf{L i n R e l}_{R} | | |
| fong-spivak-seven-sketches_FO2033 | 189 | 0.989 | (x, y) \in B | | |
| fong-spivak-seven-sketches_FO2034 | 189 | 0.730 | (r * x, r * y) \in R^{m} \times R^{n} | | |
| fong-spivak-seven-sketches_FO2035 | 189 | 0.730 | \left(x^{\prime}, y^{\prime}\right) \in B | | |
| fong-spivak-seven-sketches_FO2036 | 189 | 0.730 | \left(x+x^{\prime}, y+y^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO2037 | 189 | 0.512 | G_{R} \sqcup G_{R}^{\mathrm{op}} | | |
| fong-spivak-seven-sketches_FO2038 | 189 | 1.000 | \eta_{1} | | |
| fong-spivak-seven-sketches_FO2039 | 189 | 1.000 | \epsilon_{1} | | |
| fong-spivak-seven-sketches_FO2040 | 190 | 1.000 | n=n^{*} | | |
| fong-spivak-seven-sketches_FO2041 | 190 | 1.000 | \mathbb{R}\left[s, s^{-1}\right] | | |
| fong-spivak-seven-sketches_FO2042 | 190 | 1.000 | s^{-1} | | |
| fong-spivak-seven-sketches_FO2043 | 190 | 1.000 | s s^{-1}=s^{-1} s=1 | | |
| fong-spivak-seven-sketches_FO2044 | 190 | 1.000 | a * u+b * v | | |
| fong-spivak-seven-sketches_FO2045 | 190 | 1.000 | \{(F, u, v) \mid u=v\} | | |
| fong-spivak-seven-sketches_FO2046 | 195 | 0.872 | \mathcal{C}^{\prime} | | |
| fong-spivak-seven-sketches_FO2047 | 195 | 0.444 | \operatorname{Cospan}_{\mathrm{C}} | | |
| fong-spivak-seven-sketches_FO2048 | 195 | 1.000 | s_{m, n}: m \rightarrow n | | |
| fong-spivak-seven-sketches_FO2049 | 197 | 1.000 | \varnothing \in \mathcal{C} | | |
| fong-spivak-seven-sketches_FO2050 | 197 | 1.000 | !_{T}: \varnothing \rightarrow T | | |
| fong-spivak-seven-sketches_FO2051 | 197 | 0.844 | A, \leq | | |
| fong-spivak-seven-sketches_FO2052 | 197 | 0.994 | (A, \leq) | | |
| fong-spivak-seven-sketches_FO2053 | 197 | 1.000 | \varnothing \rightarrow T | | |
| fong-spivak-seven-sketches_FO2054 | 197 | 0.615 | T \in \mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO2055 | 197 | 0.615 | T=A | | |
| fong-spivak-seven-sketches_FO2056 | 197 | 0.615 | T=B | | |
| fong-spivak-seven-sketches_FO2057 | 197 | 0.999 | \varnothing={ }^{?} A | | |
| fong-spivak-seven-sketches_FO2058 | 197 | 1.000 | \varnothing={ }^{?} B | | |
| fong-spivak-seven-sketches_FO2059 | 197 | 1.000 | B \rightarrow A | | |
| fong-spivak-seven-sketches_FO2060 | 197 | 0.374 | \boxed{\bullet} | | |
| fong-spivak-seven-sketches_FO2061 | 197 | 0.816 | \stackrel{a}{\bullet} \rightarrow \stackrel{b}{\bullet} \rightarrow \stackrel{c}{\bullet} | | |
| fong-spivak-seven-sketches_FO2062 | 197 | 0.981 | \stackrel{a}{\bullet} \stackrel{b}{\bullet} | | |
| fong-spivak-seven-sketches_FO2063 | 197 | 0.985 | \left(R, 0_{R},+_{R}, 1_{R}, *_{R}\right) | | |
| fong-spivak-seven-sketches_FO2064 | 197 | 0.985 | \left(S, 0_{S},+_{S}, 1_{S}, *_{S}\right) | | |
| fong-spivak-seven-sketches_FO2065 | 197 | 0.985 | f: R \rightarrow S | | |
| fong-spivak-seven-sketches_FO2066 | 197 | 0.985 | f\left(0_{R}\right)=0_{S} | | |
| fong-spivak-seven-sketches_FO2067 | 197 | 0.979 | f\left(r_{1}+{ }_{R} r_{2}\right)=f\left(r_{1}\right)+s f\left(r_{2}\right) | | |
| fong-spivak-seven-sketches_FO2068 | 198 | 1.000 | c_{1} | | |
| fong-spivak-seven-sketches_FO2069 | 198 | 1.000 | c_{2} | | |
| fong-spivak-seven-sketches_FO2070 | 198 | 0.978 | \iota_{A}: A \rightarrow | | |
| fong-spivak-seven-sketches_FO2071 | 198 | 0.977 | A+B, \iota_{B}: B \rightarrow A+B | | |
| fong-spivak-seven-sketches_FO2072 | 198 | 0.977 | (f: A \rightarrow | | |
| fong-spivak-seven-sketches_FO2073 | 198 | 0.994 | T, g: B \rightarrow T | | |
| fong-spivak-seven-sketches_FO2074 | 198 | 0.994 | [f, g]: A+B \rightarrow T | | |
| fong-spivak-seven-sketches_FO2075 | 198 | 0.999 | [f, g] | | |
| fong-spivak-seven-sketches_FO2076 | 198 | 1.000 | \iota_{A}: A \longrightarrow A \sqcup B | | |
| fong-spivak-seven-sketches_FO2077 | 199 | 1.000 | \iota_{B}: B \rightarrow A \sqcup B | | |
| fong-spivak-seven-sketches_FO2078 | 199 | 1.000 | f: A \rightarrow T | | |
| fong-spivak-seven-sketches_FO2079 | 199 | 1.000 | g: B \rightarrow T | | |
| fong-spivak-seven-sketches_FO2080 | 199 | 0.591 | [f, g]: A \sqcup B \rightarrow T | | |
| fong-spivak-seven-sketches_FO2081 | 199 | 0.591 | \iota_{A} | | |
| fong-spivak-seven-sketches_FO2082 | 199 | 0.591 | [f, g]=f | | |
| fong-spivak-seven-sketches_FO2083 | 199 | 0.591 | \iota_{B} | | |
| fong-spivak-seven-sketches_FO2084 | 199 | 0.591 | [f, g]=g | | |
| fong-spivak-seven-sketches_FO2085 | 199 | 0.976 | x \in A \sqcup B | | |
| fong-spivak-seven-sketches_FO2086 | 199 | 0.976 | x=\iota_{A}(a) | | |
| fong-spivak-seven-sketches_FO2087 | 199 | 1.000 | b \in B | | |
| fong-spivak-seven-sketches_FO2088 | 199 | 1.000 | x=\iota_{B}(b) | | |
| fong-spivak-seven-sketches_FO2089 | 199 | 1.000 | T=\{a, b, c, \ldots, z\} | | |
| fong-spivak-seven-sketches_FO2090 | 199 | 1.000 | f( | | |
| fong-spivak-seven-sketches_FO2091 | 199 | 1.000 | )=a | | |
| fong-spivak-seven-sketches_FO2092 | 199 | 0.999 | g( | | |
| fong-spivak-seven-sketches_FO2093 | 199 | 0.999 | )=e | | |
| fong-spivak-seven-sketches_FO2094 | 199 | 1.000 | [f, g](x) | | |
| fong-spivak-seven-sketches_FO2095 | 199 | 1.000 | f: A \rightarrow C, g: B \rightarrow C | | |
| fong-spivak-seven-sketches_FO2096 | 199 | 0.538 | \iota_{A} ~:[f, g]=f | | |
| fong-spivak-seven-sketches_FO2097 | 199 | 0.982 | \iota_{B}:[f, g]=g | | |
| fong-spivak-seven-sketches_FO2098 | 199 | 1.000 | [f, g] ; h=[f ; h, g ; h] | | |
| fong-spivak-seven-sketches_FO2099 | 199 | 1.000 | \left[\iota_{A}, \iota_{B}\right]=\mathrm{id}_{A+B} | | |
| fong-spivak-seven-sketches_FO2100 | 199 | 0.914 | (\mathcal{C},+, \varnothing) | | |
| fong-spivak-seven-sketches_FO2101 | 199 | 0.827 | \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C} | | |
| fong-spivak-seven-sketches_FO2102 | 199 | 0.771 | \mathcal{C} \times \mathcal{C} | | |
| fong-spivak-seven-sketches_FO2103 | 199 | 1.000 | A+\varnothing \rightarrow A | | |
| fong-spivak-seven-sketches_FO2104 | 199 | 1.000 | \varnothing+A \rightarrow A | | |
| fong-spivak-seven-sketches_FO2105 | 199 | 1.000 | (A+B)+C \rightarrow A+(B+C) | | |
| fong-spivak-seven-sketches_FO2106 | 199 | 1.000 | A+B \rightarrow B+A | | |
| fong-spivak-seven-sketches_FO2107 | 200 | 1.000 | f: A \rightarrow X | | |
| fong-spivak-seven-sketches_FO2108 | 200 | 1.000 | g: A \rightarrow Y | | |
| fong-spivak-seven-sketches_FO2109 | 200 | 0.786 | X+{ }_{A} Y | | |
| fong-spivak-seven-sketches_FO2110 | 200 | 0.857 | X+_{A} Y | | |
| fong-spivak-seven-sketches_FO2111 | 200 | 0.857 | \iota_{X}: X \rightarrow | | |
| fong-spivak-seven-sketches_FO2112 | 200 | 0.905 | \iota_{Y}: Y \rightarrow X+{ }_{A} Y | | |
| fong-spivak-seven-sketches_FO2113 | 200 | 0.371 | \ulcorner | | |
| fong-spivak-seven-sketches_FO2114 | 200 | 1.000 | x: X \rightarrow T, y: Y \rightarrow T | | |
| fong-spivak-seven-sketches_FO2115 | 200 | 0.933 | t: X+{ }_{A} Y \rightarrow T | | |
| fong-spivak-seven-sketches_FO2116 | 201 | 0.989 | \iota_{X} | | |
| fong-spivak-seven-sketches_FO2117 | 201 | 0.989 | \iota_{Y} | | |
| fong-spivak-seven-sketches_FO2118 | 201 | 1.000 | B \leftarrow A \rightarrow C | | |
| fong-spivak-seven-sketches_FO2119 | 201 | 1.000 | B \sqcup C | | |
| fong-spivak-seven-sketches_FO2120 | 201 | 1.000 | B \vee C | | |
| fong-spivak-seven-sketches_FO2121 | 201 | 0.867 | i: A \rightarrow A^{\prime} | | |
| fong-spivak-seven-sketches_FO2122 | 201 | 0.867 | j: X \rightarrow X^{\prime} | | |
| fong-spivak-seven-sketches_FO2123 | 201 | 0.867 | X^{\prime} | | |
| fong-spivak-seven-sketches_FO2124 | 201 | 0.867 | X+{ }_{A} A^{\prime} | | |
| fong-spivak-seven-sketches_FO2125 | 201 | 0.847 | f^{\prime}:=i^{-1}: f: j | | |
| fong-spivak-seven-sketches_FO2126 | 201 | 0.997 | f ; x=i ; a | | |
| fong-spivak-seven-sketches_FO2127 | 201 | 0.997 | x^{\prime}: X \rightarrow T | | |
| fong-spivak-seven-sketches_FO2128 | 201 | 0.997 | x^{\prime}:=j^{-1} ; x | | |
| fong-spivak-seven-sketches_FO2129 | 201 | 0.560 | f^{\prime} ; x^{\prime}=i^{-1} ; f ; j ; j^{-1} ; x=i^{-1} ; i ; a=a | | |
| fong-spivak-seven-sketches_FO2130 | 201 | 0.998 | _{S} | | |
| fong-spivak-seven-sketches_FO2131 | 202 | 1.000 | \{f(a) \sim g(a) \mid a \in A\} | | |
| fong-spivak-seven-sketches_FO2132 | 202 | 1.000 | f(a) | | |
| fong-spivak-seven-sketches_FO2133 | 202 | 1.000 | g(a) | | |
| fong-spivak-seven-sketches_FO2134 | 202 | 1.000 | f: \underline{4} \rightarrow \underline{5} | | |
| fong-spivak-seven-sketches_FO2135 | 202 | 1.000 | g: \underline{4} \rightarrow \underline{3} | | |
| fong-spivak-seven-sketches_FO2136 | 202 | 1.000 | X, Y \in | | |
| fong-spivak-seven-sketches_FO2137 | 202 | 0.785 | f: \varnothing \rightarrow X | | |
| fong-spivak-seven-sketches_FO2138 | 202 | 0.785 | g: \varnothing \rightarrow Y | | |
| fong-spivak-seven-sketches_FO2139 | 202 | 0.884 | X \stackrel{f}{\leftarrow} \varnothing \xrightarrow{g} Y | | |
| fong-spivak-seven-sketches_FO2140 | 202 | 1.000 | X+Y | | |
| fong-spivak-seven-sketches_FO2141 | 202 | 0.897 | X+Y \rightarrow T | | |
| fong-spivak-seven-sketches_FO2142 | 202 | 0.615 | X+\varnothing Y \cong X+Y | | |
| fong-spivak-seven-sketches_FO2143 | 202 | 0.986 | X+\varnothing Y | | |
| fong-spivak-seven-sketches_FO2144 | 202 | 0.991 | A=X=Y=\mathbb{N} | | |
| fong-spivak-seven-sketches_FO2145 | 203 | 1.000 | f(a):=\lfloor a / 2\rfloor | | |
| fong-spivak-seven-sketches_FO2146 | 203 | 1.000 | g(a):=\lfloor(a+1) / 2\rfloor | | |
| fong-spivak-seven-sketches_FO2147 | 203 | 1.000 | x: X \rightarrow T | | |
| fong-spivak-seven-sketches_FO2148 | 203 | 1.000 | y: Y \rightarrow T | | |
| fong-spivak-seven-sketches_FO2149 | 203 | 1.000 | f \cong x=g \cong y | | |
| fong-spivak-seven-sketches_FO2150 | 203 | 0.781 | f(1)=0 | | |
| fong-spivak-seven-sketches_FO2151 | 203 | 0.686 | g(1)=1 | | |
| fong-spivak-seven-sketches_FO2152 | 203 | 0.686 | t=x(0)=x(f(1))=y(g(1))=y(1) | | |
| fong-spivak-seven-sketches_FO2153 | 203 | 1.000 | g(2)=1 | | |
| fong-spivak-seven-sketches_FO2154 | 203 | 1.000 | f(2)=1 | | |
| fong-spivak-seven-sketches_FO2155 | 203 | 1.000 | t=g(1)=y(g(2))= | | |
| fong-spivak-seven-sketches_FO2156 | 203 | 0.900 | x(f(2))=x(1) | | |
| fong-spivak-seven-sketches_FO2157 | 203 | 0.996 | X \sqcup_{A} Y \cong\{1\} | | |
| fong-spivak-seven-sketches_FO2158 | 203 | 0.832 | j \in \operatorname{Ob} \mathcal{J} | | |
| fong-spivak-seven-sketches_FO2159 | 203 | 0.832 | \operatorname{id}_{j} | | |
| fong-spivak-seven-sketches_FO2160 | 204 | 0.996 | \operatorname{colim}_{\mathcal{J}} D | | |
| fong-spivak-seven-sketches_FO2161 | 204 | 0.890 | \mathbf{0} \rightarrow \mathcal{C} | | |
| fong-spivak-seven-sketches_FO2162 | 205 | 1.000 | (v, d) \sim(w, e) | | |
| fong-spivak-seven-sketches_FO2163 | 205 | 1.000 | a: v \rightarrow w | | |
| fong-spivak-seven-sketches_FO2164 | 205 | 0.996 | D(a)(d)=e | | |
| fong-spivak-seven-sketches_FO2165 | 205 | 0.996 | \iota_{v}: D(v) \rightarrow \operatorname{colim}_{\mathcal{J}} D | | |
| fong-spivak-seven-sketches_FO2166 | 205 | 1.000 | d \in D(v) | | |
| fong-spivak-seven-sketches_FO2167 | 205 | 0.654 | \mathfrak{J}= | | |
| fong-spivak-seven-sketches_FO2168 | 205 | 0.654 | D: \mathfrak{J} \rightarrow | | |
| fong-spivak-seven-sketches_FO2169 | 205 | 1.000 | X:=D\left(v_{1}\right) | | |
| fong-spivak-seven-sketches_FO2170 | 205 | 1.000 | Y:=D\left(v_{2}\right) | | |
| fong-spivak-seven-sketches_FO2171 | 206 | 0.716 | \mathbf{1}=\boxed{\bullet} | | |
| fong-spivak-seven-sketches_FO2172 | 206 | 1.000 | D(v) | | |
| fong-spivak-seven-sketches_FO2173 | 206 | 0.824 | X: \mathbf{1} \boldsymbol{\rightarrow} | | |
| fong-spivak-seven-sketches_FO2174 | 206 | 1.000 | X \stackrel{f}{\leftarrow} N \xrightarrow{g} Y | | |
| fong-spivak-seven-sketches_FO2175 | 206 | 0.994 | X \xrightarrow[g]{f} Y | | |
| fong-spivak-seven-sketches_FO2176 | 206 | 1.000 | y \sim y^{\prime} | | |
| fong-spivak-seven-sketches_FO2177 | 206 | 1.000 | f(x)=y | | |
| fong-spivak-seven-sketches_FO2178 | 206 | 1.000 | g(x)=y^{\prime} | | |
| fong-spivak-seven-sketches_FO2179 | 206 | 0.777 | f, g: X \rightarrow Y | | |
| fong-spivak-seven-sketches_FO2180 | 206 | 1.000 | A \rightarrow N \leftarrow B | | |
| fong-spivak-seven-sketches_FO2181 | 207 | 0.749 | N \stackrel{g}{\leftarrow} B \xrightarrow{h} P | | |
| fong-spivak-seven-sketches_FO2182 | 207 | 1.000 | C \in \mathrm{Ob} \mathcal{C} | | |
| fong-spivak-seven-sketches_FO2183 | 207 | 1.000 | C \rightarrow C \leftarrow C | | |
| fong-spivak-seven-sketches_FO2184 | 207 | 1.000 | A \rightarrow B \rightarrow C | | |
| fong-spivak-seven-sketches_FO2185 | 207 | 1.000 | A \rightarrow P \leftarrow B | | |
| fong-spivak-seven-sketches_FO2186 | 207 | 1.000 | A \rightarrow P^{\prime} \leftarrow B | | |
| fong-spivak-seven-sketches_FO2187 | 207 | 1.000 | P \cong P^{\prime} | | |
| fong-spivak-seven-sketches_FO2188 | 207 | 0.761 | { }_{\text {C }} | | |
| fong-spivak-seven-sketches_FO2189 | 207 | 0.743 | \operatorname{Cospan}_{\mathcal{C}} | | |
| fong-spivak-seven-sketches_FO2190 | 208 | 0.839 | _{\mathcal{C}} | | |
| fong-spivak-seven-sketches_FO2191 | 208 | 0.839 | \operatorname{Ob}\left(\right. | | |
| fong-spivak-seven-sketches_FO2192 | 208 | 0.839 | \left._{\mathcal{C}}\right)=\operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO2193 | 208 | 0.370 | _{\mathrm{C}}, \varnothing,+ | | |
| fong-spivak-seven-sketches_FO2194 | 208 | 1.000 | A+\varnothing \cong A | | |
| fong-spivak-seven-sketches_FO2195 | 208 | 0.699 | { }_{\mathrm{C}}, \varnothing,+ | | |
| fong-spivak-seven-sketches_FO2196 | 208 | 0.955 | _{\text {FinSet }} | | |
| fong-spivak-seven-sketches_FO2197 | 208 | 1.000 | { }_{\text {FinSet }} | | |
| fong-spivak-seven-sketches_FO2198 | 208 | 0.958 | A \xrightarrow{f} N \stackrel{g}{\leftarrow} B | | |
| fong-spivak-seven-sketches_FO2199 | 208 | 1.000 | g(b) | | |
| fong-spivak-seven-sketches_FO2200 | 209 | 0.904 | A+B \rightarrow B+C | | |
| fong-spivak-seven-sketches_FO2201 | 209 | 0.904 | \operatorname{Cospan}_{\text {FinSet }} | | |
| fong-spivak-seven-sketches_FO2202 | 210 | 0.992 | (X, \mu, \eta) | | |
| fong-spivak-seven-sketches_FO2205 | 210 | 0.963 | X \otimes X | | |
| fong-spivak-seven-sketches_FO2206 | 210 | 0.963 | (X, \delta, \epsilon) | | |
| fong-spivak-seven-sketches_FO2207 | 210 | 1.000 | \delta: X \rightarrow X \otimes X, \epsilon: X \rightarrow I | | |
| fong-spivak-seven-sketches_FO2208 | 210 | 1.000 | \mu, \eta, \delta | | |
| fong-spivak-seven-sketches_FO2209 | 210 | 1.000 | \epsilon | | |
| fong-spivak-seven-sketches_FO2210 | 210 | 1.000 | \mu | | |
| fong-spivak-seven-sketches_FO2211 | 210 | 1.000 | \delta | | |
| fong-spivak-seven-sketches_FO2212 | 210 | 0.953 | (\mathcal{C}, \otimes, I) | | |
| fong-spivak-seven-sketches_FO2213 | 210 | 0.574 | (\mu, \eta, \delta, \epsilon) | | |
| fong-spivak-seven-sketches_FO2214 | 211 | 1.000 | X^{\otimes m} \rightarrow X^{\otimes n} | | |
| fong-spivak-seven-sketches_FO2215 | 211 | 0.961 | \mu, \delta | | |
| fong-spivak-seven-sketches_FO2216 | 211 | 0.508 | X, \mu, \eta, \delta, \epsilon | | |
| fong-spivak-seven-sketches_FO2217 | 211 | 0.999 | s_{m, n}: X^{\otimes m} \rightarrow X^{\otimes n} | | |
| fong-spivak-seven-sketches_FO2218 | 211 | 0.799 | (m-1) \mu | | |
| fong-spivak-seven-sketches_FO2219 | 211 | 0.799 | (n-1) \delta | | |
| fong-spivak-seven-sketches_FO2220 | 211 | 1.000 | m=0 | | |
| fong-spivak-seven-sketches_FO2221 | 211 | 0.892 | s_{m, n} | | |
| fong-spivak-seven-sketches_FO2222 | 211 | 1.000 | \mu, \eta, \delta, \epsilon, \sigma | | |
| fong-spivak-seven-sketches_FO2223 | 211 | 0.781 | (X, \mu, \eta, \delta, \epsilon) | | |
| fong-spivak-seven-sketches_FO2224 | 211 | 1.000 | f: X^{\otimes m} \rightarrow X^{\otimes n} | | |
| fong-spivak-seven-sketches_FO2225 | 211 | 1.000 | \sigma: X^{\otimes 2} \rightarrow X^{\otimes 2} | | |
| fong-spivak-seven-sketches_FO2226 | 211 | 1.000 | f=s_{m, n} | | |
| fong-spivak-seven-sketches_FO2227 | 211 | 1.000 | X \otimes X \rightarrow X \otimes X \otimes X | | |
| fong-spivak-seven-sketches_FO2228 | 212 | 0.937 | G:=\{\mu, \eta, \delta, \epsilon\} | | |
| fong-spivak-seven-sketches_FO2229 | 212 | 0.937 | G \rightarrow | | |
| fong-spivak-seven-sketches_FO2230 | 212 | 0.976 | (G, E) | | |
| fong-spivak-seven-sketches_FO2231 | 213 | 0.985 | X, \mu_{X}, \delta_{X}, \eta_{X}, \epsilon_{X} | | |
| fong-spivak-seven-sketches_FO2232 | 214 | 0.991 | \eta_{I}=\mathrm{id}_{I}=\epsilon_{I} | | |
| fong-spivak-seven-sketches_FO2233 | 214 | 0.978 | \mu_{X}, \delta_{X}, \eta_{X}, \epsilon_{X} | | |
| fong-spivak-seven-sketches_FO2234 | 214 | 0.794 | \underline{1} | | |
| fong-spivak-seven-sketches_FO2235 | 214 | 0.792 | \varnothing, \sqcup | | |
| fong-spivak-seven-sketches_FO2236 | 214 | 0.578 | \mu_{X}: X \sqcup X \rightarrow X | | |
| fong-spivak-seven-sketches_FO2237 | 214 | 0.578 | X \sqcup X \sqcup X | | |
| fong-spivak-seven-sketches_FO2238 | 214 | 1.000 | \delta_{X}: X \rightarrow X \sqcup X | | |
| fong-spivak-seven-sketches_FO2239 | 214 | 1.000 | \eta_{X}: \varnothing \rightarrow X | | |
| fong-spivak-seven-sketches_FO2240 | 214 | 1.000 | \epsilon_{X}: X \rightarrow \varnothing | | |
| fong-spivak-seven-sketches_FO2241 | 214 | 1.000 | X=\varnothing \sqcup X=X \sqcup \varnothing | | |
| fong-spivak-seven-sketches_FO2242 | 214 | 1.000 | \left(X, \mu_{X}, \eta_{X}, \delta_{X}, \epsilon_{X}\right) | | |
| fong-spivak-seven-sketches_FO2243 | 216 | 0.384 | \left(\mathcal{C}, I_{\mathcal{C}}, \otimes_{\mathcal{C}}\right) | | |
| fong-spivak-seven-sketches_FO2244 | 216 | 0.384 | \left(\mathcal{D}, I_{\mathcal{D}}, \otimes_{\mathcal{D}}\right) | | |
| fong-spivak-seven-sketches_FO2245 | 216 | 0.994 | (F, \varphi) | | |
| fong-spivak-seven-sketches_FO2246 | 216 | 0.984 | \varphi_{I}: I_{\mathcal{D}} \rightarrow F\left(I_{\mathcal{C}}\right) | | |
| fong-spivak-seven-sketches_FO2247 | 216 | 0.899 | c_{1}, c_{2} \in \operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO2248 | 216 | 1.000 | \varphi | | |
| fong-spivak-seven-sketches_FO2249 | 216 | 1.000 | \varphi_{I} | | |
| fong-spivak-seven-sketches_FO2250 | 216 | 1.000 | \varphi_{c_{1}, c_{2}} | | |
| fong-spivak-seven-sketches_FO2251 | 216 | 1.000 | c_{1}, c_{2} | | |
| fong-spivak-seven-sketches_FO2252 | 216 | 1.000 | S \in | | |
| fong-spivak-seven-sketches_FO2253 | 216 | 1.000 | \mathrm{P}(S):=\{R \subseteq S\} | | |
| fong-spivak-seven-sketches_FO2254 | 216 | 1.000 | \operatorname{im}_{f}: \mathrm{P}(S) \rightarrow \mathrm{P}(T) | | |
| fong-spivak-seven-sketches_FO2255 | 216 | 1.000 | R \subseteq S | | |
| fong-spivak-seven-sketches_FO2256 | 216 | 1.000 | \{f(r) \mid r \in R\} \subseteq T | | |
| fong-spivak-seven-sketches_FO2257 | 216 | 1.000 | (\{1\}, \times) | | |
| fong-spivak-seven-sketches_FO2258 | 216 | 1.000 | \varphi_{I}:\{1\} \rightarrow | | |
| fong-spivak-seven-sketches_FO2259 | 216 | 0.997 | \mathrm{P}(\{1\}) | | |
| fong-spivak-seven-sketches_FO2260 | 216 | 0.997 | \varphi_{S, T}: \mathrm{P}(S) \times \mathrm{P}(T) \rightarrow \mathrm{P}(S \times T) | | |
| fong-spivak-seven-sketches_FO2261 | 216 | 1.000 | \varphi_{I}(1) | | |
| fong-spivak-seven-sketches_FO2262 | 216 | 1.000 | \{1\} \subseteq\{1\} | | |
| fong-spivak-seven-sketches_FO2263 | 216 | 1.000 | A \subseteq S | | |
| fong-spivak-seven-sketches_FO2264 | 216 | 1.000 | B \subseteq T | | |
| fong-spivak-seven-sketches_FO2265 | 216 | 1.000 | \varphi_{S, T}(A, B) | | |
| fong-spivak-seven-sketches_FO2266 | 216 | 1.000 | A \times B \subseteq S \times T | | |
| fong-spivak-seven-sketches_FO2267 | 216 | 0.957 | (\mathrm{P}, \varphi) | | |
| fong-spivak-seven-sketches_FO2268 | 216 | 1.000 | \varphi_{S, T} | | |
| fong-spivak-seven-sketches_FO2269 | 217 | 1.000 | 2 \Omega | | |
| fong-spivak-seven-sketches_FO2270 | 217 | 1.000 | A \rightarrow N | | |
| fong-spivak-seven-sketches_FO2271 | 217 | 1.000 | B \rightarrow N | | |
| fong-spivak-seven-sketches_FO2272 | 218 | 1.000 | (F, \varphi):(\mathcal{C},+) \longrightarrow( | | |
| fong-spivak-seven-sketches_FO2273 | 218 | 1.000 | \times) | | |
| fong-spivak-seven-sketches_FO2274 | 218 | 0.672 | A \xrightarrow{i} N \stackrel{o}{\leftarrow} B | | |
| fong-spivak-seven-sketches_FO2275 | 218 | 0.672 | s \in F(N) .{ }^{5} | | |
| fong-spivak-seven-sketches_FO2276 | 218 | 0.596 | \mathcal{C}= | | |
| fong-spivak-seven-sketches_FO2277 | 218 | 0.596 | N \in | | |
| fong-spivak-seven-sketches_FO2278 | 218 | 1.000 | F(N) | | |
| fong-spivak-seven-sketches_FO2279 | 218 | 0.980 | (A \xrightarrow{f} N \stackrel{g}{\leftarrow} B, s) | | |
| fong-spivak-seven-sketches_FO2280 | 218 | 0.980 | (B \xrightarrow{h} P \stackrel{k}{\leftarrow} C, t) | | |
| fong-spivak-seven-sketches_FO2281 | 218 | 1.000 | B \xrightarrow{h} P \stackrel{k}{\leftarrow} C | | |
| fong-spivak-seven-sketches_FO2282 | 219 | 0.998 | F\left(N+{ }_{B} P\right) | | |
| fong-spivak-seven-sketches_FO2283 | 219 | 0.875 | _{F} | | |
| fong-spivak-seven-sketches_FO2284 | 219 | 0.581 | _{\text {C }} | | |
| fong-spivak-seven-sketches_FO2285 | 219 | 0.876 | _{\mathrm{e}} | | |
| fong-spivak-seven-sketches_FO2286 | 219 | 0.579 | \operatorname{Cospan}_{F} | | |
| fong-spivak-seven-sketches_FO2287 | 219 | 1.000 | F(c):=\{*\} | | |
| fong-spivak-seven-sketches_FO2288 | 219 | 1.000 | \mathbb{R}^{+} | | |
| fong-spivak-seven-sketches_FO2289 | 219 | 1.000 | \ell: A \rightarrow C | | |
| fong-spivak-seven-sketches_FO2290 | 219 | 1.000 | V=\{1,2\}, A=\{e\}, s(e)=1 | | |
| fong-spivak-seven-sketches_FO2291 | 219 | 0.562 | t(e)=2 | | |
| fong-spivak-seven-sketches_FO2292 | 219 | 0.562 | \ell(e)=3 \Omega | | |
| fong-spivak-seven-sketches_FO2293 | 219 | 1.000 | 3 \Omega | | |
| fong-spivak-seven-sketches_FO2294 | 220 | 1.000 | V=\{1,2\}, A=\{e\} | | |
| fong-spivak-seven-sketches_FO2295 | 220 | 1.000 | s(e)=2, t(e)=1 | | |
| fong-spivak-seven-sketches_FO2296 | 220 | 1.000 | \ell(e)=3 F | | |
| fong-spivak-seven-sketches_FO2297 | 220 | 0.983 | V, A, s, t, \ell | | |
| fong-spivak-seven-sketches_FO2298 | 220 | 0.951 | \psi | | |
| fong-spivak-seven-sketches_FO2299 | 220 | 1.000 | f: V \rightarrow V^{\prime} | | |
| fong-spivak-seven-sketches_FO2300 | 220 | 0.693 | c \in \operatorname{Circ}(\underline{4}) | | |
| fong-spivak-seven-sketches_FO2301 | 220 | 1.000 | f: \underline{4} \rightarrow \underline{3} | | |
| fong-spivak-seven-sketches_FO2302 | 220 | 1.000 | \operatorname{Circ}(f)(c) | | |
| fong-spivak-seven-sketches_FO2303 | 220 | 1.000 | \psi_{V, V^{\prime}} | | |
| fong-spivak-seven-sketches_FO2304 | 220 | 1.000 | V, V^{\prime} | | |
| fong-spivak-seven-sketches_FO2305 | 220 | 1.000 | V+V^{\prime} | | |
| fong-spivak-seven-sketches_FO2306 | 220 | 1.000 | \psi_{\underline{2}, \underline{2}}(b, s) \in \operatorname{Circ}(\underline{2}+\underline{2})=\operatorname{Circ}(\underline{4}) | | |
| fong-spivak-seven-sketches_FO2307 | 221 | 0.339 | ( | | |
| fong-spivak-seven-sketches_FO2308 | 221 | 0.339 | \psi):( | | |
| fong-spivak-seven-sketches_FO2309 | 221 | 0.339 | ) \rightarrow( | | |
| fong-spivak-seven-sketches_FO2310 | 221 | 0.945 | { }_{\text {Circ }} | | |
| fong-spivak-seven-sketches_FO2311 | 221 | 0.147 | \underline{2} | | |
| fong-spivak-seven-sketches_FO2312 | 221 | 0.147 | _{\text {Circ }} | | |
| fong-spivak-seven-sketches_FO2313 | 221 | 0.999 | \underline{m} \rightarrow \underline{p} \leftarrow \underline{n} | | |
| fong-spivak-seven-sketches_FO2314 | 221 | 0.999 | \operatorname{Circ}(\underline{p}) | | |
| fong-spivak-seven-sketches_FO2315 | 221 | 1.000 | \underline{1} \xrightarrow{i_{1}} \underline{2} \stackrel{i_{2}}{\leftarrow} \underline{1} | | |
| fong-spivak-seven-sketches_FO2316 | 221 | 1.000 | i_{1}(1)=1 | | |
| fong-spivak-seven-sketches_FO2317 | 221 | 1.000 | i_{2}(1)=2 | | |
| fong-spivak-seven-sketches_FO2318 | 221 | 0.760 | (V, A, s, t, \ell) \in \operatorname{Circ}(\underline{2}) | | |
| fong-spivak-seven-sketches_FO2319 | 221 | 0.578 | \underline{1} \rightarrow \underline{1} | | |
| fong-spivak-seven-sketches_FO2320 | 221 | 1.000 | \operatorname{Circ}\left(\left[\iota_{N}, \iota_{P}\right]\right)\left(\psi_{N, P}(s, t)\right) | | |
| fong-spivak-seven-sketches_FO2321 | 222 | 0.962 | \underline{1}=\{\circ\} | | |
| fong-spivak-seven-sketches_FO2322 | 222 | 0.307 | \psi_{2, \underline{2}} | | |
| fong-spivak-seven-sketches_FO2323 | 222 | 0.834 | \operatorname{Circ}(\underline{4}) | | |
| fong-spivak-seven-sketches_FO2324 | 222 | 0.834 | \psi_{N, P}(s, t) | | |
| fong-spivak-seven-sketches_FO2325 | 222 | 0.834 | \left[\iota_{N}, \iota_{P}\right] | | |
| fong-spivak-seven-sketches_FO2326 | 222 | 0.963 | \operatorname{Cospan}_{\text {Circ }} | | |
| fong-spivak-seven-sketches_FO2327 | 222 | 0.930 | A \rightarrow M \leftarrow B | | |
| fong-spivak-seven-sketches_FO2328 | 222 | 0.930 | C \rightarrow N \leftarrow D | | |
| fong-spivak-seven-sketches_FO2329 | 222 | 1.000 | A+C \rightarrow M+N \leftarrow B+D | | |
| fong-spivak-seven-sketches_FO2330 | 222 | 0.997 | \psi_{M, N}: \operatorname{Circ}(M) \times \operatorname{Circ}(N) \rightarrow \operatorname{Circ}(M+N) | | |
| fong-spivak-seven-sketches_FO2331 | 223 | 1.000 | \eta: 0 \rightarrow 2 | | |
| fong-spivak-seven-sketches_FO2332 | 223 | 0.915 | \eta: 2 \rightarrow 0 | | |
| fong-spivak-seven-sketches_FO2333 | 223 | 0.967 | (\underline{1}, \varnothing,!,!,!) \in \operatorname{Circ}(\underline{1}) .{ }^{6} | | |
| fong-spivak-seven-sketches_FO2334 | 223 | 0.782 | \eta ; x ; \epsilon | | |
| fong-spivak-seven-sketches_FO2335 | 223 | 0.782 | \underline{0} \rightarrow \underline{0} | | |
| fong-spivak-seven-sketches_FO2336 | 223 | 0.574 | \mathcal{O} | | |
| fong-spivak-seven-sketches_FO2337 | 225 | 1.000 | f, g, h, k | | |
| fong-spivak-seven-sketches_FO2338 | 225 | 0.997 | \mathcal{M} \rightarrow | | |
| fong-spivak-seven-sketches_FO2339 | 225 | 0.992 | _{\text {FinSet }} \rightarrow | | |
| fong-spivak-seven-sketches_FO2340 | 225 | 0.725 | t_{1}, \ldots, t_{n}, t | | |
| fong-spivak-seven-sketches_FO2341 | 225 | 0.725 | \mathcal{O}\left(t_{1}, \ldots, t_{n} ; t\right) | | |
| fong-spivak-seven-sketches_FO2342 | 225 | 0.623 | \left(t_{1}, \ldots, t_{n} ; t\right) | | |
| fong-spivak-seven-sketches_FO2343 | 225 | 0.988 | \left(s_{1}, \ldots, s_{m}, t_{i}\right) | | |
| fong-spivak-seven-sketches_FO2344 | 225 | 0.988 | \left(t_{1}, \ldots, t_{n}, t\right) | | |
| fong-spivak-seven-sketches_FO2345 | 225 | 0.739 | \mathrm{id}_{t} \in O(t ; t) | | |
| fong-spivak-seven-sketches_FO2346 | 225 | 1.000 | m+n-1 | | |
| fong-spivak-seven-sketches_FO2347 | 225 | 1.000 | n-1 | | |
| fong-spivak-seven-sketches_FO2348 | 226 | 0.943 | \circ_{i} | | |
| fong-spivak-seven-sketches_FO2349 | 226 | 0.943 | f \circ_{i} g \in \mathcal{O}(m+n-1) | | |
| fong-spivak-seven-sketches_FO2350 | 226 | 0.835 | X_{1} \times \cdots \times X_{n} \rightarrow Y | | |
| fong-spivak-seven-sketches_FO2351 | 226 | 0.835 | X_{1}, \ldots, X_{n} ; Y | | |
| fong-spivak-seven-sketches_FO2352 | 227 | 0.985 | f \in \operatorname{Set}\left(W_{1}, \ldots, W_{m} ; X_{i}\right), g \in \operatorname{Set}\left(X_{1}, \ldots, X_{n} ; Y\right) | | |
| fong-spivak-seven-sketches_FO2353 | 227 | 0.985 | g \circ_{i} f | | |
| fong-spivak-seven-sketches_FO2354 | 227 | 0.994 | \operatorname{id}_{X} \in \operatorname{Set}(X ; X) | | |
| fong-spivak-seven-sketches_FO2355 | 227 | 0.994 | \operatorname{id}_{X}: X \rightarrow X | | |
| fong-spivak-seven-sketches_FO2356 | 227 | 1.000 | a \in \mathbb{N} | | |
| fong-spivak-seven-sketches_FO2357 | 227 | 0.998 | \underline{a_{1}}+\cdots+\underline{a_{n}} \rightarrow \underline{p} \leftarrow \underline{b} | | |
| fong-spivak-seven-sketches_FO2358 | 227 | 0.998 | \left(a_{1}, \ldots, a_{n} ; b\right) | | |
| fong-spivak-seven-sketches_FO2359 | 227 | 0.180 | \operatorname{id}_{a} \in \operatorname{Set}(a ; a) | | |
| fong-spivak-seven-sketches_FO2360 | 227 | 0.180 | \operatorname{cospan} \underline{a} \xrightarrow{\mathrm{id}_{\underline{a}}} \underline{a} \stackrel{\mathrm{id}_{\underline{a}}}{\text { a }} \underline{a} | | |
| fong-spivak-seven-sketches_FO2361 | 227 | 0.894 | _{\text {FinSet }}, 0,+ | | |
| fong-spivak-seven-sketches_FO2362 | 227 | 0.844 | \underline{3}, \underline{3}, \underline{4}, \underline{2} ; \underline{3} | | |
| fong-spivak-seven-sketches_FO2363 | 227 | 0.809 | 3,3,4,2 | | |
| fong-spivak-seven-sketches_FO2364 | 227 | 1.000 | \underline{3}+\underline{3}+\underline{4}+\underline{2} \xrightarrow{a} \underline{p} \stackrel{b}{\leftarrow} \underline{3} | | |
| fong-spivak-seven-sketches_FO2365 | 227 | 1.000 | \underline{p} | | |
| fong-spivak-seven-sketches_FO2366 | 227 | 0.590 | \underline{7} | | |
| fong-spivak-seven-sketches_FO2367 | 228 | 0.986 | f \in \operatorname{Cospan}(2,2 ; 2) | | |
| fong-spivak-seven-sketches_FO2368 | 228 | 1.000 | g \in \operatorname{Cospan}(2,2,2 ; 0) | | |
| fong-spivak-seven-sketches_FO2369 | 228 | 1.000 | g \circ_{1} f | | |
| fong-spivak-seven-sketches_FO2370 | 228 | 0.461 | \mathcal{O}_{\mathcal{C}} | | |
| fong-spivak-seven-sketches_FO2371 | 228 | 0.535 | C_{1} \otimes \cdots \otimes C_{n} \rightarrow D | | |
| fong-spivak-seven-sketches_FO2372 | 228 | 0.535 | C_{1}, \ldots, C_{n} ; D | | |
| fong-spivak-seven-sketches_FO2373 | 228 | 0.639 | \operatorname{id}_{a} \in \mathcal{O}_{\mathcal{C}}(a ; a) | | |
| fong-spivak-seven-sketches_FO2374 | 228 | 0.639 | \operatorname{id}_{a} | | |
| fong-spivak-seven-sketches_FO2375 | 229 | 1.000 | F: \mathcal{O} \rightarrow \mathcal{P} | | |
| fong-spivak-seven-sketches_FO2376 | 229 | 1.000 | f: T \rightarrow U | | |
| fong-spivak-seven-sketches_FO2377 | 229 | 1.000 | F: \mathcal{O} \rightarrow | | |
| fong-spivak-seven-sketches_FO2378 | 229 | 0.998 | F(t) | | |
| fong-spivak-seven-sketches_FO2379 | 229 | 1.000 | f \in \mathcal{O}\left(t_{1}, \ldots, t_{n} ; t\right) | | |
| fong-spivak-seven-sketches_FO2380 | 229 | 1.000 | F(f) | | |
| fong-spivak-seven-sketches_FO2381 | 229 | 1.000 | F\left(t_{i}\right) | | |
| fong-spivak-seven-sketches_FO2382 | 229 | 1.000 | t_{1}, \ldots, t_{n} | | |
| fong-spivak-seven-sketches_FO2383 | 229 | 0.854 | T=\operatorname{Ob}( | | |
| fong-spivak-seven-sketches_FO2384 | 229 | 1.000 | \operatorname{Circ}(t) | | |
| fong-spivak-seven-sketches_FO2385 | 229 | 0.967 | \operatorname{Circ}(\varphi) \in \operatorname{Set}(\operatorname{Circ}(2), \operatorname{Circ}(2), \operatorname{Circ}(2) ; \operatorname{Circ}(0)) | | |
| fong-spivak-seven-sketches_FO2386 | 233 | 1.000 | \in \mathbb{R} | | |
| fong-spivak-seven-sketches_FO2387 | 233 | 1.000 | S^{\prime} | | |
| fong-spivak-seven-sketches_FO2388 | 234 | 1.000 | S \geq | | |
| fong-spivak-seven-sketches_FO2389 | 234 | 1.000 | S< | | |
| fong-spivak-seven-sketches_FO2390 | 235 | 1.000 | A= | | |
| fong-spivak-seven-sketches_FO2391 | 236 | 1.000 | [a, b] | | |
| fong-spivak-seven-sketches_FO2392 | 236 | 1.000 | [a, b] \rightarrow \mathbb{R} | | |
| fong-spivak-seven-sketches_FO2393 | 236 | 0.809 | \left\{\left(S^{\prime}, A\right) \mid E q\right. | | |
| fong-spivak-seven-sketches_FO2394 | 236 | 0.809 | \left(S^{\prime}, A\right) | | |
| fong-spivak-seven-sketches_FO2395 | 236 | 0.982 | \in P | | |
| fong-spivak-seven-sketches_FO2396 | 237 | 0.994 | \mathcal{E} | | |
| fong-spivak-seven-sketches_FO2397 | 237 | 1.000 | 1 \xrightarrow{\text { true }} \Omega | | |
| fong-spivak-seven-sketches_FO2398 | 237 | 0.736 | \mathcal{E}= | | |
| fong-spivak-seven-sketches_FO2399 | 237 | 0.999 | (B, C, E, F) | | |
| fong-spivak-seven-sketches_FO2400 | 237 | 0.939 | (A, B, D, E) | | |
| fong-spivak-seven-sketches_FO2401 | 237 | 0.939 | (A, C, D, F) | | |
| fong-spivak-seven-sketches_FO2402 | 237 | 0.980 | \left(B, C, B^{\prime}, C^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO2403 | 237 | 0.574 | \left(A, B, A^{\prime}, B^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO2404 | 237 | 0.574 | \left(A, C, A^{\prime}, C^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO2405 | 238 | 1.000 | i: B^{\prime} \rightarrow B | | |
| fong-spivak-seven-sketches_FO2406 | 238 | 1.000 | i^{\prime} | | |
| fong-spivak-seven-sketches_FO2407 | 238 | 1.000 | f^{\prime} | | |
| fong-spivak-seven-sketches_FO2408 | 239 | 1.000 | f: C \rightarrow D | | |
| fong-spivak-seven-sketches_FO2409 | 239 | 1.000 | \operatorname{im}(f) | | |
| fong-spivak-seven-sketches_FO2410 | 239 | 0.865 | C \rightarrow \operatorname{im}(f) | | |
| fong-spivak-seven-sketches_FO2411 | 239 | 0.865 | \operatorname{im}(f) \succ D | | |
| fong-spivak-seven-sketches_FO2412 | 239 | 1.000 | \operatorname{im}(f)= | | |
| fong-spivak-seven-sketches_FO2413 | 239 | 1.000 | \{f(c) \mid c \in C\} | | |
| fong-spivak-seven-sketches_FO2414 | 239 | 1.000 | f: \underline{3} \rightarrow \underline{3} | | |
| fong-spivak-seven-sketches_FO2415 | 239 | 1.000 | f=(e ; m) | | |
| fong-spivak-seven-sketches_FO2416 | 239 | 1.000 | C, D \in \mathcal{C} | | |
| fong-spivak-seven-sketches_FO2417 | 239 | 1.000 | D^{C} \in \mathcal{C} | | |
| fong-spivak-seven-sketches_FO2418 | 239 | 0.998 | A \in \mathcal{C} | | |
| fong-spivak-seven-sketches_FO2419 | 239 | 1.000 | f(-,-): A \times C \rightarrow D | | |
| fong-spivak-seven-sketches_FO2420 | 239 | 1.000 | f(a, c) | | |
| fong-spivak-seven-sketches_FO2421 | 239 | 0.993 | f(a,-): C \rightarrow D | | |
| fong-spivak-seven-sketches_FO2422 | 239 | 0.993 | D^{C} | | |
| fong-spivak-seven-sketches_FO2423 | 239 | 0.962 | v, w, x \in V | | |
| fong-spivak-seven-sketches_FO2424 | 239 | 1.000 | v \leq I | | |
| fong-spivak-seven-sketches_FO2425 | 239 | 1.000 | v \otimes w \leq v | | |
| fong-spivak-seven-sketches_FO2426 | 239 | 1.000 | v \otimes w \leq w | | |
| fong-spivak-seven-sketches_FO2427 | 239 | 1.000 | x \leq v | | |
| fong-spivak-seven-sketches_FO2428 | 239 | 1.000 | x \leq w | | |
| fong-spivak-seven-sketches_FO2429 | 239 | 1.000 | x \leq v \otimes w | | |
| fong-spivak-seven-sketches_FO2430 | 240 | 0.924 | X \mapsto Y | | |
| fong-spivak-seven-sketches_FO2431 | 240 | 0.925 | \Omega \in \mathcal{E} | | |
| fong-spivak-seven-sketches_FO2432 | 240 | 1.000 | 1 \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2433 | 240 | 0.899 | m: X \succ Y | | |
| fong-spivak-seven-sketches_FO2434 | 240 | 0.899 | \ulcorner m\urcorner: Y \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2435 | 240 | 1.000 | \ulcorner m\urcorner | | |
| fong-spivak-seven-sketches_FO2436 | 240 | 1.000 | p: Y \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2437 | 240 | 1.000 | Y \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2438 | 241 | 0.999 | 1 \rightarrow | | |
| fong-spivak-seven-sketches_FO2439 | 241 | 0.980 | m: X \hookrightarrow Y | | |
| fong-spivak-seven-sketches_FO2440 | 241 | 0.998 | \ulcorner m\urcorner: Y \rightarrow\{ | | |
| fong-spivak-seven-sketches_FO2441 | 241 | 1.000 | \ulcorner m\urcorner(y) | | |
| fong-spivak-seven-sketches_FO2442 | 241 | 1.000 | y \in X | | |
| fong-spivak-seven-sketches_FO2443 | 241 | 1.000 | p: Y \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO2444 | 241 | 1.000 | \{Y \mid p\} | | |
| fong-spivak-seven-sketches_FO2445 | 241 | 0.998 | X=\mathbb{N}=\{0,1,2, \ldots\} | | |
| fong-spivak-seven-sketches_FO2446 | 241 | 0.998 | Y=\mathbb{Z}=\{\ldots,-1,0,1,2, \ldots\} | | |
| fong-spivak-seven-sketches_FO2447 | 241 | 0.985 | \ulcorner m\urcorner: Y \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO2448 | 241 | 1.000 | \ulcorner m\urcorner(-5) \in \mathbb{B} | | |
| fong-spivak-seven-sketches_FO2449 | 241 | 1.000 | \ulcorner m\urcorner(0) \in \mathbb{B} | | |
| fong-spivak-seven-sketches_FO2450 | 241 | 0.915 | \operatorname{id}_{\mathbb{N}}: \mathbb{N} \rightarrow \mathbb{N} | | |
| fong-spivak-seven-sketches_FO2451 | 241 | 0.746 | \left\ulcorner\operatorname{id}_{\mathbb{N}}\right\urcorner: \mathbb{N} \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO2452 | 241 | 0.746 | \left\ulcorner\operatorname{id}_{\mathbb{N}}\right\urcorner | | |
| fong-spivak-seven-sketches_FO2453 | 242 | 0.990 | !_{\mathbb{N}}: \varnothing \rightarrow \mathbb{N} | | |
| fong-spivak-seven-sketches_FO2454 | 242 | 0.470 | \ulcorner!\sqrt{ }\urcorner: \mathbb{N} \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO2455 | 242 | 1.000 | 1 \rightarrow \mathbb{B} \times \mathbb{B} | | |
| fong-spivak-seven-sketches_FO2456 | 242 | 0.257 | { }^{\urcorner} | | |
| fong-spivak-seven-sketches_FO2457 | 242 | 0.257 | \mathbb{B} \times \mathbb{B} \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO2458 | 242 | 0.257 | \mathbb{B} \times \mathbb{B} | | |
| fong-spivak-seven-sketches_FO2459 | 242 | 0.990 | \ulcorner( | | |
| fong-spivak-seven-sketches_FO2460 | 242 | 0.990 | )\urcorner(P, Q) \in \mathbb{B} | | |
| fong-spivak-seven-sketches_FO2462 | 242 | 1.000 | P \wedge Q | | |
| fong-spivak-seven-sketches_FO2463 | 242 | 0.965 | )\urcorner=\wedge | | |
| fong-spivak-seven-sketches_FO2464 | 242 | 0.965 | \wedge: \mathbb{B} \times \mathbb{B} \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO2465 | 242 | 1.000 | x \wedge y:=\wedge(x, y) | | |
| fong-spivak-seven-sketches_FO2466 | 242 | 1.000 | \vee: \mathbb{B} \times \mathbb{B} \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO2468 | 242 | 1.000 | m: X \subseteq \mathbb{B} \times \mathbb{B} | | |
| fong-spivak-seven-sketches_FO2469 | 242 | 0.801 | \mathbb{B} \times\{ | | |
| fong-spivak-seven-sketches_FO2470 | 242 | 1.000 | \Omega \times \Omega | | |
| fong-spivak-seven-sketches_FO2471 | 242 | 1.000 | \Omega \times \Omega \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2472 | 243 | 1.000 | \neg: \mathbb{B} \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO2473 | 243 | 1.000 | P \Rightarrow Q | | |
| fong-spivak-seven-sketches_FO2474 | 243 | 1.000 | P=(P \wedge Q) | | |
| fong-spivak-seven-sketches_FO2475 | 243 | 1.000 | m: \mathbb{B} \times \mathbb{B} \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO2476 | 243 | 1.000 | E:=\{n \in \mathbb{N} \mid n | | |
| fong-spivak-seven-sketches_FO2477 | 243 | 1.000 | \}, P:=\{n \in \mathbb{N} \mid n | | |
| fong-spivak-seven-sketches_FO2478 | 243 | 1.000 | T:=\{n \in \mathbb{N} \mid n \geq 10\} | | |
| fong-spivak-seven-sketches_FO2479 | 243 | 1.000 | \mathbb{N} \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO2480 | 243 | 1.000 | \ulcorner E\urcorner(17) | | |
| fong-spivak-seven-sketches_FO2481 | 243 | 1.000 | \ulcorner P\urcorner(17) | | |
| fong-spivak-seven-sketches_FO2482 | 243 | 1.000 | \ulcorner T\urcorner(17) | | |
| fong-spivak-seven-sketches_FO2483 | 243 | 0.995 | (\ulcorner E\urcorner \wedge\ulcorner P\urcorner) \vee\ulcorner T\urcorner | | |
| fong-spivak-seven-sketches_FO2484 | 243 | 1.000 | \wedge, \vee, \Rightarrow: \Omega \times \Omega \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2485 | 243 | 1.000 | \neg: \Omega \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2486 | 244 | 1.000 | \mathbb{R}^{2} | | |
| fong-spivak-seven-sketches_FO2487 | 244 | 0.729 | \mathcal{C}^{\mathrm{op}} \rightarrow | | |
| fong-spivak-seven-sketches_FO2488 | 244 | 0.988 | P(c) | | |
| fong-spivak-seven-sketches_FO2489 | 244 | 1.000 | P(f): P(c) \rightarrow P\left(c^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO2490 | 244 | 1.000 | s \in P(c) | | |
| fong-spivak-seven-sketches_FO2491 | 244 | 1.000 | P(f)(s) \in P\left(c^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO2492 | 244 | 1.000 | \left.s\right|_{f} | | |
| fong-spivak-seven-sketches_FO2493 | 245 | 1.000 | \alpha: P \rightarrow Q | | |
| fong-spivak-seven-sketches_FO2494 | 245 | 0.852 | { }^{\text {op }} \rightarrow | | |
| fong-spivak-seven-sketches_FO2495 | 245 | 0.532 | \mathbf{A r S h p}^{\text {op }} | | |
| fong-spivak-seven-sketches_FO2496 | 245 | 0.532 | { }^{\text {op }} | | |
| fong-spivak-seven-sketches_FO2497 | 246 | 1.000 | \mathrm{P}(X)=\{U \subseteq X\} | | |
| fong-spivak-seven-sketches_FO2498 | 246 | 0.672 | \mathbf{O p} \subseteq \mathbf{P}(X) | | |
| fong-spivak-seven-sketches_FO2499 | 246 | 1.000 | X \subseteq X | | |
| fong-spivak-seven-sketches_FO2500 | 246 | 1.000 | X \in \mathbf{O p} | | |
| fong-spivak-seven-sketches_FO2501 | 246 | 0.871 | U, V \in \mathbf{O p} | | |
| fong-spivak-seven-sketches_FO2502 | 246 | 0.871 | (U \cap V) \in \mathbf{O p} | | |
| fong-spivak-seven-sketches_FO2503 | 246 | 0.743 | U_{i} \in \mathbf{O} \mathbf{p} | | |
| fong-spivak-seven-sketches_FO2504 | 246 | 0.999 | \left(\bigcup_{i \in I} U_{i}\right) \in \mathbf{O p} | | |
| fong-spivak-seven-sketches_FO2505 | 246 | 1.000 | I=\varnothing | | |
| fong-spivak-seven-sketches_FO2506 | 246 | 1.000 | \varnothing \in \mathbf{O p} | | |
| fong-spivak-seven-sketches_FO2507 | 246 | 1.000 | U=\bigcup_{i \in I} U_{i} | | |
| fong-spivak-seven-sketches_FO2508 | 246 | 1.000 | \left(U_{i}\right)_{i \in I} | | |
| fong-spivak-seven-sketches_FO2509 | 246 | 0.630 | (X, \mathbf{O p}) | | |
| fong-spivak-seven-sketches_FO2510 | 246 | 0.630 | \mathbf{O p} | | |
| fong-spivak-seven-sketches_FO2511 | 246 | 0.914 | X, \mathbf{O p}_{X} | | |
| fong-spivak-seven-sketches_FO2512 | 246 | 0.914 | Y, \mathbf{O} \mathbf{p}_{Y} | | |
| fong-spivak-seven-sketches_FO2513 | 246 | 0.650 | U \in \mathbf{O p}_{Y} | | |
| fong-spivak-seven-sketches_FO2514 | 246 | 0.650 | f^{-1}(U) | | |
| fong-spivak-seven-sketches_FO2515 | 246 | 0.650 | \mathbf{O p}_{X} | | |
| fong-spivak-seven-sketches_FO2516 | 246 | 1.000 | \epsilon \in \mathbb{R} | | |
| fong-spivak-seven-sketches_FO2517 | 246 | 1.000 | \epsilon>0 | | |
| fong-spivak-seven-sketches_FO2518 | 246 | 1.000 | p=(x, y) \in \mathbb{R}^{2} | | |
| fong-spivak-seven-sketches_FO2519 | 246 | 1.000 | U \in \mathbf{O p} | | |
| fong-spivak-seven-sketches_FO2520 | 247 | 0.999 | B(x, y ; \epsilon) | | |
| fong-spivak-seven-sketches_FO2521 | 247 | 0.974 | U \subseteq \mathbb{R}^{2} | | |
| fong-spivak-seven-sketches_FO2522 | 247 | 0.974 | \in U | | |
| fong-spivak-seven-sketches_FO2523 | 247 | 1.000 | B(x, y ; \epsilon) \subseteq U | | |
| fong-spivak-seven-sketches_FO2524 | 247 | 1.000 | p=(x, y) \in U | | |
| fong-spivak-seven-sketches_FO2525 | 247 | 1.000 | d\left(x_{1}, x_{2}\right):=\left|x_{1}-x_{2}\right| | | |
| fong-spivak-seven-sketches_FO2526 | 247 | 1.000 | x \in \mathbb{R} | | |
| fong-spivak-seven-sketches_FO2527 | 247 | 1.000 | B(x, \epsilon) | | |
| fong-spivak-seven-sketches_FO2528 | 247 | 1.000 | U \subseteq \mathbb{R} | | |
| fong-spivak-seven-sketches_FO2529 | 247 | 1.000 | U_{1}, U_{2} | | |
| fong-spivak-seven-sketches_FO2530 | 247 | 1.000 | \left(U_{i}\right)_{i \in\{1,2\}} | | |
| fong-spivak-seven-sketches_FO2531 | 247 | 0.803 | \mathbf{O p}_{\text {crse }}=(\varnothing, X) | | |
| fong-spivak-seven-sketches_FO2532 | 247 | 0.592 | \mathbf{O p}_{\text {fine }}=\mathrm{P}(X) | | |
| fong-spivak-seven-sketches_FO2533 | 247 | 0.592 | (X, \mathrm{P}(X)) | | |
| fong-spivak-seven-sketches_FO2534 | 247 | 0.995 | \mathbf{O p}_{\text {crse }} | | |
| fong-spivak-seven-sketches_FO2535 | 247 | 0.998 | \mathbf{O p}_{\text {fine }} | | |
| fong-spivak-seven-sketches_FO2536 | 247 | 0.762 | \left(Y, \mathbf{O p}_{Y}\right) | | |
| fong-spivak-seven-sketches_FO2537 | 247 | 1.000 | X \rightarrow Y | | |
| fong-spivak-seven-sketches_FO2538 | 247 | 1.000 | X=\{1,2\} | | |
| fong-spivak-seven-sketches_FO2539 | 247 | 1.000 | d\left((x, y),\left(x^{\prime}, y^{\prime}\right)\right):=\sqrt{\left(x-x^{\prime}\right)^{2}+\left(y-y^{\prime}\right)^{2}} | | |
| fong-spivak-seven-sketches_FO2540 | 248 | 0.817 | \mathbf{O} \mathbf{p}_{\text {fine }} | | |
| fong-spivak-seven-sketches_FO2541 | 248 | 0.943 | \{1,2\}, \mathbf{O} \mathbf{p}_{1} | | |
| fong-spivak-seven-sketches_FO2542 | 248 | 0.943 | \{1,2\}, \mathbf{O} \mathbf{p}_{2} | | |
| fong-spivak-seven-sketches_FO2543 | 248 | 0.931 | (\mathbf{O p}, \subseteq) | | |
| fong-spivak-seven-sketches_FO2544 | 248 | 1.000 | U \subseteq U | | |
| fong-spivak-seven-sketches_FO2545 | 248 | 1.000 | U \subseteq V | | |
| fong-spivak-seven-sketches_FO2546 | 248 | 1.000 | V \subseteq W | | |
| fong-spivak-seven-sketches_FO2547 | 248 | 1.000 | U \subseteq W | | |
| fong-spivak-seven-sketches_FO2548 | 248 | 1.000 | U, V | | |
| fong-spivak-seven-sketches_FO2549 | 248 | 1.000 | \mathbf{O p}(U, V) | | |
| fong-spivak-seven-sketches_FO2550 | 248 | 1.000 | U \nsubseteq V | | |
| fong-spivak-seven-sketches_FO2551 | 248 | 0.761 | X, \mathbf{O p}_{1} | | |
| fong-spivak-seven-sketches_FO2552 | 248 | 0.686 | X, \mathbf{O p} | | |
| fong-spivak-seven-sketches_FO2553 | 248 | 0.686 | Y \subseteq X | | |
| fong-spivak-seven-sketches_FO2554 | 248 | 0.955 | \mathbf{O p}_{? \cap \gamma} | | |
| fong-spivak-seven-sketches_FO2555 | 248 | 0.955 | A \subseteq Y | | |
| fong-spivak-seven-sketches_FO2556 | 248 | 0.955 | A \in \mathbf{O p}_{? \cap Y} | | |
| fong-spivak-seven-sketches_FO2557 | 248 | 1.000 | B \in \mathbf{O p} | | |
| fong-spivak-seven-sketches_FO2558 | 248 | 1.000 | A=B \cap Y | | |
| fong-spivak-seven-sketches_FO2559 | 248 | 1.000 | Y \in \mathbf{O p}_{? \cap Y} | | |
| fong-spivak-seven-sketches_FO2560 | 248 | 1.000 | \mathbf{O p}_{? \cap Y} | | |
| fong-spivak-seven-sketches_FO2561 | 248 | 1.000 | Y \hookrightarrow X | | |
| fong-spivak-seven-sketches_FO2562 | 248 | 0.555 | \mathbf{O p}, \subseteq | | |
| fong-spivak-seven-sketches_FO2563 | 248 | 0.983 | (\mathbf{O p}, \subseteq, X, \cap) | | |
| fong-spivak-seven-sketches_FO2564 | 248 | 0.638 | (X, \mathrm{Op}) | | |
| fong-spivak-seven-sketches_FO2565 | 248 | 0.997 | U_{i} \subseteq X | | |
| fong-spivak-seven-sketches_FO2566 | 248 | 0.997 | \left(\bigcup_{i \in I} U_{i}\right) \cap V= | | |
| fong-spivak-seven-sketches_FO2567 | 248 | 0.798 | \bigcup_{i \in I}\left(U_{i} \cap V\right) | | |
| fong-spivak-seven-sketches_FO2568 | 248 | 0.798 | U, V, W \subseteq X | | |
| fong-spivak-seven-sketches_FO2569 | 248 | 0.798 | (U \cap W) \cap(V \cap W)=(U \cap V) \cap W | | |
| fong-spivak-seven-sketches_FO2570 | 249 | 0.770 | P: \mathbf{O p}^{\text {op }} \rightarrow | | |
| fong-spivak-seven-sketches_FO2571 | 249 | 0.770 | P(U) | | |
| fong-spivak-seven-sketches_FO2572 | 249 | 1.000 | V \subseteq U | | |
| fong-spivak-seven-sketches_FO2573 | 249 | 1.000 | P(U) \rightarrow P(V) | | |
| fong-spivak-seven-sketches_FO2574 | 249 | 1.000 | s \in P(U) | | |
| fong-spivak-seven-sketches_FO2575 | 249 | 1.000 | \left.s\right|_{V} | | |
| fong-spivak-seven-sketches_FO2576 | 249 | 0.613 | P: \mathbf{O p}^{\mathrm{op}} \rightarrow | | |
| fong-spivak-seven-sketches_FO2577 | 249 | 1.000 | U_{i} \in \mathbf{O p} | | |
| fong-spivak-seven-sketches_FO2578 | 249 | 1.000 | \left(s_{i}\right)_{i \in I} | | |
| fong-spivak-seven-sketches_FO2579 | 249 | 1.000 | s_{i} \in P\left(U_{i}\right) | | |
| fong-spivak-seven-sketches_FO2580 | 249 | 1.000 | i \in I | | |
| fong-spivak-seven-sketches_FO2581 | 249 | 1.000 | i, j \in I | | |
| fong-spivak-seven-sketches_FO2582 | 249 | 1.000 | \left.s\right|_{U_{i}}=s_{i} | | |
| fong-spivak-seven-sketches_FO2583 | 249 | 0.992 | \boldsymbol{\operatorname { S h v }}(X, \mathbf{O p}) | | |
| fong-spivak-seven-sketches_FO2584 | 249 | 1.000 | U=\varnothing \subseteq X | | |
| fong-spivak-seven-sketches_FO2585 | 249 | 0.999 | P(\varnothing) \cong\{()\} | | |
| fong-spivak-seven-sketches_FO2586 | 249 | 0.999 | P(\varnothing) | | |
| fong-spivak-seven-sketches_FO2587 | 250 | 0.990 | f\left(a_{1}\right)=f\left(a_{2}\right)=a, f\left(b_{1}\right)=b | | |
| fong-spivak-seven-sketches_FO2588 | 250 | 0.999 | f^{-1}(y) \subseteq X | | |
| fong-spivak-seven-sketches_FO2589 | 250 | 1.000 | f^{\prime}: X \rightarrow Y | | |
| fong-spivak-seven-sketches_FO2590 | 250 | 0.886 | \operatorname{Sec}_{f}: \mathbf{O p}(Y)^{\text {op }} \rightarrow | | |
| fong-spivak-seven-sketches_FO2591 | 250 | 1.000 | \operatorname{Sec}_{f} | | |
| fong-spivak-seven-sketches_FO2592 | 250 | 1.000 | U \subseteq Y | | |
| fong-spivak-seven-sketches_FO2593 | 250 | 1.000 | \operatorname{Sec}_{f}(U) | | |
| fong-spivak-seven-sketches_FO2594 | 250 | 1.000 | s \in \operatorname{Sec}_{f}(U) | | |
| fong-spivak-seven-sketches_FO2595 | 250 | 0.999 | U=\{a, b\} | | |
| fong-spivak-seven-sketches_FO2596 | 251 | 0.996 | V_{1}=\{a, b, c\} | | |
| fong-spivak-seven-sketches_FO2597 | 251 | 0.996 | \operatorname{Sec}_{f}\left(V_{1}\right) | | |
| fong-spivak-seven-sketches_FO2598 | 251 | 1.000 | V_{2}=\{a, b, c, d\} | | |
| fong-spivak-seven-sketches_FO2599 | 251 | 1.000 | \operatorname{Sec}_{f}\left(V_{2}\right) | | |
| fong-spivak-seven-sketches_FO2600 | 251 | 0.999 | V_{3}=\{a, b, d, e\} | | |
| fong-spivak-seven-sketches_FO2601 | 251 | 0.999 | \operatorname{Sec}_{f}\left(V_{3}\right) | | |
| fong-spivak-seven-sketches_FO2602 | 251 | 0.974 | \mathrm{Sec}_{f} | | |
| fong-spivak-seven-sketches_FO2603 | 251 | 1.000 | V=\{a\} | | |
| fong-spivak-seven-sketches_FO2604 | 251 | 1.000 | \operatorname{Sec}_{f}(\{a, b, c\}) | | |
| fong-spivak-seven-sketches_FO2605 | 251 | 1.000 | \operatorname{Sec}_{f}(\{a, c\}) | | |
| fong-spivak-seven-sketches_FO2606 | 251 | 1.000 | U_{1}=\{a, b\} | | |
| fong-spivak-seven-sketches_FO2607 | 251 | 1.000 | U_{2}=\{b, e\} | | |
| fong-spivak-seven-sketches_FO2608 | 251 | 1.000 | U=\{a, b, e\}=U_{1} \cup U_{2} | | |
| fong-spivak-seven-sketches_FO2609 | 251 | 1.000 | U_{1} | | |
| fong-spivak-seven-sketches_FO2610 | 251 | 1.000 | U_{2} | | |
| fong-spivak-seven-sketches_FO2611 | 251 | 1.000 | U_{1} \cap U_{2}=\{b\} | | |
| fong-spivak-seven-sketches_FO2612 | 252 | 1.000 | s_{1} \in \operatorname{Sec}_{f}\left(U_{1}\right) | | |
| fong-spivak-seven-sketches_FO2613 | 252 | 1.000 | s_{2} \in \operatorname{Sec}_{f}\left(U_{2}\right) | | |
| fong-spivak-seven-sketches_FO2614 | 252 | 1.000 | g_{1} | | |
| fong-spivak-seven-sketches_FO2615 | 252 | 1.000 | g_{2} | | |
| fong-spivak-seven-sketches_FO2616 | 252 | 1.000 | b_{2} | | |
| fong-spivak-seven-sketches_FO2617 | 252 | 1.000 | U=\{a, b, e\} | | |
| fong-spivak-seven-sketches_FO2618 | 252 | 0.968 | \left(s_{1}, s_{2}\right) | | |
| fong-spivak-seven-sketches_FO2619 | 252 | 0.968 | s \in \operatorname{Sec}_{f}\left(U_{1} \cup U_{2}\right) | | |
| fong-spivak-seven-sketches_FO2620 | 252 | 1.000 | \left.s\right|_{U_{1}}=s_{1} | | |
| fong-spivak-seven-sketches_FO2621 | 252 | 1.000 | \left.s\right|_{U_{2}}=s_{2} | | |
| fong-spivak-seven-sketches_FO2622 | 252 | 0.683 | h_{1} \in \operatorname{Sec}_{f}\left(U_{1}\right) | | |
| fong-spivak-seven-sketches_FO2623 | 252 | 0.683 | h_{2} \in \operatorname{Sec}_{f}\left(U_{2}\right) | | |
| fong-spivak-seven-sketches_FO2624 | 252 | 0.683 | d o | | |
| fong-spivak-seven-sketches_FO2625 | 252 | 0.975 | h \in \operatorname{Sec}_{f}\left(U_{1} \cup U_{2}\right) | | |
| fong-spivak-seven-sketches_FO2626 | 252 | 0.975 | \left.h\right|_{U_{1}}=h_{1} | | |
| fong-spivak-seven-sketches_FO2627 | 252 | 0.975 | \left.h\right|_{U_{2}}=h_{2} | | |
| fong-spivak-seven-sketches_FO2628 | 252 | 0.984 | f:\left(X, \mathbf{O p}_{X}\right) \rightarrow\left(Y, \mathbf{O p}_{Y}\right) | | |
| fong-spivak-seven-sketches_FO2629 | 252 | 0.901 | \operatorname{Sec}_{f}: \mathbf{O p}_{Y}^{\text {op }} \rightarrow | | |
| fong-spivak-seven-sketches_FO2630 | 253 | 0.623 | \mathbf{O p}_{Y} | | |
| fong-spivak-seven-sketches_FO2631 | 253 | 0.623 | g: U \rightarrow X | | |
| fong-spivak-seven-sketches_FO2632 | 253 | 0.996 | T M | | |
| fong-spivak-seven-sketches_FO2633 | 253 | 0.996 | (m, v) | | |
| fong-spivak-seven-sketches_FO2634 | 253 | 0.996 | m \in M | | |
| fong-spivak-seven-sketches_FO2635 | 253 | 1.000 | \pi: T M \rightarrow | | |
| fong-spivak-seven-sketches_FO2636 | 253 | 0.984 | (m, v) \mapsto m | | |
| fong-spivak-seven-sketches_FO2637 | 253 | 0.984 | \pi | | |
| fong-spivak-seven-sketches_FO2638 | 253 | 0.960 | \mathrm{Sec}_{\pi} | | |
| fong-spivak-seven-sketches_FO2639 | 253 | 0.960 | M^{\prime} | | |
| fong-spivak-seven-sketches_FO2640 | 253 | 1.000 | U \subseteq M | | |
| fong-spivak-seven-sketches_FO2641 | 253 | 1.000 | v \in \operatorname{Sec}_{\pi}(U) | | |
| fong-spivak-seven-sketches_FO2642 | 253 | 1.000 | v(u) | | |
| fong-spivak-seven-sketches_FO2643 | 253 | 1.000 | U \neq M | | |
| fong-spivak-seven-sketches_FO2644 | 254 | 1.000 | U \neq X | | |
| fong-spivak-seven-sketches_FO2645 | 254 | 1.000 | U=X | | |
| fong-spivak-seven-sketches_FO2646 | 254 | 0.988 | \boldsymbol{\operatorname { S h v }}(M) | | |
| fong-spivak-seven-sketches_FO2647 | 254 | 0.485 | X, \mathrm{Op} | | |
| fong-spivak-seven-sketches_FO2648 | 254 | 0.928 | \{1,2\}, \mathbf{O p}_{1} | | |
| fong-spivak-seven-sketches_FO2649 | 254 | 0.896 | X=\mathbf{1} | | |
| fong-spivak-seven-sketches_FO2650 | 255 | 1.000 | \wedge, \vee, \neg, \Rightarrow, \exists, \forall | | |
| fong-spivak-seven-sketches_FO2651 | 255 | 0.975 | \Omega=\mathbb{B} | | |
| fong-spivak-seven-sketches_FO2652 | 255 | 0.975 | \mathcal{E}=\mathbf{S h v}(X, \mathbf{O p}) | | |
| fong-spivak-seven-sketches_FO2653 | 255 | 0.854 | \Omega: \mathbf{O p}^{\mathrm{op}} \rightarrow | | |
| fong-spivak-seven-sketches_FO2654 | 255 | 0.854 | \Omega(U) | | |
| fong-spivak-seven-sketches_FO2655 | 255 | 1.000 | \Omega(U) \rightarrow \Omega(V) | | |
| fong-spivak-seven-sketches_FO2656 | 256 | 1.000 | X=\{1\} | | |
| fong-spivak-seven-sketches_FO2657 | 256 | 1.000 | W \subseteq V \subseteq U | | |
| fong-spivak-seven-sketches_FO2658 | 256 | 1.000 | \Omega(V) \rightarrow \Omega(W) | | |
| fong-spivak-seven-sketches_FO2659 | 256 | 1.000 | \Omega(U) \rightarrow \Omega(W) | | |
| fong-spivak-seven-sketches_FO2660 | 256 | 1.000 | V_{i} \in \Omega\left(U_{i}\right) | | |
| fong-spivak-seven-sketches_FO2661 | 256 | 1.000 | V_{i} \subseteq U_{i} | | |
| fong-spivak-seven-sketches_FO2662 | 256 | 1.000 | V_{i} \cap U_{j}=V_{j} \cap U_{i} | | |
| fong-spivak-seven-sketches_FO2663 | 256 | 0.995 | V:=\bigcup_{i \in I} V_{i} | | |
| fong-spivak-seven-sketches_FO2664 | 256 | 1.000 | \left(V_{i}\right)_{i \in I} | | |
| fong-spivak-seven-sketches_FO2665 | 256 | 1.000 | V \cap U_{j}=V_{j} | | |
| fong-spivak-seven-sketches_FO2666 | 256 | 1.000 | j \in I | | |
| fong-spivak-seven-sketches_FO2667 | 256 | 1.000 | \{1\} \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2668 | 256 | 0.623 | \{1\}: \mathbf{O p}^{\text {op }} \rightarrow | | |
| fong-spivak-seven-sketches_FO2669 | 256 | 1.000 | \{1\}=\{1\}(U) \rightarrow \Omega(U) | | |
| fong-spivak-seven-sketches_FO2670 | 256 | 0.997 | 1 \mapsto U | | |
| fong-spivak-seven-sketches_FO2671 | 256 | 1.000 | U=\varnothing | | |
| fong-spivak-seven-sketches_FO2672 | 257 | 1.000 | (V, V ; A) | | |
| fong-spivak-seven-sketches_FO2673 | 257 | 1.000 | i: H \subseteq G | | |
| fong-spivak-seven-sketches_FO2674 | 257 | 0.931 | \ulcorner H\urcorner | | |
| fong-spivak-seven-sketches_FO2675 | 257 | 0.931 | G \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2676 | 257 | 1.000 | G^{\prime \prime} | | |
| fong-spivak-seven-sketches_FO2677 | 257 | 0.589 | (V, 0 ; 0),(0, V ; 0),(V, V ; 0) | | |
| fong-spivak-seven-sketches_FO2678 | 257 | 0.589 | 0 ; 0 | | |
| fong-spivak-seven-sketches_FO2679 | 257 | 1.000 | H \subseteq G | | |
| fong-spivak-seven-sketches_FO2680 | 257 | 0.907 | \ulcorner H\urcorner: G \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2681 | 257 | 0.615 | U, V \in \Omega(X) | | |
| fong-spivak-seven-sketches_FO2682 | 257 | 1.000 | \cap | | |
| fong-spivak-seven-sketches_FO2683 | 257 | 1.000 | \cup | | |
| fong-spivak-seven-sketches_FO2684 | 257 | 1.000 | U \Rightarrow V | | |
| fong-spivak-seven-sketches_FO2685 | 257 | 1.000 | R \cap U \subseteq V | | |
| fong-spivak-seven-sketches_FO2686 | 257 | 0.668 | \neg U:= | | |
| fong-spivak-seven-sketches_FO2687 | 257 | 0.668 | U \Rightarrow | | |
| fong-spivak-seven-sketches_FO2688 | 257 | 1.000 | R \cap U=\varnothing | | |
| fong-spivak-seven-sketches_FO2689 | 257 | 1.000 | \neg U | | |
| fong-spivak-seven-sketches_FO2690 | 258 | 1.000 | X=\mathbb{R} | | |
| fong-spivak-seven-sketches_FO2691 | 258 | 1.000 | U=\{x \in \mathbb{R} \mid x<3\} | | |
| fong-spivak-seven-sketches_FO2692 | 258 | 1.000 | V=\{x \in \mathbb{R} \mid-4<x<4\} | | |
| fong-spivak-seven-sketches_FO2693 | 258 | 1.000 | U=(-\infty, 3) | | |
| fong-spivak-seven-sketches_FO2694 | 258 | 1.000 | V=(-4,4) | | |
| fong-spivak-seven-sketches_FO2695 | 258 | 1.000 | U \wedge V=(-4,3) | | |
| fong-spivak-seven-sketches_FO2696 | 258 | 1.000 | U \vee V=(-\infty, 4) | | |
| fong-spivak-seven-sketches_FO2697 | 258 | 1.000 | \neg U=(3, \infty) | | |
| fong-spivak-seven-sketches_FO2698 | 258 | 1.000 | \neg V=(-\infty,-4) \cup(4, \infty) | | |
| fong-spivak-seven-sketches_FO2699 | 258 | 0.999 | (U \Rightarrow V)=(-4, \infty) | | |
| fong-spivak-seven-sketches_FO2700 | 258 | 1.000 | (V \Rightarrow U)=U | | |
| fong-spivak-seven-sketches_FO2701 | 258 | 1.000 | U=\mathbb{R}-\{0\} | | |
| fong-spivak-seven-sketches_FO2702 | 258 | 1.000 | \neg \neg U | | |
| fong-spivak-seven-sketches_FO2703 | 258 | 1.000 | U \subseteq \neg \neg U | | |
| fong-spivak-seven-sketches_FO2704 | 258 | 1.000 | \neg \neg U \subseteq U | | |
| fong-spivak-seven-sketches_FO2705 | 258 | 0.942 | \wedge U)=U | | |
| fong-spivak-seven-sketches_FO2706 | 258 | 0.877 | \vee U)= | | |
| fong-spivak-seven-sketches_FO2707 | 258 | 0.877 | (U \Rightarrow | | |
| fong-spivak-seven-sketches_FO2708 | 258 | 0.559 | )= | | |
| fong-spivak-seven-sketches_FO2709 | 258 | 0.559 | \Rightarrow U)=U | | |
| fong-spivak-seven-sketches_FO2710 | 258 | 0.974 | \vee U)=U | | |
| fong-spivak-seven-sketches_FO2711 | 258 | 1.000 | \wedge U | | |
| fong-spivak-seven-sketches_FO2712 | 258 | 1.000 | \pi: E \rightarrow X | | |
| fong-spivak-seven-sketches_FO2713 | 258 | 0.978 | \operatorname{Sec}_{\pi} | | |
| fong-spivak-seven-sketches_FO2714 | 258 | 0.978 | i_{U}: U \subseteq X | | |
| fong-spivak-seven-sketches_FO2715 | 258 | 0.999 | s: U \rightarrow E | | |
| fong-spivak-seven-sketches_FO2716 | 258 | 0.999 | s ; \pi=i_{U} | | |
| fong-spivak-seven-sketches_FO2717 | 258 | 1.000 | \pi: T M \rightarrow M | | |
| fong-spivak-seven-sketches_FO2718 | 258 | 0.991 | \operatorname{VF}(U) | | |
| fong-spivak-seven-sketches_FO2719 | 258 | 0.996 | \mathrm{VF} \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2720 | 258 | 0.991 | \operatorname{Grad}(v) | | |
| fong-spivak-seven-sketches_FO2721 | 259 | 0.585 | p: S \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2722 | 259 | 0.585 | S \in \mathcal{E} | | |
| fong-spivak-seven-sketches_FO2723 | 259 | 1.000 | p(U): S(U) \rightarrow \Omega(U) | | |
| fong-spivak-seven-sketches_FO2724 | 259 | 0.972 | \operatorname{Bob} \in S(U) | | |
| fong-spivak-seven-sketches_FO2725 | 259 | 0.987 | p(\mathrm{Bob}) | | |
| fong-spivak-seven-sketches_FO2726 | 259 | 0.979 | s \in S(U) | | |
| fong-spivak-seven-sketches_FO2727 | 259 | 0.979 | s= | | |
| fong-spivak-seven-sketches_FO2728 | 259 | 1.000 | \{S \mid p\} \subseteq S | | |
| fong-spivak-seven-sketches_FO2729 | 259 | 1.000 | \left|\Omega^{E}\right|=\mathcal{E}(S, \Omega) | | |
| fong-spivak-seven-sketches_FO2730 | 259 | 0.917 | p, q: S \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2731 | 259 | 0.917 | p \leq^{S} q | | |
| fong-spivak-seven-sketches_FO2732 | 259 | 0.997 | p(s) \subseteq U | | |
| fong-spivak-seven-sketches_FO2733 | 259 | 0.997 | q(s) \subseteq U | | |
| fong-spivak-seven-sketches_FO2734 | 259 | 0.997 | p(s) \subseteq q(s) | | |
| fong-spivak-seven-sketches_FO2735 | 259 | 0.999 | \leq^{S} | | |
| fong-spivak-seven-sketches_FO2736 | 260 | 1.000 | S=1 | | |
| fong-spivak-seven-sketches_FO2737 | 260 | 1.000 | \left|\Omega^{S}\right| | | |
| fong-spivak-seven-sketches_FO2738 | 260 | 1.000 | |\Omega| | | |
| fong-spivak-seven-sketches_FO2739 | 260 | 1.000 | p \in|\Omega| | | |
| fong-spivak-seven-sketches_FO2740 | 260 | 1.000 | p: 1 \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2741 | 260 | 1.000 | p=q | | |
| fong-spivak-seven-sketches_FO2742 | 260 | 1.000 | U, V \subseteq X | | |
| fong-spivak-seven-sketches_FO2743 | 260 | 1.000 | U=V | | |
| fong-spivak-seven-sketches_FO2744 | 260 | 0.976 | S \in \mathbf{S h v}(X) | | |
| fong-spivak-seven-sketches_FO2745 | 260 | 1.000 | p(s) \vdash_{s: S} q(s) | | |
| fong-spivak-seven-sketches_FO2746 | 260 | 1.000 | \wedge, \vee, \Rightarrow, \neg | | |
| fong-spivak-seven-sketches_FO2747 | 260 | 1.000 | (p \wedge q): S \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2748 | 260 | 1.000 | (p \wedge q)(s):=p(s) \wedge q(s) | | |
| fong-spivak-seven-sketches_FO2749 | 260 | 1.000 | n+1 | | |
| fong-spivak-seven-sketches_FO2750 | 260 | 1.000 | S, T \in \mathbf{S h v}(X, \mathbf{O p}) | | |
| fong-spivak-seven-sketches_FO2751 | 260 | 1.000 | p: S \times | | |
| fong-spivak-seven-sketches_FO2752 | 260 | 1.000 | T \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2753 | 260 | 1.000 | t \in T(U) | | |
| fong-spivak-seven-sketches_FO2754 | 260 | 1.000 | p(s, t) | | |
| fong-spivak-seven-sketches_FO2755 | 260 | 0.876 | T, \ldots | | |
| fong-spivak-seven-sketches_FO2756 | 260 | 1.000 | t^{\prime \prime} | | |
| fong-spivak-seven-sketches_FO2757 | 260 | 1.000 | \exists(t: T) . p(s, t) | | |
| fong-spivak-seven-sketches_FO2758 | 261 | 1.000 | p: \mathbb{N} \times \mathbb{Z} \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO2759 | 261 | 0.999 | \forall(z: \mathbb{Z}) . p(n, z) | | |
| fong-spivak-seven-sketches_FO2760 | 261 | 1.000 | \exists(z: \mathbb{Z}) . p(n, z) | | |
| fong-spivak-seven-sketches_FO2761 | 261 | 0.999 | z \in \mathbb{Z} | | |
| fong-spivak-seven-sketches_FO2762 | 261 | 0.999 | \forall(n: \mathbb{N}) . p(n, z) | | |
| fong-spivak-seven-sketches_FO2763 | 261 | 0.998 | \exists(n: \mathbb{N}) . p(n, z) | | |
| fong-spivak-seven-sketches_FO2764 | 261 | 1.000 | \forall(t | | |
| fong-spivak-seven-sketches_FO2765 | 261 | 1.000 | S \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2766 | 261 | 1.000 | p: S \times T \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2767 | 261 | 0.997 | \forall(t: T) . p(s, t) | | |
| fong-spivak-seven-sketches_FO2768 | 261 | 1.000 | V \in \Omega(U) | | |
| fong-spivak-seven-sketches_FO2769 | 261 | 1.000 | p\left(\left.s\right|_{V}, t\right)=V | | |
| fong-spivak-seven-sketches_FO2770 | 261 | 1.000 | t \in T(V) | | |
| fong-spivak-seven-sketches_FO2771 | 261 | 0.926 | p(s, t)= | | |
| fong-spivak-seven-sketches_FO2772 | 261 | 1.000 | T(V) | | |
| fong-spivak-seven-sketches_FO2773 | 261 | 1.000 | p^{\prime}: S \rightarrow \Omega^{T} | | |
| fong-spivak-seven-sketches_FO2774 | 261 | 1.000 | S \times T \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2775 | 261 | 1.000 | { }^{T} | | |
| fong-spivak-seven-sketches_FO2776 | 261 | 0.982 | 1 \times T \xrightarrow{!} 1 \xrightarrow{\text { true }} \Omega | | |
| fong-spivak-seven-sketches_FO2777 | 261 | 1.000 | V=\bigcup_{i} V_{i} | | |
| fong-spivak-seven-sketches_FO2778 | 261 | 1.000 | V_{i} | | |
| fong-spivak-seven-sketches_FO2779 | 261 | 1.000 | t_{i} \in T\left(V_{i}\right) | | |
| fong-spivak-seven-sketches_FO2780 | 261 | 1.000 | p\left(\left.s\right|_{V_{i}}, t_{i}\right)=V_{i} | | |
| fong-spivak-seven-sketches_FO2781 | 261 | 1.000 | p(t) | | |
| fong-spivak-seven-sketches_FO2782 | 261 | 0.516 | U=U U_{i} | | |
| fong-spivak-seven-sketches_FO2783 | 261 | 0.516 | t_{i} \in T\left(U_{i}\right) | | |
| fong-spivak-seven-sketches_FO2784 | 262 | 0.951 | t^{\prime} | | |
| fong-spivak-seven-sketches_FO2785 | 262 | 1.000 | \{(s, t) \in S \times T \mid p(s, t)\} \subseteq S \times T | | |
| fong-spivak-seven-sketches_FO2786 | 262 | 0.982 | \pi_{S}: S \times T \rightarrow S | | |
| fong-spivak-seven-sketches_FO2787 | 262 | 0.983 | \left(\left|\Omega^{S}\right|, \leq^{S}\right) | | |
| fong-spivak-seven-sketches_FO2788 | 262 | 0.983 | \left|\Omega^{S}\right|=\mathcal{E}(S, \Omega) | | |
| fong-spivak-seven-sketches_FO2789 | 262 | 0.999 | \boldsymbol{\operatorname { S h v }}(X) | | |
| fong-spivak-seven-sketches_FO2790 | 262 | 0.999 | j: \Omega \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2791 | 262 | 1.000 | p, q \in \Omega(U) | | |
| fong-spivak-seven-sketches_FO2792 | 262 | 1.000 | (j ; j)(p) \leq j(p) | | |
| fong-spivak-seven-sketches_FO2793 | 262 | 1.000 | j(p \wedge q)=j(p) \wedge j(q) | | |
| fong-spivak-seven-sketches_FO2794 | 262 | 1.000 | p \in \Omega(U) | | |
| fong-spivak-seven-sketches_FO2795 | 262 | 1.000 | q \in \Omega(U) | | |
| fong-spivak-seven-sketches_FO2796 | 262 | 1.000 | j(j(q)) \leq j(q) | | |
| fong-spivak-seven-sketches_FO2797 | 262 | 1.000 | j(j(q))=j(q) | | |
| fong-spivak-seven-sketches_FO2798 | 262 | 1.000 | \Omega \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2799 | 262 | 1.000 | p \vee q | | |
| fong-spivak-seven-sketches_FO2800 | 262 | 0.997 | (q \Rightarrow p) \Rightarrow p | | |
| fong-spivak-seven-sketches_FO2801 | 263 | 1.000 | p(s) | | |
| fong-spivak-seven-sketches_FO2802 | 263 | 1.000 | j(p(s)) | | |
| fong-spivak-seven-sketches_FO2803 | 263 | 1.000 | p(s) \leq j(p(s)) | | |
| fong-spivak-seven-sketches_FO2804 | 263 | 1.000 | j(j(p(s)))=j(p(s)) | | |
| fong-spivak-seven-sketches_FO2805 | 263 | 1.000 | q: S \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2806 | 264 | 0.500 | s \in S(c) | | |
| fong-spivak-seven-sketches_FO2807 | 264 | 1.000 | t \in T(c) | | |
| fong-spivak-seven-sketches_FO2808 | 264 | 1.000 | f(s)=t | | |
| fong-spivak-seven-sketches_FO2809 | 264 | 1.000 | \left(U_{i} \subseteq U\right)_{i \in I} | | |
| fong-spivak-seven-sketches_FO2810 | 264 | 1.000 | f\left(t_{i}\right)=\left.s\right|_{U_{i}} | | |
| fong-spivak-seven-sketches_FO2811 | 264 | 1.000 | U_{i} | | |
| fong-spivak-seven-sketches_FO2812 | 264 | 0.912 | \boldsymbol{\operatorname { S h v }}(\llbracket \mathbb{R}) | | |
| fong-spivak-seven-sketches_FO2813 | 264 | 0.773 | \mathbb{I} \mathbb{R} | | |
| fong-spivak-seven-sketches_FO2814 | 264 | 1.000 | a<b | | |
| fong-spivak-seven-sketches_FO2815 | 264 | 1.000 | o_{[a, b]} | | |
| fong-spivak-seven-sketches_FO2816 | 264 | 1.000 | o_{[a, b]}:=\{[d, u] \in \mathbb{R} \mid a<d \leq u<b\} | | |
| fong-spivak-seven-sketches_FO2817 | 264 | 1.000 | o_{[0,5]} | | |
| fong-spivak-seven-sketches_FO2818 | 264 | 1.000 | [d, u] | | |
| fong-spivak-seven-sketches_FO2819 | 264 | 1.000 | \{x \in \mathbb{R} \mid 0<x<5\} | | |
| fong-spivak-seven-sketches_FO2820 | 264 | 1.000 | o_{[4,8]} | | |
| fong-spivak-seven-sketches_FO2821 | 264 | 0.965 | o_{[0,5]} \cup o_{[4,8]} \neq o_{[0,8]} | | |
| fong-spivak-seven-sketches_FO2822 | 265 | 1.000 | [2,6] \in o_{[0,8]} | | |
| fong-spivak-seven-sketches_FO2823 | 265 | 0.996 | [2,6] \notin o_{[0,5]} \cup o_{[4,8]} | | |
| fong-spivak-seven-sketches_FO2824 | 265 | 0.964 | \mathbb{R} \subseteq \mathbb{I} \mathbb{R} | | |
| fong-spivak-seven-sketches_FO2825 | 265 | 1.000 | U \cap \mathbb{R} | | |
| fong-spivak-seven-sketches_FO2826 | 265 | 0.280 | \mathbf{B T}=\mathbf{S h v}(\mathbb{I} \mathbb{R}, \mathbf{O p}) | | |
| fong-spivak-seven-sketches_FO2827 | 265 | 0.591 | \mathbb{I R}, \mathbf{O p} | | |
| fong-spivak-seven-sketches_FO2828 | 265 | 0.864 | S: \mathbf{O p}^{\text {op }} \rightarrow | | |
| fong-spivak-seven-sketches_FO2829 | 265 | 0.864 | S(U) | | |
| fong-spivak-seven-sketches_FO2830 | 265 | 0.997 | \left(s_{i}\right) | | |
| fong-spivak-seven-sketches_FO2831 | 265 | 0.835 | U \subseteq \mathbb{I} \mathbb{R} | | |
| fong-spivak-seven-sketches_FO2832 | 265 | 1.000 | S\left(o_{[a, b]}\right) | | |
| fong-spivak-seven-sketches_FO2833 | 265 | 0.753 | \mathrm{A} \in \mathbf{S h v}(\llbracket \mathbb{R}) | | |
| fong-spivak-seven-sketches_FO2834 | 265 | 1.000 | \mathrm{A}(U):=A | | |
| fong-spivak-seven-sketches_FO2835 | 265 | 1.000 | 7 \in \mathbb{N} | | |
| fong-spivak-seven-sketches_FO2836 | 266 | 0.881 | \left(X, \mathbf{O p}_{X}\right) | | |
| fong-spivak-seven-sketches_FO2837 | 266 | 0.881 | F_{X} | | |
| fong-spivak-seven-sketches_FO2838 | 266 | 0.614 | U \in \mathbf{O p}_{\mathbb{R}} | | |
| fong-spivak-seven-sketches_FO2839 | 266 | 0.614 | F_{X}(U):= | | |
| fong-spivak-seven-sketches_FO2840 | 266 | 0.875 | \{f: U \rightarrow X \mid f | | |
| fong-spivak-seven-sketches_FO2841 | 266 | 0.875 | G_{X} | | |
| fong-spivak-seven-sketches_FO2842 | 266 | 0.960 | \mathbb{R} \subseteq \mathbb{R} | | |
| fong-spivak-seven-sketches_FO2843 | 266 | 0.960 | G_{X}(U):= | | |
| fong-spivak-seven-sketches_FO2844 | 266 | 0.994 | \{f: U \cap \mathbb{R} \rightarrow X \mid f | | |
| fong-spivak-seven-sketches_FO2845 | 266 | 0.995 | R \subseteq \mathbb{R} | | |
| fong-spivak-seven-sketches_FO2846 | 266 | 0.995 | H_{X}(U):=\{f: U \cap R \rightarrow | | |
| fong-spivak-seven-sketches_FO2847 | 266 | 1.000 | X \mid f | | |
| fong-spivak-seven-sketches_FO2848 | 266 | 1.000 | H_{X} | | |
| fong-spivak-seven-sketches_FO2849 | 266 | 0.715 | (d, d+3) | | |
| fong-spivak-seven-sketches_FO2850 | 266 | 1.000 | L(d, d+3) | | |
| fong-spivak-seven-sketches_FO2851 | 266 | 0.585 | (\mathbb{I} \mathbb{R}, \mathbf{O p}) | | |
| fong-spivak-seven-sketches_FO2852 | 267 | 1.000 | (0,5) | | |
| fong-spivak-seven-sketches_FO2853 | 267 | 0.975 | (0,8) | | |
| fong-spivak-seven-sketches_FO2854 | 267 | 0.833 | (4,8) | | |
| fong-spivak-seven-sketches_FO2855 | 267 | 1.000 | \theta \in \mathbb{R}_{\geq 0} | | |
| fong-spivak-seven-sketches_FO2856 | 267 | 0.925 | \rightarrow \Omega | | |
| fong-spivak-seven-sketches_FO2857 | 268 | 1.000 | p(D, T) | | |
| fong-spivak-seven-sketches_FO2858 | 268 | 0.997 | @_{t} | | |
| fong-spivak-seven-sketches_FO2859 | 268 | 0.988 | @_{t}(q) | | |
| fong-spivak-seven-sketches_FO2860 | 268 | 0.998 | D \in | | |
| fong-spivak-seven-sketches_FO2861 | 268 | 0.962 | (U) | | |
| fong-spivak-seven-sketches_FO2862 | 268 | 0.962 | T \in | | |
| fong-spivak-seven-sketches_FO2863 | 271 | 1.000 | f(x)=x+5 | | |
| fong-spivak-seven-sketches_FO2864 | 271 | 1.000 | x+5 \leq y+5 | | |
| fong-spivak-seven-sketches_FO2865 | 271 | 0.416 | g(x):=-x | | |
| fong-spivak-seven-sketches_FO2866 | 271 | 0.416 | 1 \leq 2 | | |
| fong-spivak-seven-sketches_FO2867 | 271 | 0.416 | -1 \leq ?-2 | | |
| fong-spivak-seven-sketches_FO2868 | 271 | 0.986 | f(x):=x+5 | | |
| fong-spivak-seven-sketches_FO2869 | 271 | 0.986 | |x-y|=|(x+5)-(y+5)| | | |
| fong-spivak-seven-sketches_FO2870 | 271 | 1.000 | g(x):=2 * x | | |
| fong-spivak-seven-sketches_FO2871 | 271 | 1.000 | x=1 | | |
| fong-spivak-seven-sketches_FO2872 | 271 | 1.000 | y=2 | | |
| fong-spivak-seven-sketches_FO2873 | 271 | 1.000 | |x-y|=1 | | |
| fong-spivak-seven-sketches_FO2874 | 271 | 1.000 | |2 x-2 y|=2 | | |
| fong-spivak-seven-sketches_FO2875 | 271 | 1.000 | f(x):=3 * x | | |
| fong-spivak-seven-sketches_FO2876 | 271 | 1.000 | 3 *(x+y)=(3 * x)+(3 * y) | | |
| fong-spivak-seven-sketches_FO2877 | 271 | 1.000 | g(x):=x+1 | | |
| fong-spivak-seven-sketches_FO2878 | 271 | 1.000 | x=0 | | |
| fong-spivak-seven-sketches_FO2879 | 271 | 1.000 | y=0 | | |
| fong-spivak-seven-sketches_FO2880 | 271 | 1.000 | g(x+y)=1 | | |
| fong-spivak-seven-sketches_FO2881 | 271 | 1.000 | g(x)+g(y)=2 | | |
| fong-spivak-seven-sketches_FO2882 | 272 | 1.000 | A=(12)(3)(4) | | |
| fong-spivak-seven-sketches_FO2883 | 272 | 1.000 | B=(13)(2)(4) | | |
| fong-spivak-seven-sketches_FO2884 | 272 | 1.000 | A \vee B=(123)(4) | | |
| fong-spivak-seven-sketches_FO2885 | 272 | 0.622 | \{1,2,3\} \cup\{1\}=\{1,2,3\} | | |
| fong-spivak-seven-sketches_FO2886 | 272 | 0.992 | \{h, 1\} \times\{1,2,3\} | | |
| fong-spivak-seven-sketches_FO2887 | 272 | 1.000 | \{h, 1\} \sqcup\{1,2,3\} | | |
| fong-spivak-seven-sketches_FO2888 | 272 | 1.000 | \{h, 1\} \cup\{1,2,3\} | | |
| fong-spivak-seven-sketches_FO2889 | 272 | 1.000 | 0 \notin\{n \in \mathbb{Z} \mid n \geq 1\} | | |
| fong-spivak-seven-sketches_FO2890 | 273 | 0.994 | p_{1}^{\prime}, p_{2}^{\prime} \in P^{\prime} | | |
| fong-spivak-seven-sketches_FO2891 | 273 | 0.994 | A_{p}=A_{p_{1}^{\prime}}^{\prime} | | |
| fong-spivak-seven-sketches_FO2892 | 273 | 0.994 | A_{p}=A_{p_{2}^{\prime}}^{\prime} | | |
| fong-spivak-seven-sketches_FO2893 | 273 | 0.994 | A_{p_{1}^{\prime}}=A_{p_{2}^{\prime}} | | |
| fong-spivak-seven-sketches_FO2894 | 273 | 1.000 | A_{p_{1}^{\prime}} \cap A_{p_{2}^{\prime}}=A_{p_{1}^{\prime}} | | |
| fong-spivak-seven-sketches_FO2895 | 273 | 1.000 | A_{p_{1}^{\prime}} \neq \varnothing | | |
| fong-spivak-seven-sketches_FO2896 | 273 | 1.000 | p_{1} \neq p_{2} | | |
| fong-spivak-seven-sketches_FO2897 | 273 | 0.998 | A_{p_{1}^{\prime}} \cap A_{p_{2}^{\prime}}=\varnothing | | |
| fong-spivak-seven-sketches_FO2898 | 273 | 0.998 | p_{1}^{\prime}=p_{2}^{\prime} | | |
| fong-spivak-seven-sketches_FO2899 | 273 | 1.000 | A_{p^{\prime}}^{\prime} \neq \varnothing | | |
| fong-spivak-seven-sketches_FO2900 | 273 | 1.000 | a \in A_{p^{\prime}}^{\prime} | | |
| fong-spivak-seven-sketches_FO2901 | 273 | 1.000 | A_{p^{\prime}}^{\prime} \subseteq A | | |
| fong-spivak-seven-sketches_FO2902 | 273 | 1.000 | p^{\prime \prime} \in P^{\prime} | | |
| fong-spivak-seven-sketches_FO2903 | 273 | 1.000 | A_{p}=A_{p^{\prime \prime}}^{\prime} | | |
| fong-spivak-seven-sketches_FO2904 | 273 | 1.000 | a \in A_{p^{\prime \prime}}^{\prime} | | |
| fong-spivak-seven-sketches_FO2905 | 273 | 1.000 | a \in A_{p^{\prime}}^{\prime} \cap A_{p^{\prime \prime}}^{\prime} | | |
| fong-spivak-seven-sketches_FO2906 | 273 | 1.000 | p^{\prime}=p^{\prime \prime} | | |
| fong-spivak-seven-sketches_FO2907 | 273 | 1.000 | A_{p}=A_{p^{\prime}} | | |
| fong-spivak-seven-sketches_FO2908 | 273 | 0.987 | a \in A_{p} \cap A_{q} | | |
| fong-spivak-seven-sketches_FO2909 | 273 | 1.000 | A_{p}=A_{q} | | |
| fong-spivak-seven-sketches_FO2910 | 273 | 1.000 | a^{\prime} \in A_{p} | | |
| fong-spivak-seven-sketches_FO2911 | 273 | 1.000 | a^{\prime} \in A_{q} | | |
| fong-spivak-seven-sketches_FO2912 | 273 | 1.000 | a^{\prime} \in A | | |
| fong-spivak-seven-sketches_FO2913 | 273 | 1.000 | a \sim a^{\prime} | | |
| fong-spivak-seven-sketches_FO2914 | 273 | 1.000 | X:=\left\{a^{\prime} \in A \mid a^{\prime} \sim a\right\} | | |
| fong-spivak-seven-sketches_FO2915 | 273 | 0.996 | a^{\prime} \in X | | |
| fong-spivak-seven-sketches_FO2916 | 273 | 0.996 | b \sim a^{\prime} | | |
| fong-spivak-seven-sketches_FO2917 | 273 | 0.996 | b \sim a | | |
| fong-spivak-seven-sketches_FO2918 | 273 | 0.983 | b \in X | | |
| fong-spivak-seven-sketches_FO2919 | 273 | 0.983 | b, c \in X | | |
| fong-spivak-seven-sketches_FO2920 | 273 | 0.988 | c \sim a | | |
| fong-spivak-seven-sketches_FO2921 | 273 | 0.988 | a \in X | | |
| fong-spivak-seven-sketches_FO2922 | 273 | 1.000 | \varnothing \rightarrow\{1\} | | |
| fong-spivak-seven-sketches_FO2923 | 273 | 1.000 | \{a, b\} \rightarrow\{1\} | | |
| fong-spivak-seven-sketches_FO2924 | 273 | 1.000 | F \subseteq A \times \varnothing | | |
| fong-spivak-seven-sketches_FO2925 | 273 | 0.995 | b \in \varnothing | | |
| fong-spivak-seven-sketches_FO2926 | 273 | 0.995 | (a, b) \in F | | |
| fong-spivak-seven-sketches_FO2927 | 273 | 0.995 | b \in^{?} \varnothing | | |
| fong-spivak-seven-sketches_FO2928 | 274 | 0.986 | \{\bullet, *, \circ\} | | |
| fong-spivak-seven-sketches_FO2929 | 274 | 0.963 | P:=V=\{1,2,3,4\} | | |
| fong-spivak-seven-sketches_FO2930 | 275 | 1.000 | 4 \not \leq 6 | | |
| fong-spivak-seven-sketches_FO2931 | 275 | 1.000 | 6 \not \leq 4 | | |
| fong-spivak-seven-sketches_FO2932 | 275 | 1.000 | a, b \in \mathbb{R} | | |
| fong-spivak-seven-sketches_FO2933 | 275 | 0.996 | !: S \rightarrow\{1\} | | |
| fong-spivak-seven-sketches_FO2934 | 275 | 0.967 | \operatorname{id}_{S}: S \rightarrow S | | |
| fong-spivak-seven-sketches_FO2935 | 275 | 0.505 | \mathrm{U}(X) | | |
| fong-spivak-seven-sketches_FO2936 | 275 | 1.000 | \mathrm{U}(X)=\mathrm{P}(X) | | |
| fong-spivak-seven-sketches_FO2937 | 276 | 1.000 | 0 \leq \cdots \leq 3 | | |
| fong-spivak-seven-sketches_FO2938 | 276 | 1.000 | q \in \uparrow p | | |
| fong-spivak-seven-sketches_FO2939 | 276 | 1.000 | p \leq q^{\prime} | | |
| fong-spivak-seven-sketches_FO2940 | 276 | 0.991 | q^{\prime} \in \uparrow p | | |
| fong-spivak-seven-sketches_FO2941 | 276 | 0.589 | q \leq^{\mathrm{op}} p | | |
| fong-spivak-seven-sketches_FO2942 | 276 | 0.589 | \uparrow q \subseteq \uparrow p | | |
| fong-spivak-seven-sketches_FO2943 | 276 | 0.997 | q^{\prime} \in \uparrow q | | |
| fong-spivak-seven-sketches_FO2944 | 276 | 0.997 | p \not \leq p^{\prime} | | |
| fong-spivak-seven-sketches_FO2945 | 276 | 0.997 | \uparrow\left(p^{\prime}\right) \nsubseteq \uparrow(p) | | |
| fong-spivak-seven-sketches_FO2946 | 276 | 0.997 | p^{\prime} \in \uparrow\left(p^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO2947 | 276 | 0.998 | p^{\prime} \notin \uparrow(p) | | |
| fong-spivak-seven-sketches_FO2948 | 276 | 0.290 | \uparrow: P^{\mathrm{op}} \rightarrow \mathrm{U}(P) | | |
| fong-spivak-seven-sketches_FO2949 | 276 | 1.000 | p_{1} \leq_{P} p_{2} | | |
| fong-spivak-seven-sketches_FO2950 | 276 | 1.000 | f\left(p_{1}\right) \leq_{Q} f\left(p_{2}\right) | | |
| fong-spivak-seven-sketches_FO2951 | 276 | 1.000 | f\left(p_{1}\right)=f\left(p_{2}\right) | | |
| fong-spivak-seven-sketches_FO2952 | 276 | 1.000 | f\left(p_{1}\right) \leq f\left(p_{2}\right) | | |
| fong-spivak-seven-sketches_FO2953 | 276 | 1.000 | X=\mathbb{Z}=\{\ldots,-2,-1,0,1, \ldots\} | | |
| fong-spivak-seven-sketches_FO2954 | 276 | 1.000 | Y=\{n, z, p\} | | |
| fong-spivak-seven-sketches_FO2955 | 276 | 1.000 | P:=(n z)(p) | | |
| fong-spivak-seven-sketches_FO2956 | 276 | 1.000 | Q:=(n p)(z) | | |
| fong-spivak-seven-sketches_FO2957 | 276 | 1.000 | \{\{n, z\},\{p\}\} | | |
| fong-spivak-seven-sketches_FO2958 | 276 | 1.000 | \{\{n, p\},\{z\}\} | | |
| fong-spivak-seven-sketches_FO2959 | 276 | 1.000 | f^{*} P=(\ldots,-2,-1,0)(1,2, \ldots) | | |
| fong-spivak-seven-sketches_FO2960 | 276 | 1.000 | f^{*} Q=(0)(\ldots,-2,-1,1,2, \ldots) | | |
| fong-spivak-seven-sketches_FO2961 | 277 | 1.000 | \left(P, \leq_{P}\right),\left(Q, \leq_{Q}\right) | | |
| fong-spivak-seven-sketches_FO2962 | 277 | 0.544 | \mathrm{id}_{P} | | |
| fong-spivak-seven-sketches_FO2963 | 277 | 0.544 | \operatorname{id}_{P}\left(p_{1}\right) \leq \operatorname{id}_{P}\left(p_{2}\right) | | |
| fong-spivak-seven-sketches_FO2964 | 277 | 1.000 | \operatorname{id}_{P}(p)=p | | |
| fong-spivak-seven-sketches_FO2965 | 277 | 1.000 | q_{1} \leq_{Q} q_{2} | | |
| fong-spivak-seven-sketches_FO2966 | 277 | 1.000 | g\left(q_{1}\right) \leq_{R} g\left(q_{2}\right) | | |
| fong-spivak-seven-sketches_FO2967 | 277 | 1.000 | g\left(f\left(p_{1}\right)\right) \leq_{R} g\left(f\left(p_{2}\right)\right) | | |
| fong-spivak-seven-sketches_FO2968 | 277 | 1.000 | (f ; g) | | |
| fong-spivak-seven-sketches_FO2969 | 277 | 1.000 | p_{2} \leq p_{1} | | |
| fong-spivak-seven-sketches_FO2970 | 277 | 0.323 | \operatorname{map} \Phi: \operatorname{Prt}(\{\bullet, *, \circ\}) \rightarrow | | |
| fong-spivak-seven-sketches_FO2971 | 277 | 0.998 | \Phi(P)= | | |
| fong-spivak-seven-sketches_FO2972 | 277 | 0.998 | \Phi(Q)= | | |
| fong-spivak-seven-sketches_FO2973 | 277 | 0.997 | x \sim_{P} y | | |
| fong-spivak-seven-sketches_FO2974 | 277 | 0.997 | x \sim_{Q} y | | |
| fong-spivak-seven-sketches_FO2975 | 277 | 0.997 | x, y \in\{\bullet, *, \circ\} | | |
| fong-spivak-seven-sketches_FO2976 | 277 | 0.997 | \bullet, * | | |
| fong-spivak-seven-sketches_FO2977 | 277 | 0.642 | U \mapsto f^{-1}(U) | | |
| fong-spivak-seven-sketches_FO2978 | 277 | 1.000 | f^{*}(U) | | |
| fong-spivak-seven-sketches_FO2979 | 277 | 1.000 | q \mapsto | | |
| fong-spivak-seven-sketches_FO2980 | 277 | 0.890 | p \mapsto | | |
| fong-spivak-seven-sketches_FO2981 | 277 | 0.890 | f(p) \in U | | |
| fong-spivak-seven-sketches_FO2982 | 277 | 1.000 | \{p \in P \mid f(p) \in U\} | | |
| fong-spivak-seven-sketches_FO2983 | 277 | 1.000 | S=\left\{\left.\frac{1}{n+1} \right\rvert\, n \in \mathbb{N}\right\} | | |
| fong-spivak-seven-sketches_FO2984 | 277 | 1.000 | 0 \leq \frac{1}{n+1} | | |
| fong-spivak-seven-sketches_FO2985 | 277 | 1.000 | b \leq 0 | | |
| fong-spivak-seven-sketches_FO2986 | 277 | 1.000 | 0<b | | |
| fong-spivak-seven-sketches_FO2987 | 277 | 1.000 | 1 / b | | |
| fong-spivak-seven-sketches_FO2988 | 277 | 1.000 | 1 / b<n<n+1 | | |
| fong-spivak-seven-sketches_FO2989 | 277 | 1.000 | 1<b(n+1) | | |
| fong-spivak-seven-sketches_FO2990 | 277 | 1.000 | \frac{1}{n+1}<b | | |
| fong-spivak-seven-sketches_FO2991 | 277 | 1.000 | a=p | | |
| fong-spivak-seven-sketches_FO2992 | 277 | 1.000 | 4 \wedge 6=2 | | |
| fong-spivak-seven-sketches_FO2993 | 277 | 1.000 | 4 \vee 6=12 | | |
| fong-spivak-seven-sketches_FO2994 | 278 | 1.000 | a \leq a \vee b | | |
| fong-spivak-seven-sketches_FO2995 | 278 | 1.000 | b \leq a \vee b | | |
| fong-spivak-seven-sketches_FO2996 | 278 | 1.000 | f(a) \leq f(a \vee b) | | |
| fong-spivak-seven-sketches_FO2997 | 278 | 1.000 | f(b) \leq f(a \vee b) | | |
| fong-spivak-seven-sketches_FO2998 | 278 | 0.771 | \lfloor-/ 3\rfloor | | |
| fong-spivak-seven-sketches_FO2999 | 278 | 1.000 | r \in \mathbb{R} | | |
| fong-spivak-seven-sketches_FO3000 | 278 | 1.000 | z \leq\lfloor r / 3\rfloor | | |
| fong-spivak-seven-sketches_FO3001 | 278 | 1.000 | 3 * z \leq r | | |
| fong-spivak-seven-sketches_FO3002 | 278 | 1.000 | r / 3 | | |
| fong-spivak-seven-sketches_FO3003 | 278 | 1.000 | z^{\prime}:=\lfloor r / 3\rfloor | | |
| fong-spivak-seven-sketches_FO3004 | 278 | 1.000 | z \leq z^{\prime} | | |
| fong-spivak-seven-sketches_FO3005 | 278 | 1.000 | 3 * z \leq 3 * z^{\prime} \leq 3 * r / 3=r | | |
| fong-spivak-seven-sketches_FO3006 | 278 | 0.943 | z=3 * z / 3 \leq r / 3 | | |
| fong-spivak-seven-sketches_FO3007 | 278 | 1.000 | \lfloor r / 3\rfloor | | |
| fong-spivak-seven-sketches_FO3008 | 278 | 1.000 | \{(p, q) \mid 1 \leq p \leq 3 | | |
| fong-spivak-seven-sketches_FO3009 | 278 | 1.000 | 1 \leq q \leq 3\} | | |
| fong-spivak-seven-sketches_FO3010 | 278 | 1.000 | p=q=1 | | |
| fong-spivak-seven-sketches_FO3011 | 278 | 1.000 | f(p)=1 | | |
| fong-spivak-seven-sketches_FO3012 | 278 | 1.000 | g(q)=2 | | |
| fong-spivak-seven-sketches_FO3013 | 278 | 1.000 | f(p)=1 \leq 1=q | | |
| fong-spivak-seven-sketches_FO3014 | 278 | 1.000 | p=1 \leq g(q) | | |
| fong-spivak-seven-sketches_FO3015 | 278 | 0.616 | (p, q) | | |
| fong-spivak-seven-sketches_FO3016 | 278 | 0.616 | (1,2),(1,3),(2,1),(2,2),(2,3) | | |
| fong-spivak-seven-sketches_FO3017 | 278 | 0.616 | (3,3) | | |
| fong-spivak-seven-sketches_FO3018 | 278 | 1.000 | p=3, q=1 | | |
| fong-spivak-seven-sketches_FO3019 | 278 | 1.000 | p=3, q=2 | | |
| fong-spivak-seven-sketches_FO3020 | 278 | 1.000 | f(p)=3 | | |
| fong-spivak-seven-sketches_FO3021 | 278 | 1.000 | f(p) \not \leq q | | |
| fong-spivak-seven-sketches_FO3022 | 278 | 1.000 | p \not \leq g(q) | | |
| fong-spivak-seven-sketches_FO3023 | 278 | 1.000 | f(2) \not \leq 1 | | |
| fong-spivak-seven-sketches_FO3024 | 278 | 1.000 | 2 \leq g(1) | | |
| fong-spivak-seven-sketches_FO3025 | 278 | 1.000 | C(r):=\lceil r / 3\rceil | | |
| fong-spivak-seven-sketches_FO3026 | 278 | 1.000 | L(z) \leq r | | |
| fong-spivak-seven-sketches_FO3027 | 278 | 1.000 | z \leq C(r) | | |
| fong-spivak-seven-sketches_FO3028 | 278 | 1.000 | z=1 | | |
| fong-spivak-seven-sketches_FO3029 | 278 | 1.000 | r=.01 | | |
| fong-spivak-seven-sketches_FO3030 | 278 | 1.000 | \lceil r / 3\rceil=1 | | |
| fong-spivak-seven-sketches_FO3031 | 278 | 1.000 | L(1) \leq 0.01 | | |
| fong-spivak-seven-sketches_FO3032 | 278 | 1.000 | L(1) \leq r | | |
| fong-spivak-seven-sketches_FO3033 | 278 | 1.000 | r>0 | | |
| fong-spivak-seven-sketches_FO3034 | 278 | 1.000 | L(1) \leq 0 | | |
| fong-spivak-seven-sketches_FO3035 | 278 | 1.000 | 1 \leq C(0)=\lceil 0 / 3\rceil=0 | | |
| fong-spivak-seven-sketches_FO3036 | 278 | 1.000 | c_{1}, c_{2}, c_{3}, c_{4} | | |
| fong-spivak-seven-sketches_FO3037 | 279 | 0.995 | g_{!}\left(c_{1}\right), g_{!}\left(c_{2}\right), g_{!}\left(c_{3}\right) | | |
| fong-spivak-seven-sketches_FO3038 | 279 | 0.995 | g_{!}\left(c_{4}\right) | | |
| fong-spivak-seven-sketches_FO3039 | 279 | 1.000 | S=\{1,2,3,4\} | | |
| fong-spivak-seven-sketches_FO3040 | 279 | 1.000 | f: P \leftrightarrows Q: g | | |
| fong-spivak-seven-sketches_FO3041 | 279 | 1.000 | g(q) \leq g(q) | | |
| fong-spivak-seven-sketches_FO3042 | 279 | 1.000 | p:=g(q) | | |
| fong-spivak-seven-sketches_FO3043 | 279 | 1.000 | f(p) \leq f(g(q)) | | |
| fong-spivak-seven-sketches_FO3044 | 280 | 1.000 | g, g^{\prime}: Q \rightarrow P | | |
| fong-spivak-seven-sketches_FO3045 | 280 | 1.000 | g(q) \leq g^{\prime}(q) | | |
| fong-spivak-seven-sketches_FO3046 | 280 | 1.000 | g^{\prime}(q) \leq g(q) | | |
| fong-spivak-seven-sketches_FO3047 | 280 | 1.000 | p \leq g^{\prime}(f(p)) | | |
| fong-spivak-seven-sketches_FO3048 | 280 | 0.999 | j:=\vee A | | |
| fong-spivak-seven-sketches_FO3049 | 280 | 1.000 | f(a) \leq f(j) | | |
| fong-spivak-seven-sketches_FO3050 | 280 | 1.000 | f(j) | | |
| fong-spivak-seven-sketches_FO3051 | 280 | 1.000 | f(a) \leq b | | |
| fong-spivak-seven-sketches_FO3052 | 280 | 1.000 | a \leq g(b) | | |
| fong-spivak-seven-sketches_FO3053 | 280 | 1.000 | j \leq g(b) | | |
| fong-spivak-seven-sketches_FO3054 | 280 | 1.000 | f(j) \leq b | | |
| fong-spivak-seven-sketches_FO3055 | 281 | 1.000 | f(p) | | |
| fong-spivak-seven-sketches_FO3058 | 281 | 0.999 | B_{1}:=\{a, b\} | | |
| fong-spivak-seven-sketches_FO3059 | 281 | 0.999 | B_{2}:=\{c\} | | |
| fong-spivak-seven-sketches_FO3060 | 281 | 0.999 | f^{*}\left(B_{1}\right)=\left\{a_{1}\right\} | | |
| fong-spivak-seven-sketches_FO3061 | 281 | 0.999 | f^{*}\left(B_{2}\right)=\left\{c_{1}, c_{2}\right\} | | |
| fong-spivak-seven-sketches_FO3062 | 281 | 1.000 | A_{1}:=\varnothing | | |
| fong-spivak-seven-sketches_FO3063 | 281 | 1.000 | A_{2}:=\left\{a_{1}, c_{1}\right\} | | |
| fong-spivak-seven-sketches_FO3064 | 281 | 1.000 | f_{!}\left(A_{1}\right)=\varnothing | | |
| fong-spivak-seven-sketches_FO3065 | 281 | 1.000 | f_{!}\left(A_{2}\right)=\{a, c\} | | |
| fong-spivak-seven-sketches_FO3066 | 281 | 1.000 | A_{1} | | |
| fong-spivak-seven-sketches_FO3067 | 281 | 1.000 | A_{2} | | |
| fong-spivak-seven-sketches_FO3068 | 281 | 1.000 | f_{*}\left(A_{1}\right)=\{b\} | | |
| fong-spivak-seven-sketches_FO3069 | 281 | 1.000 | f_{*}\left(A_{2}\right)=\{a, b\} | | |
| fong-spivak-seven-sketches_FO3070 | 281 | 1.000 | g(f(p)) | | |
| fong-spivak-seven-sketches_FO3071 | 281 | 1.000 | (f ; g)(p) | | |
| fong-spivak-seven-sketches_FO3072 | 281 | 1.000 | g(f(g(f(p)))) \leq g(f(p)) | | |
| fong-spivak-seven-sketches_FO3073 | 281 | 1.000 | g(f(p)) \leq g(f(g(f(p)))) | | |
| fong-spivak-seven-sketches_FO3074 | 281 | 1.000 | p^{\prime} \leq g\left(f\left(p^{\prime}\right)\right) | | |
| fong-spivak-seven-sketches_FO3075 | 281 | 1.000 | f(g(f(p))) \leq f(p) | | |
| fong-spivak-seven-sketches_FO3076 | 281 | 0.911 | a b | | |
| fong-spivak-seven-sketches_FO3077 | 281 | 0.911 | \{(1,1),(1,2),(2,1)\} | | |
| fong-spivak-seven-sketches_FO3078 | 281 | 1.000 | S:=\{1,2,3\} | | |
| fong-spivak-seven-sketches_FO3079 | 281 | 1.000 | 1 \leq 1,2 \leq 2 | | |
| fong-spivak-seven-sketches_FO3080 | 281 | 1.000 | 3 \leq 3 | | |
| fong-spivak-seven-sketches_FO3081 | 281 | 1.000 | U(\leq)= | | |
| fong-spivak-seven-sketches_FO3082 | 281 | 0.999 | \{(1,1),(1,2),(2,2),(3,3)\} | | |
| fong-spivak-seven-sketches_FO3083 | 281 | 1.000 | Q:=\{(1,1)\} | | |
| fong-spivak-seven-sketches_FO3084 | 281 | 1.000 | Q^{\prime}:=\{(2,1)\} | | |
| fong-spivak-seven-sketches_FO3085 | 281 | 0.825 | \mathrm{Cl}(Q) | | |
| fong-spivak-seven-sketches_FO3086 | 281 | 0.825 | (1,1) | | |
| fong-spivak-seven-sketches_FO3087 | 281 | 0.825 | \mathrm{Cl}(Q)=\{(1,1),(2,2),(3,3)\} | | |
| fong-spivak-seven-sketches_FO3088 | 281 | 0.793 | \mathrm{Cl}\left(Q^{\prime}\right)=\{(1,1),(2,1),(2,2),(3,3)\} | | |
| fong-spivak-seven-sketches_FO3089 | 281 | 0.999 | (2,1) | | |
| fong-spivak-seven-sketches_FO3090 | 281 | 0.999 | \mathrm{Cl}\left(Q^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO3091 | 282 | 1.000 | x_{1}=-1, x_{2}=0, y_{1}=-1 | | |
| fong-spivak-seven-sketches_FO3092 | 282 | 1.000 | y_{2}=1 | | |
| fong-spivak-seven-sketches_FO3093 | 282 | 0.901 | -1 \leq-1 | | |
| fong-spivak-seven-sketches_FO3094 | 282 | 0.901 | -1 * 0=0 \not \leq-1=-1 * 1 | | |
| fong-spivak-seven-sketches_FO3095 | 282 | 0.940 | (\operatorname{Disc}(M),=, *, e) | | |
| fong-spivak-seven-sketches_FO3096 | 282 | 1.000 | x_{1} \otimes x_{2}=x_{1} \otimes x_{2} | | |
| fong-spivak-seven-sketches_FO3097 | 282 | 1.000 | u \leq u, v \leq v | | |
| fong-spivak-seven-sketches_FO3098 | 282 | 1.000 | z \leq z | | |
| fong-spivak-seven-sketches_FO3099 | 282 | 1.000 | t+u \leq y+z | | |
| fong-spivak-seven-sketches_FO3100 | 282 | 1.000 | z \rightarrow w | | |
| fong-spivak-seven-sketches_FO3101 | 282 | 1.000 | x+z \rightarrow y+w | | |
| fong-spivak-seven-sketches_FO3102 | 282 | 0.991 | 1 \mid 1 | | |
| fong-spivak-seven-sketches_FO3103 | 282 | 0.991 | 1 \mid 2 | | |
| fong-spivak-seven-sketches_FO3104 | 282 | 0.991 | (1+1) \nmid(1+2) | | |
| fong-spivak-seven-sketches_FO3105 | 283 | 0.936 | z \leq w | | |
| fong-spivak-seven-sketches_FO3106 | 283 | 0.936 | \min (x, z) \leq \min (y, w) | | |
| fong-spivak-seven-sketches_FO3107 | 283 | 0.936 | \min (x | | |
| fong-spivak-seven-sketches_FO3108 | 283 | 0.936 | =x= | | |
| fong-spivak-seven-sketches_FO3109 | 283 | 0.948 | \min ( | | |
| fong-spivak-seven-sketches_FO3110 | 283 | 0.948 | x) | | |
| fong-spivak-seven-sketches_FO3111 | 283 | 0.948 | \min (\min (x, y), z)=\min (x, \min (y, z)) | | |
| fong-spivak-seven-sketches_FO3112 | 283 | 0.948 | \min (x, y)=\min (y, z) | | |
| fong-spivak-seven-sketches_FO3113 | 283 | 0.991 | (\mathrm{P}(S), \leq, S, \cap) | | |
| fong-spivak-seven-sketches_FO3114 | 283 | 1.000 | (P \wedge Q)(n) | | |
| fong-spivak-seven-sketches_FO3115 | 283 | 0.833 | { }^{\mathbb{N}}, \leq | | |
| fong-spivak-seven-sketches_FO3116 | 283 | 0.957 | n+10 \leq 1 | | |
| fong-spivak-seven-sketches_FO3117 | 283 | 0.957 | (P \vee Q)(n) | | |
| fong-spivak-seven-sketches_FO3118 | 283 | 0.561 | \left(\operatorname{Prop}^{\mathbb{N}}, \leq\right. | | |
| fong-spivak-seven-sketches_FO3119 | 283 | 0.561 | \left.\vee\right) | | |
| fong-spivak-seven-sketches_FO3120 | 283 | 0.676 | X^{\mathrm{op}} | | |
| fong-spivak-seven-sketches_FO3121 | 283 | 0.676 | I \otimes x= | | |
| fong-spivak-seven-sketches_FO3122 | 283 | 0.964 | x=x \otimes I | | |
| fong-spivak-seven-sketches_FO3123 | 283 | 0.506 | y \otimes x | | |
| fong-spivak-seven-sketches_FO3124 | 283 | 0.506 | x^{\mathrm{op}} | | |
| fong-spivak-seven-sketches_FO3125 | 283 | 0.978 | (x \otimes y) \cong(y \otimes x) | | |
| fong-spivak-seven-sketches_FO3126 | 283 | 0.978 | (x \otimes y) \leq(y \otimes x) | | |
| fong-spivak-seven-sketches_FO3127 | 283 | 0.927 | (y \otimes x) \leq(x \otimes y) | | |
| fong-spivak-seven-sketches_FO3128 | 283 | 0.768 | \operatorname{Cost}^{\mathrm{op}} | | |
| fong-spivak-seven-sketches_FO3129 | 283 | 0.999 | =\infty \geq 0=g | | |
| fong-spivak-seven-sketches_FO3130 | 283 | 0.999 | 0 \geq 0= | | |
| fong-spivak-seven-sketches_FO3131 | 284 | 0.966 | f(n)=1 | | |
| fong-spivak-seven-sketches_FO3132 | 284 | 0.966 | f:(\mathbb{N}, \leq, 0,+) \rightarrow(\mathbb{N}, \leq 1, *) | | |
| fong-spivak-seven-sketches_FO3133 | 284 | 0.523 | \mathbb{Z}, \leq, *, 1 | | |
| fong-spivak-seven-sketches_FO3134 | 284 | 0.523 | -1 \leq 0 | | |
| fong-spivak-seven-sketches_FO3135 | 284 | 0.523 | -1 *-1 | | |
| fong-spivak-seven-sketches_FO3136 | 284 | 0.523 | \nsubseteq | | |
| fong-spivak-seven-sketches_FO3137 | 284 | 0.998 | (0 * 0) | | |
| fong-spivak-seven-sketches_FO3138 | 284 | 0.654 | X_{P} | | |
| fong-spivak-seven-sketches_FO3139 | 284 | 0.654 | X_{P}(p, q):= | | |
| fong-spivak-seven-sketches_FO3140 | 284 | 0.880 | \operatorname{Ob}\left(X_{P}\right)=P | | |
| fong-spivak-seven-sketches_FO3141 | 284 | 0.880 | X_{P}(p, q)= | | |
| fong-spivak-seven-sketches_FO3142 | 284 | 0.335 | (\operatorname{Ob}(\mathscr{X}), \leq) | | |
| fong-spivak-seven-sketches_FO3143 | 284 | 0.948 | \operatorname{Ob}(X) | | |
| fong-spivak-seven-sketches_FO3144 | 284 | 0.948 | X^{\prime}(x, y)= | | |
| fong-spivak-seven-sketches_FO3145 | 284 | 0.600 | (\mathrm{Ob}(X), \leq) | | |
| fong-spivak-seven-sketches_FO3146 | 284 | 0.600 | X^{\prime}(x, y)=\mathcal{X}(x, y) | | |
| fong-spivak-seven-sketches_FO3147 | 284 | 0.563 | \left(\mathbb{R}_{\geq 0}, \geq, 0,+\right) | | |
| fong-spivak-seven-sketches_FO3149 | 284 | 0.890 | \min (\mathcal{X}(x, y), \mathcal{X}(y, z)) \leq \mathcal{X}(x, z) | | |
| fong-spivak-seven-sketches_FO3150 | 284 | 1.000 | x, y, z | | |
| fong-spivak-seven-sketches_FO3151 | 285 | 0.978 | X(C, D) | | |
| fong-spivak-seven-sketches_FO3152 | 285 | 0.978 | C \rightarrow A \rightarrow | | |
| fong-spivak-seven-sketches_FO3153 | 285 | 1.000 | B \rightarrow D | | |
| fong-spivak-seven-sketches_FO3154 | 285 | 1.000 | C \rightarrow D | | |
| fong-spivak-seven-sketches_FO3155 | 285 | 0.946 | \mathcal{X}(x, y) \cap \mathcal{X}(y, z) \subseteq \mathcal{X}(x, z) | | |
| fong-spivak-seven-sketches_FO3156 | 285 | 1.000 | (x, y)^{\text {th }} | | |
| fong-spivak-seven-sketches_FO3158 | 285 | 1.000 | \min (M(x, y), M(y, z)) \leq M(x, y) | | |
| fong-spivak-seven-sketches_FO3159 | 285 | 1.000 | x, y, z \in\{A, B, C\} | | |
| fong-spivak-seven-sketches_FO3160 | 286 | 0.900 | d(A, B)=d(B, A)=5 | | |
| fong-spivak-seven-sketches_FO3161 | 286 | 0.210 | x_{f}= | | |
| fong-spivak-seven-sketches_FO3162 | 286 | 0.210 | x_{g}=\stackrel{A}{\bullet} \longrightarrow | | |
| fong-spivak-seven-sketches_FO3163 | 286 | 1.000 | d(x, y) \leq d(y, x) | | |
| fong-spivak-seven-sketches_FO3164 | 286 | 1.000 | d(y, x) \leq d(x, y) | | |
| fong-spivak-seven-sketches_FO3165 | 286 | 1.000 | d(y, x)=0 | | |
| fong-spivak-seven-sketches_FO3166 | 286 | 0.893 | (x, y) \in \mathcal{X} \times \mathcal{Y} | | |
| fong-spivak-seven-sketches_FO3167 | 286 | 0.893 | I \leq \mathfrak{X}(x, x) | | |
| fong-spivak-seven-sketches_FO3168 | 286 | 0.893 | I \leq y(y, y) | | |
| fong-spivak-seven-sketches_FO3169 | 286 | 0.999 | I=I \otimes I \leq X(x, x) \otimes y(y, y)=(\mathcal{X} \times y)((x, y),(x, y)) | | |
| fong-spivak-seven-sketches_FO3170 | 286 | 0.955 | y\left(y_{1}, y_{2}\right) \otimes \mathcal{X}\left(x_{2}, x_{3}\right)=\mathcal{X}\left(x_{2}, x_{3}\right) \otimes y\left(y_{1}, y_{2}\right) | | |
| fong-spivak-seven-sketches_FO3171 | 286 | 0.990 | (\mathbb{R} \times \mathbb{R})((5,6),(-1,4))=\mathbb{R}(5,-1)+\mathbb{R}(6,4)=6+2=8 | | |
| fong-spivak-seven-sketches_FO3172 | 286 | 1.000 | u \leq u^{\prime} | | |
| fong-spivak-seven-sketches_FO3173 | 286 | 1.000 | u \otimes v \leq u^{\prime} \otimes v | | |
| fong-spivak-seven-sketches_FO3174 | 286 | 0.996 | a:=(v \multimap w) | | |
| fong-spivak-seven-sketches_FO3175 | 286 | 0.995 | (v \multimap u) \otimes v \leq u \leq u^{\prime} | | |
| fong-spivak-seven-sketches_FO3176 | 286 | 1.000 | (v \multimap u) \leq\left(v \multimap u^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO3177 | 286 | 1.000 | (v \multimap-): V \rightarrow V | | |
| fong-spivak-seven-sketches_FO3178 | 286 | 0.859 | (v---) | | |
| fong-spivak-seven-sketches_FO3179 | 287 | 1.000 | x \Rightarrow y | | |
| fong-spivak-seven-sketches_FO3180 | 287 | 0.989 | (a \wedge v) \leq w | | |
| fong-spivak-seven-sketches_FO3181 | 287 | 0.989 | a \leq(v \Rightarrow w) | | |
| fong-spivak-seven-sketches_FO3182 | 287 | 0.989 | v= | | |
| fong-spivak-seven-sketches_FO3183 | 287 | 0.989 | a \wedge | | |
| fong-spivak-seven-sketches_FO3184 | 287 | 0.879 | =a | | |
| fong-spivak-seven-sketches_FO3185 | 287 | 0.879 | a \leq w | | |
| fong-spivak-seven-sketches_FO3186 | 287 | 0.879 | \Rightarrow w)=w | | |
| fong-spivak-seven-sketches_FO3187 | 287 | 0.980 | (\mathrm{P}(S), \subseteq, S, \cap) | | |
| fong-spivak-seven-sketches_FO3188 | 287 | 0.980 | B \multimap C | | |
| fong-spivak-seven-sketches_FO3189 | 287 | 1.000 | \bar{B} \cup C | | |
| fong-spivak-seven-sketches_FO3190 | 287 | 1.000 | \bar{B} | | |
| fong-spivak-seven-sketches_FO3191 | 287 | 1.000 | (A \cap B) \subseteq C | | |
| fong-spivak-seven-sketches_FO3192 | 287 | 1.000 | A \subseteq(\bar{B} \cup C) | | |
| fong-spivak-seven-sketches_FO3193 | 287 | 0.558 | (\bigvee \varnothing)= | | |
| fong-spivak-seven-sketches_FO3194 | 287 | 0.946 | (\vee \varnothing)=\infty | | |
| fong-spivak-seven-sketches_FO3195 | 287 | 0.974 | (\vee \varnothing)=0 | | |
| fong-spivak-seven-sketches_FO3196 | 287 | 1.000 | \min (x, y) | | |
| fong-spivak-seven-sketches_FO3197 | 287 | 1.000 | v \in V, 0 \otimes v=0 | | |
| fong-spivak-seven-sketches_FO3198 | 289 | 0.999 | \left(v, a_{1}, \ldots, a_{n}\right) | | |
| fong-spivak-seven-sketches_FO3199 | 289 | 0.999 | s\left(a_{1}\right)=v | | |
| fong-spivak-seven-sketches_FO3200 | 289 | 0.999 | t\left(a_{i}\right)=s\left(a_{i+1}\right) | | |
| fong-spivak-seven-sketches_FO3201 | 289 | 1.000 | i \in\{1, \ldots, n-1\} | | |
| fong-spivak-seven-sketches_FO3202 | 289 | 1.000 | s(p):=v | | |
| fong-spivak-seven-sketches_FO3203 | 289 | 1.000 | p=\left(v, a_{1}, \ldots, a_{m}\right) | | |
| fong-spivak-seven-sketches_FO3204 | 289 | 1.000 | q=\left(w, b_{1}, \ldots, b_{n}\right) | | |
| fong-spivak-seven-sketches_FO3205 | 289 | 1.000 | t(p)=s(q) | | |
| fong-spivak-seven-sketches_FO3206 | 289 | 0.870 | p=(v) | | |
| fong-spivak-seven-sketches_FO3207 | 289 | 0.551 | (v) | | |
| fong-spivak-seven-sketches_FO3208 | 289 | 0.551 | \left(w, b_{1}, \ldots, b_{n}\right) | | |
| fong-spivak-seven-sketches_FO3209 | 289 | 1.000 | v=w | | |
| fong-spivak-seven-sketches_FO3210 | 289 | 1.000 | \left(v, a_{1}, \ldots, a_{m}\right) | | |
| fong-spivak-seven-sketches_FO3211 | 289 | 1.000 | (w) | | |
| fong-spivak-seven-sketches_FO3212 | 289 | 1.000 | w=t\left(a_{m}\right) | | |
| fong-spivak-seven-sketches_FO3213 | 289 | 0.996 | r=\left(x, c_{1}, \ldots, c_{o}\right) | | |
| fong-spivak-seven-sketches_FO3214 | 289 | 0.996 | (p ; q) ; r | | |
| fong-spivak-seven-sketches_FO3215 | 289 | 0.952 | p ;(q ; r) | | |
| fong-spivak-seven-sketches_FO3216 | 289 | 0.543 | \boxtimes | | |
| fong-spivak-seven-sketches_FO3218 | 292 | 1.000 | f_{2} | | |
| fong-spivak-seven-sketches_FO3221 | 289 | 1.000 | 1+2+\cdots+n | | |
| fong-spivak-seven-sketches_FO3222 | 289 | 0.818 | n-i | | |
| fong-spivak-seven-sketches_FO3223 | 289 | 1.000 | 0 \leq i \leq n | | |
| fong-spivak-seven-sketches_FO3224 | 290 | 0.999 | m+n | | |
| fong-spivak-seven-sketches_FO3225 | 290 | 0.663 | h=j=A | | |
| fong-spivak-seven-sketches_FO3226 | 290 | 0.485 | z, s, s | | |
| fong-spivak-seven-sketches_FO3227 | 290 | 0.485 | s ; s | | |
| fong-spivak-seven-sketches_FO3228 | 291 | 1.000 | (f(1), f(2)) | | |
| fong-spivak-seven-sketches_FO3229 | 291 | 0.985 | n! | | |
| fong-spivak-seven-sketches_FO3230 | 291 | 0.999 | 5 * 4 * 3 * 2 * 1=120 | | |
| fong-spivak-seven-sketches_FO3231 | 291 | 0.999 | \{1,2,3,4,5\} | | |
| fong-spivak-seven-sketches_FO3232 | 291 | 0.999 | \{a, b, c, d, e\} | | |
| fong-spivak-seven-sketches_FO3233 | 291 | 0.566 | f: c \rightarrow c | | |
| fong-spivak-seven-sketches_FO3234 | 291 | 0.761 | \mathrm{id}_{c}=\mathrm{id}_{c} | | |
| fong-spivak-seven-sketches_FO3235 | 291 | 0.761 | f=\mathrm{id}_{c} | | |
| fong-spivak-seven-sketches_FO3236 | 291 | 1.000 | s^{n} | | |
| fong-spivak-seven-sketches_FO3237 | 291 | 1.000 | s^{n+1} | | |
| fong-spivak-seven-sketches_FO3238 | 291 | 1.000 | s^{0} . p | | |
| fong-spivak-seven-sketches_FO3239 | 291 | 0.803 | p ; q=\mathrm{id} | | |
| fong-spivak-seven-sketches_FO3240 | 291 | 0.803 | q ; p=\mathrm{id} | | |
| fong-spivak-seven-sketches_FO3241 | 291 | 0.961 | \mathbf{2} \boldsymbol{\rightarrow} \mathbf{3} | | |
| fong-spivak-seven-sketches_FO3242 | 292 | 1.000 | g^{\prime} ; i^{\prime} \mapsto f ; h | | |
| fong-spivak-seven-sketches_FO3243 | 292 | 0.546 | g^{\prime} ; i^{\prime} \mapsto g ; i | | |
| fong-spivak-seven-sketches_FO3244 | 292 | 0.546 | g ; i=f ; h | | |
| fong-spivak-seven-sketches_FO3245 | 292 | 0.449 | F, G | | |
| fong-spivak-seven-sketches_FO3246 | 292 | 0.449 | \stackrel{a}{\rightarrow} \xrightarrow{b} | | |
| fong-spivak-seven-sketches_FO3247 | 292 | 0.449 | \stackrel{a^{\prime}}{\underset{f_{2}}{f_{1}}} \stackrel{b^{\prime}}{\bullet} | | |
| fong-spivak-seven-sketches_FO3248 | 292 | 0.958 | \operatorname{id}_{\mathcal{C}}: \mathcal{C} \rightarrow \mathcal{C} | | |
| fong-spivak-seven-sketches_FO3249 | 292 | 0.958 | \operatorname{id}_{\mathcal{C}}(x)=x | | |
| fong-spivak-seven-sketches_FO3250 | 292 | 0.098 | \operatorname{identities} \mathrm{id}_{\mathcal{C}}\left(\mathrm{id}_{\mathcal{C}}\right)=\mathrm{id}_{c}=\mathrm{id}_{\operatorname{idC}(c)} | | |
| fong-spivak-seven-sketches_FO3251 | 292 | 0.995 | \operatorname{id}_{\mathcal{C}}(f ; g)=f ; g=\operatorname{id}_{\mathcal{C}}(f) ; \operatorname{id}_{\mathcal{C}(g)} | | |
| fong-spivak-seven-sketches_FO3252 | 292 | 0.995 | (F ; G)\left(\mathrm{id}_{c}\right)= | | |
| fong-spivak-seven-sketches_FO3253 | 292 | 0.515 | G\left(F\left(\mathrm{id}_{c}\right)\right)=G\left(\mathrm{id}_{F(c)}\right)=\operatorname{id}_{G(F(c))} | | |
| fong-spivak-seven-sketches_FO3254 | 292 | 0.874 | \operatorname{id}_{\mathcal{C}} | | |
| fong-spivak-seven-sketches_FO3255 | 292 | 0.989 | \mathrm{id}_{\mathcal{D}} | | |
| fong-spivak-seven-sketches_FO3256 | 292 | 1.000 | H: \mathcal{E} \rightarrow \mathcal{F} | | |
| fong-spivak-seven-sketches_FO3257 | 292 | 0.991 | (F ; G) ; H=F ;(G ; H) | | |
| fong-spivak-seven-sketches_FO3258 | 292 | 0.991 | x \in \mathcal{C} | | |
| fong-spivak-seven-sketches_FO3259 | 292 | 0.996 | F_{S}\left(\mathrm{id}_{1}\right)=\mathrm{id}_{S} | | |
| fong-spivak-seven-sketches_FO3260 | 292 | 0.996 | F_{S} | | |
| fong-spivak-seven-sketches_FO3261 | 292 | 1.000 | D(z) | | |
| fong-spivak-seven-sketches_FO3262 | 292 | 1.000 | D(s): D(z) \rightarrow D(z) | | |
| fong-spivak-seven-sketches_FO3263 | 292 | 0.999 | D_{z} | | |
| fong-spivak-seven-sketches_FO3264 | 293 | 1.000 | D(c) | | |
| fong-spivak-seven-sketches_FO3265 | 293 | 0.995 | D(b) | | |
| fong-spivak-seven-sketches_FO3266 | 293 | 0.997 | D(a) | | |
| fong-spivak-seven-sketches_FO3267 | 293 | 0.999 | d(f): D(a) \rightarrow D(b) | | |
| fong-spivak-seven-sketches_FO3268 | 293 | 0.568 | F, G, H \in \mathcal{D}^{\mathcal{C}} | | |
| fong-spivak-seven-sketches_FO3269 | 293 | 1.000 | F, G, H: \mathcal{C} \rightarrow \mathcal{D} | | |
| fong-spivak-seven-sketches_FO3270 | 293 | 1.000 | \beta: G \rightarrow H | | |
| fong-spivak-seven-sketches_FO3271 | 293 | 0.877 | c \in \mathcal{C}, \alpha | | |
| fong-spivak-seven-sketches_FO3272 | 293 | 0.877 | \beta_{c}: G(c) \rightarrow H(c) | | |
| fong-spivak-seven-sketches_FO3273 | 293 | 0.872 | (\alpha ; \beta)_{\mathcal{C}}:=\left(\alpha_{\mathcal{C}} ; \beta_{\mathcal{C}}\right) | | |
| fong-spivak-seven-sketches_FO3274 | 293 | 1.000 | F(c) \rightarrow H(c) | | |
| fong-spivak-seven-sketches_FO3275 | 293 | 0.995 | \operatorname{id}_{F} | | |
| fong-spivak-seven-sketches_FO3276 | 293 | 0.660 | \left(\mathrm{id}_{F}\right)_{c}:=\mathrm{id}_{F(c)} | | |
| fong-spivak-seven-sketches_FO3277 | 293 | 1.000 | \beta: F \rightarrow G | | |
| fong-spivak-seven-sketches_FO3278 | 293 | 0.992 | \alpha: E \rightarrow F | | |
| fong-spivak-seven-sketches_FO3279 | 293 | 0.992 | \operatorname{id}_{F} ; \beta | | |
| fong-spivak-seven-sketches_FO3280 | 293 | 0.896 | \left(\mathrm{id}_{F}\right)_{c} ; \beta_{c}=\left(\mathrm{id}_{F(c)} ; \beta_{c}\right)=\beta_{c} | | |
| fong-spivak-seven-sketches_FO3281 | 293 | 1.000 | \alpha, \beta: F \rightarrow G | | |
| fong-spivak-seven-sketches_FO3282 | 293 | 1.000 | \alpha=\beta | | |
| fong-spivak-seven-sketches_FO3283 | 293 | 1.000 | \alpha_{c}=\beta_{c} | | |
| fong-spivak-seven-sketches_FO3284 | 293 | 1.000 | \beta_{c} | | |
| fong-spivak-seven-sketches_FO3285 | 293 | 0.998 | F(c) \rightarrow G(c) | | |
| fong-spivak-seven-sketches_FO3286 | 293 | 0.421 | \mathcal{P}:=\mathbf{1} | | |
| fong-spivak-seven-sketches_FO3287 | 293 | 0.421 | \mathcal{C}:=\vec{\bullet} \xrightarrow[f_{2}]{f_{1}} \stackrel{b}{\bullet} | | |
| fong-spivak-seven-sketches_FO3288 | 293 | 0.421 | F(1):=a | | |
| fong-spivak-seven-sketches_FO3289 | 293 | 0.421 | G(1):=b | | |
| fong-spivak-seven-sketches_FO3290 | 293 | 0.421 | \alpha_{1}:=f_{1} | | |
| fong-spivak-seven-sketches_FO3291 | 293 | 0.421 | \beta_{1}:=g_{2} | | |
| fong-spivak-seven-sketches_FO3292 | 294 | 1.000 | \alpha_{\text {Arrow }}(b)=e | | |
| fong-spivak-seven-sketches_FO3293 | 294 | 1.000 | \alpha_{\text {Vertex }}(1)=4, \alpha_{\text {Vertex }}(2)=5 | | |
| fong-spivak-seven-sketches_FO3294 | 294 | 1.000 | \alpha_{\text {Vertex }}(3)=5 | | |
| fong-spivak-seven-sketches_FO3295 | 295 | 1.000 | X \times B \rightarrow Y \times B | | |
| fong-spivak-seven-sketches_FO3296 | 295 | 1.000 | f \times B | | |
| fong-spivak-seven-sketches_FO3297 | 295 | 1.000 | (x, b) | | |
| fong-spivak-seven-sketches_FO3298 | 295 | 1.000 | (f(x), b) | | |
| fong-spivak-seven-sketches_FO3299 | 295 | 0.988 | \mathrm{id}_{X \times B} | | |
| fong-spivak-seven-sketches_FO3300 | 295 | 1.000 | X^{B} \rightarrow Y^{B} | | |
| fong-spivak-seven-sketches_FO3301 | 295 | 0.988 | f^{B} | | |
| fong-spivak-seven-sketches_FO3302 | 295 | 0.988 | g: B \rightarrow X | | |
| fong-spivak-seven-sketches_FO3303 | 295 | 0.988 | (g \circ f): B \rightarrow X \rightarrow Y | | |
| fong-spivak-seven-sketches_FO3304 | 295 | 0.986 | f^{B}\left(\mathrm{id}_{X}\right)=\mathrm{id}_{X^{B}} | | |
| fong-spivak-seven-sketches_FO3305 | 295 | 1.000 | \left(f_{1} ; f_{2}\right): X \rightarrow Y \rightarrow Z | | |
| fong-spivak-seven-sketches_FO3306 | 295 | 1.000 | g \in X^{B} | | |
| fong-spivak-seven-sketches_FO3307 | 295 | 0.999 | \mathbb{N}^{\mathbb{N}} | | |
| fong-spivak-seven-sketches_FO3308 | 295 | 1.000 | p(3): \mathbb{N} \rightarrow \mathbb{N} | | |
| fong-spivak-seven-sketches_FO3309 | 295 | 1.000 | p(3)(n):=n+3 | | |
| fong-spivak-seven-sketches_FO3310 | 295 | 0.533 | !: \mathcal{C} \rightarrow \mathbf{1} | | |
| fong-spivak-seven-sketches_FO3311 | 295 | 0.533 | 1 \in \mathbf{1} | | |
| fong-spivak-seven-sketches_FO3312 | 295 | 0.590 | \mathrm{B}=\mathrm{Bob}, 3=\mathrm{Em} \_3 | | |
| fong-spivak-seven-sketches_FO3313 | 295 | 0.999 | c \rightarrow z | | |
| fong-spivak-seven-sketches_FO3314 | 296 | 0.940 | \mathcal{C} \in | | |
| fong-spivak-seven-sketches_FO3315 | 296 | 0.877 | 2 V:=\bullet \bullet | | |
| fong-spivak-seven-sketches_FO3316 | 296 | 0.877 | \mathcal{C}=\operatorname{Free}(2 V) | | |
| fong-spivak-seven-sketches_FO3317 | 296 | 1.000 | z \in \mathcal{P} | | |
| fong-spivak-seven-sketches_FO3318 | 296 | 1.000 | z \rightarrow x | | |
| fong-spivak-seven-sketches_FO3319 | 296 | 1.000 | z \rightarrow y | | |
| fong-spivak-seven-sketches_FO3320 | 296 | 1.000 | z^{\prime} | | |
| fong-spivak-seven-sketches_FO3321 | 296 | 1.000 | z^{\prime} \rightarrow x | | |
| fong-spivak-seven-sketches_FO3322 | 296 | 1.000 | z^{\prime} \rightarrow y | | |
| fong-spivak-seven-sketches_FO3323 | 296 | 1.000 | z^{\prime} \rightarrow z | | |
| fong-spivak-seven-sketches_FO3324 | 296 | 1.000 | z \leq x | | |
| fong-spivak-seven-sketches_FO3325 | 296 | 1.000 | z \leq y | | |
| fong-spivak-seven-sketches_FO3326 | 296 | 1.000 | z^{\prime} \leq x | | |
| fong-spivak-seven-sketches_FO3327 | 296 | 1.000 | z^{\prime} \leq y | | |
| fong-spivak-seven-sketches_FO3328 | 296 | 1.000 | z^{\prime} \leq z | | |
| fong-spivak-seven-sketches_FO3329 | 296 | 0.867 | z=x \wedge y | | |
| fong-spivak-seven-sketches_FO3330 | 296 | 0.989 | \left(\mathrm{id}_{c}, \mathrm{id}_{d}\right) | | |
| fong-spivak-seven-sketches_FO3331 | 296 | 0.621 | \left(\left(f_{1}, g_{1}\right) \stackrel{\circ}{\mathfrak{g}}\left(f_{2}, g_{2}\right)\right) \stackrel{\circ}{\mathfrak{g}}\left(f_{3}, g_{3}\right)=\left(f_{1}, g_{1}\right) \stackrel{\circ}{\mathfrak{g}}\left(\left(f_{2}, g_{2}\right) \stackrel{\circ}{\mathfrak{g}}\left(f_{3}, g_{3}\right)\right) | | |
| fong-spivak-seven-sketches_FO3332 | 296 | 0.400 | \left(f_{1}\right. | | |
| fong-spivak-seven-sketches_FO3333 | 296 | 0.400 | \left.f_{2}\right) | | |
| fong-spivak-seven-sketches_FO3334 | 296 | 0.400 | f_{3} | | |
| fong-spivak-seven-sketches_FO3335 | 296 | 0.030 | \left.\left.g_{2}\right) \stackrel{\circ}{9} g_{3}\right) | | |
| fong-spivak-seven-sketches_FO3336 | 296 | 0.030 | \left(f_{1} ;\left(f_{2} ; f_{3}\right), g_{1}\right. | | |
| fong-spivak-seven-sketches_FO3337 | 296 | 0.030 | \left(g_{2} ; g_{3}\right) | | |
| fong-spivak-seven-sketches_FO3338 | 296 | 0.997 | (1,2) | | |
| fong-spivak-seven-sketches_FO3339 | 296 | 0.929 | (1,1) \rightarrow(1,2) | | |
| fong-spivak-seven-sketches_FO3340 | 296 | 1.000 | \mathbf{1} \times \mathcal{C} \cong \mathcal{C} | | |
| fong-spivak-seven-sketches_FO3341 | 296 | 1.000 | X=P \times Q | | |
| fong-spivak-seven-sketches_FO3342 | 296 | 0.980 | \mathcal{X}=\mathcal{P} \times Q | | |
| fong-spivak-seven-sketches_FO3343 | 296 | 0.998 | X \stackrel{p_{X}}{\longleftarrow} Z \xrightarrow{p_{Y}} Y | | |
| fong-spivak-seven-sketches_FO3344 | 296 | 0.998 | X \stackrel{p_{X}^{\prime}}{\longleftarrow} Z^{\prime} \xrightarrow{p_{Y}^{\prime}} \Upsilon | | |
| fong-spivak-seven-sketches_FO3345 | 296 | 0.998 | f: Z^{\prime} \rightarrow Z | | |
| fong-spivak-seven-sketches_FO3346 | 296 | 1.000 | f ; p_{X}=p_{X}^{\prime} | | |
| fong-spivak-seven-sketches_FO3347 | 296 | 1.000 | f ; p_{Y}=p_{Y}^{\prime} | | |
| fong-spivak-seven-sketches_FO3348 | 296 | 0.894 | Y^{\prime \prime} | | |
| fong-spivak-seven-sketches_FO3349 | 296 | 1.000 | Z \in \operatorname{Cone}(X, Y) | | |
| fong-spivak-seven-sketches_FO3350 | 296 | 0.439 | Z^{\prime} \rightarrow Z | | |
| fong-spivak-seven-sketches_FO3351 | 296 | 0.521 | \stackrel{v_{1}}{\bullet} v_{2} | | |
| fong-spivak-seven-sketches_FO3352 | 296 | 0.521 | D\left(v_{1}\right)=A | | |
| fong-spivak-seven-sketches_FO3353 | 296 | 0.521 | D\left(v_{2}\right)=B | | |
| fong-spivak-seven-sketches_FO3354 | 297 | 0.274 | F^{\mathrm{op}}: \mathcal{C}{ }^{\mathrm{op}} \rightarrow \mathcal{D}^{\mathrm{op}} | | |
| fong-spivak-seven-sketches_FO3355 | 297 | 0.274 | c \in | | |
| fong-spivak-seven-sketches_FO3356 | 297 | 0.904 | \mathrm{Ob}\left(\mathcal{C}^{\mathrm{op}}\right)=\mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO3357 | 297 | 0.904 | F^{\mathrm{op}}(c):=F(c) | | |
| fong-spivak-seven-sketches_FO3358 | 297 | 1.000 | f^{\prime}: c_{2} \rightarrow c_{1} | | |
| fong-spivak-seven-sketches_FO3359 | 297 | 1.000 | F\left(f^{\prime}\right): F\left(c_{2}\right) \rightarrow F\left(c_{1}\right) | | |
| fong-spivak-seven-sketches_FO3360 | 297 | 0.951 | F\left(f^{\prime}\right)^{\prime}: F^{\mathrm{op}}\left(c_{1}\right) \rightarrow F^{\mathrm{op}}\left(c_{2}\right) | | |
| fong-spivak-seven-sketches_FO3361 | 297 | 0.951 | F^{\mathrm{op}}(f):=F\left(f^{\prime}\right)^{\prime} | | |
| fong-spivak-seven-sketches_FO3362 | 297 | 0.951 | \left(-{ }^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO3363 | 297 | 0.815 | F^{\mathrm{op}} | | |
| fong-spivak-seven-sketches_FO3364 | 298 | 0.666 | \mathcal{X}^{\mathrm{op}} \times \mathcal{y} | | |
| fong-spivak-seven-sketches_FO3365 | 298 | 0.299 | \Lambda: \mathcal{X} \mapsto y | | |
| fong-spivak-seven-sketches_FO3366 | 298 | 0.299 | \mathcal{X}^{\mathrm{op}} \times y \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO3367 | 298 | 0.301 | x^{\mathrm{op}} \times y | | |
| fong-spivak-seven-sketches_FO3368 | 298 | 1.000 | \Lambda | | |
| fong-spivak-seven-sketches_FO3369 | 298 | 0.348 | \Phi: \mathcal{X}^{\mathrm{op}} \times \mathcal{Y} \rightarrow \mathcal{V} | | |
| fong-spivak-seven-sketches_FO3370 | 298 | 0.348 | \Phi: \operatorname{Ob}\left(\mathcal{X}^{\mathrm{op}} \times \mathcal{Y}\right) \rightarrow \operatorname{Ob}(\mathcal{V}) | | |
| fong-spivak-seven-sketches_FO3371 | 298 | 0.932 | x^{\text {op }} \times y | | |
| fong-spivak-seven-sketches_FO3372 | 299 | 0.997 | \Phi:(T \times E) \rightarrow \$ | | |
| fong-spivak-seven-sketches_FO3373 | 299 | 0.895 | \Phi(( | | |
| fong-spivak-seven-sketches_FO3374 | 299 | 0.895 | ), p)= | | |
| fong-spivak-seven-sketches_FO3375 | 299 | 1.000 | p \in \$=\{\$ 100 \mathrm{~K}, \$ 500 \mathrm{~K}, \$ 1 \mathrm{M}\} | | |
| fong-spivak-seven-sketches_FO3376 | 299 | 0.712 | (\Phi ; \Psi)(-,-)=\mathcal{X}(-, D)+9+z(r,-) | | |
| fong-spivak-seven-sketches_FO3377 | 300 | 0.432 | U_{X} | | |
| fong-spivak-seven-sketches_FO3378 | 300 | 0.705 | I \leq \mathcal{P}(p, p) | | |
| fong-spivak-seven-sketches_FO3379 | 300 | 0.993 | \Phi(p, q) \leq \Phi(p, q) | | |
| fong-spivak-seven-sketches_FO3380 | 300 | 1.000 | x \leq x \vee y | | |
| fong-spivak-seven-sketches_FO3381 | 300 | 0.998 | I= | | |
| fong-spivak-seven-sketches_FO3382 | 300 | 0.998 | \leq \mathcal{P}(p, p) | | |
| fong-spivak-seven-sketches_FO3383 | 300 | 0.972 | \mathcal{P}(p, p)= | | |
| fong-spivak-seven-sketches_FO3384 | 300 | 1.000 | \Phi(p, q) | | |
| fong-spivak-seven-sketches_FO3385 | 300 | 1.000 | \Phi(p, q)= | | |
| fong-spivak-seven-sketches_FO3386 | 300 | 0.825 | \mathcal{P}^{\text {op }} \times \mathcal{Q} \rightarrow | | |
| fong-spivak-seven-sketches_FO3387 | 300 | 0.825 | p \leq p_{1} | | |
| fong-spivak-seven-sketches_FO3388 | 300 | 1.000 | \Phi\left(p_{1}, q\right) \leq \Phi(p, q) | | |
| fong-spivak-seven-sketches_FO3389 | 300 | 1.000 | \mathcal{P}\left(p, p_{1}\right)= | | |
| fong-spivak-seven-sketches_FO3390 | 300 | 1.000 | \Phi\left(p_{1}, q\right)=\Phi(p, q)= | | |
| fong-spivak-seven-sketches_FO3391 | 300 | 1.000 | \Phi\left(p_{1}, q\right)= | | |
| fong-spivak-seven-sketches_FO3392 | 300 | 0.772 | \bigvee_{p_{1} \in \mathcal{P}} \mathcal{P}\left(p, p_{1}\right) \wedge \Phi\left(p_{1}, q\right)=\bigvee_{p_{1} \in \mathcal{P}} | | |
| fong-spivak-seven-sketches_FO3393 | 300 | 0.644 | \Phi(p, q)=\bigvee_{p_{1} \in \mathcal{P}} \mathcal{P}\left(p, p_{1}\right) \wedge \Phi\left(p_{1}, q\right) | | |
| fong-spivak-seven-sketches_FO3394 | 301 | 1.000 | v \otimes I=v | | |
| fong-spivak-seven-sketches_FO3395 | 301 | 0.999 | I \leq Q(q, q) | | |
| fong-spivak-seven-sketches_FO3396 | 301 | 1.000 | v_{1} \leq v_{2} | | |
| fong-spivak-seven-sketches_FO3397 | 301 | 1.000 | v \otimes v_{1} \leq v \otimes v_{2} | | |
| fong-spivak-seven-sketches_FO3398 | 301 | 0.830 | \widehat{\mathrm{id}}: \mathcal{P} \mapsto \mathcal{P} | | |
| fong-spivak-seven-sketches_FO3399 | 301 | 0.830 | \widehat{\mathrm{id}}(p, q)=\mathcal{P}(p, q) | | |
| fong-spivak-seven-sketches_FO3400 | 301 | 0.963 | \widehat{\operatorname{id}}(p, q):=\mathcal{P}(\operatorname{id}(p), q)=\mathcal{P}(p, q) | | |
| fong-spivak-seven-sketches_FO3401 | 301 | 0.771 | \check{+}: \mathbb{R} \rightarrow \mathbb{R} \times \mathbb{R} \times \mathbb{R} | | |
| fong-spivak-seven-sketches_FO3402 | 301 | 0.771 | \mathbb{R}(a, b+c+d) | | |
| fong-spivak-seven-sketches_FO3403 | 301 | 0.771 | a \leq b+c+d | | |
| fong-spivak-seven-sketches_FO3404 | 301 | 0.976 | \widehat{F}(p, q)=Q(F(p), q) | | |
| fong-spivak-seven-sketches_FO3405 | 301 | 0.976 | \check{G}(p, q)=Q(p, G(q)) | | |
| fong-spivak-seven-sketches_FO3406 | 301 | 0.992 | Q(F(p), q)=Q(p, G(q)) | | |
| fong-spivak-seven-sketches_FO3407 | 301 | 0.992 | \widehat{F}=\check{G} | | |
| fong-spivak-seven-sketches_FO3408 | 301 | 0.898 | \widehat{\mathrm{id}}=\widehat{\mathrm{id}} | | |
| fong-spivak-seven-sketches_FO3409 | 301 | 1.000 | \otimes: \mathcal{P} \times \mathcal{P} \rightarrow \mathcal{P} | | |
| fong-spivak-seven-sketches_FO3410 | 302 | 1.000 | \left(x_{1}, x_{2}\right) \leq\left(y_{1}, y_{2}\right) | | |
| fong-spivak-seven-sketches_FO3411 | 302 | 1.000 | \mathcal{P} \otimes \mathcal{P} | | |
| fong-spivak-seven-sketches_FO3412 | 302 | 1.000 | g_{E}(5,3)= | | |
| fong-spivak-seven-sketches_FO3413 | 302 | 1.000 | g_{F}(5,3)=2 | | |
| fong-spivak-seven-sketches_FO3414 | 302 | 1.000 | g_{E}(3,5)= | | |
| fong-spivak-seven-sketches_FO3415 | 302 | 1.000 | g_{F}(3,5)=-2 | | |
| fong-spivak-seven-sketches_FO3416 | 302 | 0.988 | )=5 | | |
| fong-spivak-seven-sketches_FO3417 | 302 | 0.990 | )=-5 | | |
| fong-spivak-seven-sketches_FO3418 | 302 | 0.999 | )=6 | | |
| fong-spivak-seven-sketches_FO3419 | 302 | 1.000 | q_{G}(-2,3)=2, q_{F}(-2,3)=-13 | | |
| fong-spivak-seven-sketches_FO3420 | 302 | 1.000 | q_{G}(2,3)=-1, q_{F}(2,3)=7 | | |
| fong-spivak-seven-sketches_FO3421 | 302 | 0.963 | \mathcal{C}(x, y) | | |
| fong-spivak-seven-sketches_FO3422 | 302 | 0.430 | \mathrm{id}_{x}: I \rightarrow \mathcal{C}(x, x) | | |
| fong-spivak-seven-sketches_FO3423 | 302 | 0.650 | \mathrm{id}_{x}:\{1\} \rightarrow \mathcal{C}(x, x) | | |
| fong-spivak-seven-sketches_FO3424 | 302 | 0.650 | \mathrm{id}_{x}=\mathrm{id}_{x}(1) \in \mathcal{C}(x, x) | | |
| fong-spivak-seven-sketches_FO3425 | 302 | 0.975 | { }_{9}: \mathcal{C}(x, y) \otimes \mathcal{C}(y, z) \rightarrow \mathcal{C}(x, z) | | |
| fong-spivak-seven-sketches_FO3426 | 302 | 0.300 | \mathcal{C}(x, y) \times \mathcal{C}(y, z) \rightarrow \mathcal{C}(x, z) | | |
| fong-spivak-seven-sketches_FO3427 | 302 | 0.701 | I \rightarrow \mathcal{X}(x, x) | | |
| fong-spivak-seven-sketches_FO3428 | 302 | 0.937 | 0 \geq \mathcal{X}(x, x) | | |
| fong-spivak-seven-sketches_FO3429 | 302 | 0.937 | \mathcal{X}(x, x)=0 | | |
| fong-spivak-seven-sketches_FO3431 | 303 | 0.891 | x \in \mathcal{X} | | |
| fong-spivak-seven-sketches_FO3432 | 303 | 0.891 | y \in y | | |
| fong-spivak-seven-sketches_FO3433 | 303 | 0.891 | (x, y) \leq\left(x^{\prime}, y^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO3434 | 303 | 0.891 | y \leq y^{\prime} | | |
| fong-spivak-seven-sketches_FO3435 | 303 | 0.917 | \Phi: x_{1} \rightarrow x_{2} | | |
| fong-spivak-seven-sketches_FO3436 | 303 | 0.998 | y_{1} | | |
| fong-spivak-seven-sketches_FO3437 | 303 | 0.998 | x_{1} | | |
| fong-spivak-seven-sketches_FO3438 | 303 | 0.998 | y_{2} | | |
| fong-spivak-seven-sketches_FO3439 | 303 | 0.998 | x_{2} | | |
| fong-spivak-seven-sketches_FO3440 | 303 | 0.998 | \left(y_{1}, y_{2}\right) | | |
| fong-spivak-seven-sketches_FO3441 | 303 | 1.000 | \left(x_{1}, x_{2}\right) | | |
| fong-spivak-seven-sketches_FO3442 | 303 | 0.670 | \mathcal{X} \times \mathbf{1} \rightarrow \mathcal{X} | | |
| fong-spivak-seven-sketches_FO3443 | 303 | 0.670 | \alpha:(\mathcal{X} \times \mathbf{1})^{\mathrm{op}} \times \mathcal{X} \rightarrow \mathcal{V} | | |
| fong-spivak-seven-sketches_FO3444 | 303 | 0.670 | \alpha((x, 1), y):=\mathcal{X}(x, y) | | |
| fong-spivak-seven-sketches_FO3445 | 303 | 0.948 | \alpha^{-1}: x \mapsto x \times \mathbf{1} | | |
| fong-spivak-seven-sketches_FO3446 | 303 | 0.948 | \alpha^{-1}(x,(y, 1)):=x(x, y) | | |
| fong-spivak-seven-sketches_FO3447 | 303 | 0.943 | \alpha^{-1} ; \alpha=\mathrm{U}_{X} | | |
| fong-spivak-seven-sketches_FO3448 | 303 | 0.552 | \mathcal{X}(x, y) \otimes \mathcal{X}(y, z) \leq \mathcal{X}(x, z) | | |
| fong-spivak-seven-sketches_FO3449 | 303 | 0.473 | \alpha^{-1}=\mathrm{U}_{X \times 1} | | |
| fong-spivak-seven-sketches_FO3450 | 303 | 0.556 | \beta((1, x), y):=\mathcal{X}(x, y) | | |
| fong-spivak-seven-sketches_FO3451 | 303 | 0.556 | \beta: \mathbf{1} \times \mathcal{X} \mapsto X | | |
| fong-spivak-seven-sketches_FO3452 | 304 | 0.979 | \alpha^{-1} | | |
| fong-spivak-seven-sketches_FO3453 | 304 | 0.571 | \Phi(x, y) | | |
| fong-spivak-seven-sketches_FO3454 | 304 | 0.571 | (x, y) \in \mathcal{X}^{\mathrm{op}} \times \mathcal{X} | | |
| fong-spivak-seven-sketches_FO3455 | 304 | 0.857 | \mathrm{U}_{\mathcal{X}}(a, b)=\mathcal{X}(a, b) | | |
| fong-spivak-seven-sketches_FO3456 | 305 | 1.000 | (f ; g)(i)=g(f(i)) | | |
| fong-spivak-seven-sketches_FO3457 | 305 | 1.000 | 1 \leq i \leq m | | |
| fong-spivak-seven-sketches_FO3458 | 305 | 0.562 | \operatorname{id}_{m}: m \rightarrow m | | |
| fong-spivak-seven-sketches_FO3459 | 305 | 0.562 | \operatorname{id}_{m}(i)=i | | |
| fong-spivak-seven-sketches_FO3460 | 305 | 0.562 | \operatorname{id}_{2} | | |
| fong-spivak-seven-sketches_FO3461 | 305 | 0.562 | \operatorname{id}_{8} | | |
| fong-spivak-seven-sketches_FO3462 | 305 | 1.000 | \sigma_{3,5}: 8 \rightarrow 8 | | |
| fong-spivak-seven-sketches_FO3463 | 305 | 1.000 | m_{1} \leq n_{1} | | |
| fong-spivak-seven-sketches_FO3464 | 305 | 1.000 | m_{2} \leq n_{2} | | |
| fong-spivak-seven-sketches_FO3465 | 305 | 1.000 | m_{1}+m_{2} \leq n_{1}+n_{2} | | |
| fong-spivak-seven-sketches_FO3466 | 305 | 0.970 | d \in \mathbb{N} | | |
| fong-spivak-seven-sketches_FO3467 | 305 | 0.970 | d+m=n | | |
| fong-spivak-seven-sketches_FO3468 | 305 | 0.996 | m=n+d | | |
| fong-spivak-seven-sketches_FO3469 | 305 | 0.999 | q \in \mathbb{N} | | |
| fong-spivak-seven-sketches_FO3470 | 305 | 0.999 | m * q=d | | |
| fong-spivak-seven-sketches_FO3471 | 305 | 0.932 | 3 \preceq 3 | | |
| fong-spivak-seven-sketches_FO3472 | 305 | 0.932 | 5 \leq ? 7 | | |
| fong-spivak-seven-sketches_FO3473 | 305 | 1.000 | \mathbf{B i j}(m, n):=\{f: \underline{m} \rightarrow \underline{n} \mid f | | |
| fong-spivak-seven-sketches_FO3474 | 305 | 1.000 | \mathbf{B i j}(m, n)=\varnothing | | |
| fong-spivak-seven-sketches_FO3475 | 305 | 1.000 | n \rightarrow n | | |
| fong-spivak-seven-sketches_FO3476 | 305 | 1.000 | \underline{n} \rightarrow \underline{n} | | |
| fong-spivak-seven-sketches_FO3477 | 305 | 1.000 | i \mapsto i | | |
| fong-spivak-seven-sketches_FO3478 | 306 | 1.000 | m+n \rightarrow n+m | | |
| fong-spivak-seven-sketches_FO3479 | 306 | 1.000 | \sigma_{m, n}: \underline{m+n} \rightarrow \underline{n+m} | | |
| fong-spivak-seven-sketches_FO3480 | 306 | 1.000 | n \rightarrow p | | |
| fong-spivak-seven-sketches_FO3481 | 306 | 1.000 | (f+g):(m+n) \rightarrow | | |
| fong-spivak-seven-sketches_FO3482 | 306 | 1.000 | \left(m^{\prime}+n^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO3483 | 306 | 0.817 | \operatorname{Corel}(m, n) | | |
| fong-spivak-seven-sketches_FO3484 | 306 | 0.817 | \underline{m+n} | | |
| fong-spivak-seven-sketches_FO3485 | 306 | 1.000 | i \sim j | | |
| fong-spivak-seven-sketches_FO3486 | 306 | 1.000 | i=j | | |
| fong-spivak-seven-sketches_FO3487 | 306 | 0.981 | m+n+n+m | | |
| fong-spivak-seven-sketches_FO3488 | 306 | 0.999 | |i-j|=m+n+n | | |
| fong-spivak-seven-sketches_FO3489 | 306 | 0.969 | m+1 \leq i \leq m+n+n | | |
| fong-spivak-seven-sketches_FO3490 | 306 | 0.969 | m+1 \leq j \leq m+n+n | | |
| fong-spivak-seven-sketches_FO3491 | 306 | 0.969 | |i-j|=n | | |
| fong-spivak-seven-sketches_FO3492 | 306 | 1.000 | \dot{\sim} | | |
| fong-spivak-seven-sketches_FO3493 | 306 | 1.000 | n+p | | |
| fong-spivak-seven-sketches_FO3494 | 306 | 0.624 | \simeq | | |
| fong-spivak-seven-sketches_FO3495 | 306 | 0.624 | m+p | | |
| fong-spivak-seven-sketches_FO3496 | 306 | 0.624 | i \simeq k | | |
| fong-spivak-seven-sketches_FO3497 | 306 | 0.624 | \bar{j} \dot{\sim} k | | |
| fong-spivak-seven-sketches_FO3498 | 306 | 0.990 | \overline{m+n} | | |
| fong-spivak-seven-sketches_FO3499 | 306 | 0.990 | \sim^{\prime} | | |
| fong-spivak-seven-sketches_FO3500 | 306 | 0.990 | m^{\prime}+n^{\prime} | | |
| fong-spivak-seven-sketches_FO3501 | 306 | 0.931 | \left(\sim+\sim^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO3502 | 306 | 0.931 | m+n+m^{\prime}+n^{\prime} | | |
| fong-spivak-seven-sketches_FO3503 | 306 | 1.000 | i \leq m+n | | |
| fong-spivak-seven-sketches_FO3504 | 306 | 1.000 | j \leq m+n | | |
| fong-spivak-seven-sketches_FO3505 | 306 | 1.000 | m+n+1 \leq i | | |
| fong-spivak-seven-sketches_FO3506 | 306 | 1.000 | m+n+1 \leq j | | |
| fong-spivak-seven-sketches_FO3507 | 306 | 1.000 | i \sim^{\prime} j | | |
| fong-spivak-seven-sketches_FO3508 | 306 | 1.000 | \boldsymbol{\operatorname { R e l }}(m, n) | | |
| fong-spivak-seven-sketches_FO3509 | 306 | 1.000 | \underline{m} \times \underline{n} | | |
| fong-spivak-seven-sketches_FO3510 | 306 | 1.000 | \{(i, j) \in \underline{n} \times \underline{n} \mid i=j\} | | |
| fong-spivak-seven-sketches_FO3511 | 306 | 0.999 | (i, j) \in(\underline{m+n}) \times(\underline{n+m}) | | |
| fong-spivak-seven-sketches_FO3512 | 306 | 0.999 | i \leq m | | |
| fong-spivak-seven-sketches_FO3513 | 306 | 0.999 | m+1 \leq j | | |
| fong-spivak-seven-sketches_FO3514 | 306 | 0.999 | i+m=j | | |
| fong-spivak-seven-sketches_FO3515 | 306 | 1.000 | m+1 \leq i | | |
| fong-spivak-seven-sketches_FO3516 | 306 | 1.000 | j \leq m | | |
| fong-spivak-seven-sketches_FO3517 | 306 | 1.000 | j+m=i | | |
| fong-spivak-seven-sketches_FO3518 | 306 | 0.986 | R^{\prime} \subseteq \underline{m}^{\prime} \times \underline{n^{\prime}} | | |
| fong-spivak-seven-sketches_FO3519 | 306 | 0.986 | \left(R+R^{\prime}\right) \subseteq | | |
| fong-spivak-seven-sketches_FO3520 | 306 | 0.999 | \underline{m+m^{\prime}} \times \underline{n+n^{\prime}} | | |
| fong-spivak-seven-sketches_FO3521 | 306 | 0.324 | R_{1} \sqcup R_{2} \subseteq\left(\underline{m_{1}} \times \underline{n_{1}}\right) \sqcup\left(\underline{m_{2}} \times \underline{n_{2}}\right) \subseteq\left(\underline{m_{1} \sqcup} \underline{m_{2}}\right) \times\left(\underline{n_{1} \sqcup} \underline{n_{2}}\right) | | |
| fong-spivak-seven-sketches_FO3522 | 307 | 1.000 | \leq_{P} | | |
| fong-spivak-seven-sketches_FO3523 | 307 | 1.000 | x \leq_{P} y | | |
| fong-spivak-seven-sketches_FO3524 | 307 | 1.000 | x_{0}, \ldots, x_{n} | | |
| fong-spivak-seven-sketches_FO3525 | 307 | 1.000 | x_{0}=x | | |
| fong-spivak-seven-sketches_FO3526 | 307 | 1.000 | x_{n}=y | | |
| fong-spivak-seven-sketches_FO3527 | 307 | 1.000 | R\left(x_{i}, x_{i+1}\right) | | |
| fong-spivak-seven-sketches_FO3528 | 307 | 1.000 | 0 \leq i \leq n-1 | | |
| fong-spivak-seven-sketches_FO3529 | 307 | 1.000 | f\left(x_{i}\right) \leq_{Q} f\left(x_{i+1}\right) | | |
| fong-spivak-seven-sketches_FO3530 | 307 | 1.000 | f\left(x_{0}\right) \leq_{Q} f\left(x_{i}\right) | | |
| fong-spivak-seven-sketches_FO3531 | 307 | 0.994 | n=1, x_{0}=x | | |
| fong-spivak-seven-sketches_FO3532 | 307 | 1.000 | R(g(a), g(b)) | | |
| fong-spivak-seven-sketches_FO3533 | 307 | 1.000 | a \leq_{Q} b | | |
| fong-spivak-seven-sketches_FO3534 | 307 | 1.000 | (g(a), g(b)) \notin R | | |
| fong-spivak-seven-sketches_FO3535 | 307 | 1.000 | Q:=\{1\} | | |
| fong-spivak-seven-sketches_FO3536 | 307 | 1.000 | P:=\{1\} | | |
| fong-spivak-seven-sketches_FO3537 | 307 | 1.000 | R=\varnothing | | |
| fong-spivak-seven-sketches_FO3538 | 307 | 1.000 | (g(1), g(1)) \notin R | | |
| fong-spivak-seven-sketches_FO3539 | 307 | 1.000 | \operatorname{Mor}(\mathcal{C}) \rightarrow \operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO3540 | 307 | 0.992 | q \in \operatorname{Mor}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO3541 | 307 | 0.992 | q: y \rightarrow z | | |
| fong-spivak-seven-sketches_FO3542 | 307 | 0.991 | \operatorname{cod}(q):=z | | |
| fong-spivak-seven-sketches_FO3543 | 307 | 1.000 | F: \mathcal{G} \rightarrow \mathcal{C} | | |
| fong-spivak-seven-sketches_FO3544 | 307 | 1.000 | \operatorname{Ob}(\mathcal{G}) \rightarrow \operatorname{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO3545 | 307 | 0.985 | \operatorname{Ob}(\mathcal{G})=V | | |
| fong-spivak-seven-sketches_FO3546 | 307 | 0.970 | \operatorname{Mor}(\mathcal{G}) | | |
| fong-spivak-seven-sketches_FO3547 | 307 | 1.000 | A \subseteq \operatorname{Mor}(\mathcal{G}) | | |
| fong-spivak-seven-sketches_FO3548 | 307 | 1.000 | \operatorname{Mor}(\mathcal{G}) \rightarrow \operatorname{Mor}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO3549 | 307 | 1.000 | \operatorname{dom}(F(r))=F(\operatorname{dom}(r)) | | |
| fong-spivak-seven-sketches_FO3550 | 307 | 1.000 | \operatorname{cod}(F(r))=F(\operatorname{cod}(r)) | | |
| fong-spivak-seven-sketches_FO3551 | 307 | 1.000 | r: w \rightarrow x | | |
| fong-spivak-seven-sketches_FO3552 | 307 | 1.000 | r \in A | | |
| fong-spivak-seven-sketches_FO3553 | 307 | 1.000 | \operatorname{dom}(r)=s(r) | | |
| fong-spivak-seven-sketches_FO3554 | 307 | 1.000 | \operatorname{cod}(r)=t(r) | | |
| fong-spivak-seven-sketches_FO3555 | 308 | 0.561 | f: V \rightarrow \mathrm{Ob}(\mathcal{C}) | | |
| fong-spivak-seven-sketches_FO3556 | 308 | 0.561 | g: A \rightarrow | | |
| fong-spivak-seven-sketches_FO3557 | 308 | 1.000 | p:=\left(v_{0}, a_{1}, a_{2}, \ldots, a_{n}\right) | | |
| fong-spivak-seven-sketches_FO3558 | 308 | 1.000 | v_{0} \in V | | |
| fong-spivak-seven-sketches_FO3559 | 308 | 1.000 | a_{i} \in A, v_{0}=s\left(a_{1}\right) | | |
| fong-spivak-seven-sketches_FO3560 | 308 | 1.000 | 1 \leq i \leq n-1 | | |
| fong-spivak-seven-sketches_FO3561 | 308 | 1.000 | g\left(a_{i}\right) | | |
| fong-spivak-seven-sketches_FO3562 | 308 | 1.000 | f\left(v_{0}\right) | | |
| fong-spivak-seven-sketches_FO3563 | 308 | 1.000 | g\left(a_{i+1}\right) | | |
| fong-spivak-seven-sketches_FO3564 | 308 | 0.605 | F(p):=\operatorname{id}_{f\left(v_{0}\right)} ; g\left(a_{1}\right) ; \ldots ; g\left(a_{n}\right) | | |
| fong-spivak-seven-sketches_FO3565 | 308 | 1.000 | \left(f^{\prime}, g^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO3566 | 308 | 1.000 | f=f^{\prime} | | |
| fong-spivak-seven-sketches_FO3567 | 308 | 0.992 | g=g^{\prime} | | |
| fong-spivak-seven-sketches_FO3568 | 308 | 1.000 | F^{\prime}: \mathcal{G} \rightarrow \mathcal{C} | | |
| fong-spivak-seven-sketches_FO3569 | 308 | 1.000 | F^{\prime} | | |
| fong-spivak-seven-sketches_FO3570 | 308 | 0.499 | U(\mathcal{C}) \in | | |
| fong-spivak-seven-sketches_FO3571 | 308 | 1.000 | a^{i} * a^{j}=a^{i+j} | | |
| fong-spivak-seven-sketches_FO3572 | 308 | 1.000 | a^{i} \mapsto i | | |
| fong-spivak-seven-sketches_FO3573 | 308 | 0.560 | \operatorname{prop} \operatorname{Free}(G, s, t) | | |
| fong-spivak-seven-sketches_FO3574 | 308 | 0.963 | (\Gamma, \ell) | | |
| fong-spivak-seven-sketches_FO3575 | 308 | 0.997 | \ell(v) \in G | | |
| fong-spivak-seven-sketches_FO3576 | 308 | 0.997 | s(\ell(v))=i n(v) | | |
| fong-spivak-seven-sketches_FO3577 | 308 | 1.000 | \Gamma | | |
| fong-spivak-seven-sketches_FO3578 | 309 | 0.491 | \left(f+\mathrm{id}_{1}+\mathrm{id}_{1}\right) \stackrel{\circ}{9}\left(\sigma+\mathrm{id}_{1}\right) \stackrel{\circ}{9}\left(\mathrm{id}_{1}+h\right) \stackrel{\circ}{\circ} \sigma \stackrel{\circ}{g} | | |
| fong-spivak-seven-sketches_FO3579 | 309 | 0.491 | G=\{f: 1 \rightarrow | | |
| fong-spivak-seven-sketches_FO3580 | 309 | 0.867 | 1, g: 2 \rightarrow 2, h: 2 \rightarrow 1\}: | | |
| fong-spivak-seven-sketches_FO3581 | 309 | 0.996 | \{\{1\},\{2\},\{3\}\} | | |
| fong-spivak-seven-sketches_FO3582 | 309 | 0.992 | (R, 0,+, 1, *) | | |
| fong-spivak-seven-sketches_FO3583 | 309 | 1.000 | n=4 | | |
| fong-spivak-seven-sketches_FO3584 | 309 | 1.000 | A, B \in \operatorname{Mat}_{2}(\mathbb{N}) | | |
| fong-spivak-seven-sketches_FO3585 | 309 | 1.000 | A * B \neq B * A | | |
| fong-spivak-seven-sketches_FO3586 | 309 | 0.737 | (A * B)(1,1)= | | |
| fong-spivak-seven-sketches_FO3587 | 309 | 0.999 | 0 * 0+1 * 1=1 | | |
| fong-spivak-seven-sketches_FO3588 | 309 | 0.999 | (B * A)(1,1)=0 * 0+0 * 0=0 | | |
| fong-spivak-seven-sketches_FO3589 | 309 | 0.991 | (16 x+4 y, x+4 y) | | |
| fong-spivak-seven-sketches_FO3590 | 309 | 1.000 | A=\left(\begin{array}{lll}3 & 3 & 1 \\ 2 & 0 & 4\end{array}\right) | | |
| fong-spivak-seven-sketches_FO3591 | 309 | 1.000 | B=\left(\begin{array}{llll}2 & 5 & 6 & 1\end{array}\right) | | |
| fong-spivak-seven-sketches_FO3592 | 310 | 1.000 | c_{n} | | |
| fong-spivak-seven-sketches_FO3593 | 310 | 0.999 | s_{a}: 1 \rightarrow 1 | | |
| fong-spivak-seven-sketches_FO3594 | 310 | 0.993 | g_{3}: m \times n \rightarrow m \times n | | |
| fong-spivak-seven-sketches_FO3595 | 310 | 1.000 | (i-1) m+j | | |
| fong-spivak-seven-sketches_FO3596 | 310 | 1.000 | (j-1) n+i | | |
| fong-spivak-seven-sketches_FO3597 | 310 | 1.000 | 1 \leq i \leq n | | |
| fong-spivak-seven-sketches_FO3598 | 310 | 1.000 | 1 \leq j \leq m | | |
| fong-spivak-seven-sketches_FO3599 | 311 | 0.999 | a_{m} | | |
| fong-spivak-seven-sketches_FO3600 | 311 | 0.087 | g=g_{1} g_{2} g_{3} g_{4}: m \rightarrow | | |
| fong-spivak-seven-sketches_FO3601 | 312 | 0.729 | \sigma_{M, M}: \mu=\mu | | |
| fong-spivak-seven-sketches_FO3602 | 312 | 1.000 | R^{0}=\{()\} | | |
| fong-spivak-seven-sketches_FO3603 | 312 | 0.977 | \operatorname{prop}(\operatorname{Mat}(R) | | |
| fong-spivak-seven-sketches_FO3604 | 312 | 0.874 | R^{0} | | |
| fong-spivak-seven-sketches_FO3605 | 312 | 0.874 | \{1\} . U | | |
| fong-spivak-seven-sketches_FO3606 | 312 | 1.000 | R^{m} \times R^{n} \cong R^{m+n} | | |
| fong-spivak-seven-sketches_FO3607 | 312 | 0.889 | (m, \mu, \eta) | | |
| fong-spivak-seven-sketches_FO3608 | 312 | 0.889 | m \in \mathbb{N}, \mu: m+m \rightarrow m | | |
| fong-spivak-seven-sketches_FO3609 | 312 | 0.889 | \eta: 0 \rightarrow m | | |
| fong-spivak-seven-sketches_FO3610 | 312 | 1.000 | \mu(\eta, x)=x=\mu(x, \eta) | | |
| fong-spivak-seven-sketches_FO3611 | 312 | 1.000 | \mu(x, \mu(y, z))=\mu(\mu(x, y), z) | | |
| fong-spivak-seven-sketches_FO3612 | 312 | 0.970 | 0 \rightarrow m | | |
| fong-spivak-seven-sketches_FO3613 | 312 | 0.970 | m \in \mathbb{N} | | |
| fong-spivak-seven-sketches_FO3614 | 312 | 1.000 | m=2, \mu | | |
| fong-spivak-seven-sketches_FO3615 | 313 | 0.999 | U(\eta): R^{0} \rightarrow R^{m} | | |
| fong-spivak-seven-sketches_FO3616 | 313 | 0.999 | U(\eta)(1):= | | |
| fong-spivak-seven-sketches_FO3617 | 313 | 0.985 | (0, \ldots, 0) | | |
| fong-spivak-seven-sketches_FO3618 | 313 | 0.985 | U(\mu): R^{m} \times R^{m} \rightarrow R^{m} | | |
| fong-spivak-seven-sketches_FO3619 | 313 | 1.000 | R^{m} | | |
| fong-spivak-seven-sketches_FO3620 | 313 | 0.860 | (1, \supset-, \sim) | | |
| fong-spivak-seven-sketches_FO3621 | 313 | 0.860 | (5,3) \mapsto 8 | | |
| fong-spivak-seven-sketches_FO3622 | 313 | 0.989 | \left\{(x, y, z) \in R^{3} \mid x=\right. | | |
| fong-spivak-seven-sketches_FO3623 | 313 | 1.000 | y+z\} | | |
| fong-spivak-seven-sketches_FO3624 | 313 | 0.395 | \left\{(x, y, z) \in R^{3} \mid x=y=\right. | | |
| fong-spivak-seven-sketches_FO3625 | 313 | 1.000 | z\} | | |
| fong-spivak-seven-sketches_FO3626 | 313 | 0.926 | B+C \subseteq R^{m+p} \times R^{n+q} | | |
| fong-spivak-seven-sketches_FO3627 | 313 | 0.624 | \mathrm{B}\left(g{ }^{\circ}\left(h^{\mathrm{op}}\right)\right)=\mathrm{B}(g) ; \mathrm{B}\left(h^{\mathrm{op}}\right) | | |
| fong-spivak-seven-sketches_FO3628 | 313 | 1.000 | S(h) | | |
| fong-spivak-seven-sketches_FO3629 | 313 | 0.829 | g^{\mathrm{op}}: n \rightarrow m | | |
| fong-spivak-seven-sketches_FO3630 | 313 | 0.829 | h: m \rightarrow p | | |
| fong-spivak-seven-sketches_FO3631 | 313 | 0.882 | \mathrm{B}\left(\left(g^{\mathrm{op}}\right) ; h\right)=\mathrm{B}\left(g^{\mathrm{op}}\right) ; \mathrm{B}(h) | | |
| fong-spivak-seven-sketches_FO3632 | 313 | 0.597 | \{y \in R \mid y=0\} | | |
| fong-spivak-seven-sketches_FO3633 | 313 | 1.000 | \left\{y \in R^{n} \mid y=0\right\} | | |
| fong-spivak-seven-sketches_FO3634 | 313 | 1.000 | S(g) \subseteq R^{m} \times R^{n} | | |
| fong-spivak-seven-sketches_FO3635 | 313 | 1.000 | \left\{x \in R^{m} \mid S(g)=0\right\} | | |
| fong-spivak-seven-sketches_FO3636 | 313 | 0.494 | \{x \in R\} | | |
| fong-spivak-seven-sketches_FO3637 | 313 | 1.000 | R^{n} \subseteq R^{n} | | |
| fong-spivak-seven-sketches_FO3638 | 313 | 0.999 | \left\{y \in R^{n} \mid\right. | | |
| fong-spivak-seven-sketches_FO3639 | 313 | 0.999 | \left.S(g)(x)=y\right\} | | |
| fong-spivak-seven-sketches_FO3640 | 314 | 0.994 | \mathrm{B}(g)=\{(x, y) \mid S(g)(x)=y\} | | |
| fong-spivak-seven-sketches_FO3641 | 314 | 1.000 | S(g)(x)=y | | |
| fong-spivak-seven-sketches_FO3642 | 314 | 1.000 | S(g)(r x)=r y | | |
| fong-spivak-seven-sketches_FO3643 | 314 | 1.000 | S(g)\left(x_{1}+x_{2}\right)=S(g)\left(x_{1}\right)+S(g)\left(x_{2}\right) | | |
| fong-spivak-seven-sketches_FO3644 | 314 | 1.000 | (x, y) \in \mathrm{B}(g) | | |
| fong-spivak-seven-sketches_FO3645 | 314 | 1.000 | (r x, r y) \in \mathrm{B}(g) | | |
| fong-spivak-seven-sketches_FO3646 | 314 | 0.999 | \left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right) \in \mathrm{B}(g) | | |
| fong-spivak-seven-sketches_FO3647 | 314 | 0.999 | \left(x_{1}+x_{2}, y_{1}+y_{2}\right) \in \mathrm{B}(g) | | |
| fong-spivak-seven-sketches_FO3648 | 314 | 1.000 | \mathrm{B}\left(g^{\mathrm{op}}\right) | | |
| fong-spivak-seven-sketches_FO3649 | 314 | 1.000 | C \subseteq R^{n} \times R^{p} | | |
| fong-spivak-seven-sketches_FO3650 | 314 | 1.000 | \left(x_{1}, z_{1}\right),\left(x_{2}, z_{2}\right) \in B ; C | | |
| fong-spivak-seven-sketches_FO3651 | 314 | 1.000 | y_{1}, y_{2} \in R^{n} | | |
| fong-spivak-seven-sketches_FO3652 | 314 | 1.000 | \left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right) \in B | | |
| fong-spivak-seven-sketches_FO3653 | 314 | 1.000 | \left(y_{1}, z_{1}\right),\left(y_{2}, z_{2}\right) \in C | | |
| fong-spivak-seven-sketches_FO3654 | 314 | 1.000 | \left(r x_{1}, r y_{1}\right) \in B | | |
| fong-spivak-seven-sketches_FO3655 | 314 | 1.000 | \left(r y_{1}, r z_{1}\right) \in C | | |
| fong-spivak-seven-sketches_FO3656 | 314 | 1.000 | \left(x_{1}+x_{2}, y_{1}+y_{2}\right) \in B | | |
| fong-spivak-seven-sketches_FO3657 | 314 | 0.996 | \left(y_{1}+y_{2}, z_{1}+z_{2}\right) \in C | | |
| fong-spivak-seven-sketches_FO3658 | 314 | 0.996 | \left(r x_{1}, r z_{1}\right) \in(B ; C) | | |
| fong-spivak-seven-sketches_FO3659 | 314 | 0.996 | \left(x_{1}+x_{2}, z_{1}+z_{2}\right) \in(B ; C) | | |
| fong-spivak-seven-sketches_FO3660 | 314 | 0.996 | (B ; C) \subseteq R^{m} \times R^{p} | | |
| fong-spivak-seven-sketches_FO3661 | 314 | 0.999 | (x, z) | | |
| fong-spivak-seven-sketches_FO3662 | 314 | 0.999 | (y, z) \in C | | |
| fong-spivak-seven-sketches_FO3663 | 314 | 1.000 | (B ; C) | | |
| fong-spivak-seven-sketches_FO3664 | 314 | 0.976 | (x, z) \in(B ; C) | | |
| fong-spivak-seven-sketches_FO3665 | 314 | 0.976 | (r * x, r * y) \in B | | |
| fong-spivak-seven-sketches_FO3666 | 314 | 0.997 | (r * y, r * z) \in C | | |
| fong-spivak-seven-sketches_FO3667 | 314 | 0.997 | (r * x, r * z) \in(B ; C) | | |
| fong-spivak-seven-sketches_FO3668 | 314 | 1.000 | \left(x^{\prime}, z^{\prime}\right) \in(B ; C) | | |
| fong-spivak-seven-sketches_FO3669 | 314 | 1.000 | y^{\prime} \in R^{n} | | |
| fong-spivak-seven-sketches_FO3670 | 314 | 1.000 | \left(y^{\prime}, z^{\prime}\right) \in C | | |
| fong-spivak-seven-sketches_FO3671 | 314 | 0.998 | \left(x+x^{\prime}, y+y^{\prime}\right) \in B | | |
| fong-spivak-seven-sketches_FO3672 | 314 | 0.998 | \left(y+y^{\prime}, z+z^{\prime}\right) \in C | | |
| fong-spivak-seven-sketches_FO3673 | 314 | 0.998 | \left(x+x^{\prime}, z+z^{\prime}\right) \in(B ; C) | | |
| fong-spivak-seven-sketches_FO3674 | 315 | 0.361 | \stackrel{a}{\bullet} \stackrel{b}{\bullet}, \quad \stackrel{a}{\bullet \rightarrow \stackrel{b}{\bullet}}, \quad \stackrel{a}{\bullet \leftrightarrows} \stackrel{b}{\bullet} | | |
| fong-spivak-seven-sketches_FO3675 | 315 | 0.998 | a \not \leq b | | |
| fong-spivak-seven-sketches_FO3676 | 315 | 0.998 | b \not \leq a | | |
| fong-spivak-seven-sketches_FO3677 | 315 | 1.000 | f\left(1_{R}\right)=1_{S} | | |
| fong-spivak-seven-sketches_FO3678 | 315 | 1.000 | f\left(r_{1} *_{R} r_{2}\right)=f\left(r_{1}\right) *_{S} f\left(r_{2}\right) | | |
| fong-spivak-seven-sketches_FO3679 | 315 | 0.966 | R=\left(R, 0_{R},+_{R}, 1_{R},{ }^{*} R\right) | | |
| fong-spivak-seven-sketches_FO3680 | 315 | 1.000 | f: \mathbb{N} \rightarrow R | | |
| fong-spivak-seven-sketches_FO3681 | 315 | 1.000 | 0_{R}, 1 | | |
| fong-spivak-seven-sketches_FO3682 | 315 | 1.000 | 1_{R} | | |
| fong-spivak-seven-sketches_FO3683 | 315 | 1.000 | f(m+n)=f(m)+R f(n) | | |
| fong-spivak-seven-sketches_FO3684 | 315 | 0.610 | f(m * n)=f(m) *_{R} f(n) | | |
| fong-spivak-seven-sketches_FO3685 | 315 | 1.000 | f(m * n) | | |
| fong-spivak-seven-sketches_FO3686 | 315 | 1.000 | m n | | |
| fong-spivak-seven-sketches_FO3687 | 315 | 1.000 | \varnothing \rightarrow c | | |
| fong-spivak-seven-sketches_FO3688 | 315 | 1.000 | c_{1} \rightarrow c | | |
| fong-spivak-seven-sketches_FO3689 | 315 | 1.000 | c_{2} \rightarrow c_{1} | | |
| fong-spivak-seven-sketches_FO3690 | 316 | 0.705 | c_{1} \rightarrow c_{1} | | |
| fong-spivak-seven-sketches_FO3691 | 316 | 0.705 | \mathrm{id}_{c_{1}} | | |
| fong-spivak-seven-sketches_FO3692 | 316 | 0.705 | f ; g | | |
| fong-spivak-seven-sketches_FO3693 | 316 | 0.735 | f ; g=\operatorname{id}_{c_{1}} | | |
| fong-spivak-seven-sketches_FO3694 | 316 | 0.735 | g ; f=\operatorname{id}_{c_{2}} | | |
| fong-spivak-seven-sketches_FO3695 | 316 | 1.000 | p+q | | |
| fong-spivak-seven-sketches_FO3696 | 316 | 1.000 | p \leq p+q | | |
| fong-spivak-seven-sketches_FO3697 | 316 | 1.000 | q \leq p+q | | |
| fong-spivak-seven-sketches_FO3698 | 316 | 1.000 | p+q \leq x | | |
| fong-spivak-seven-sketches_FO3699 | 316 | 1.000 | [f, g]: A \sqcup B \longrightarrow T | | |
| fong-spivak-seven-sketches_FO3700 | 316 | 0.729 | o | | |
| fong-spivak-seven-sketches_FO3701 | 316 | 0.768 | \iota_{A} ;[f, g]=f | | |
| fong-spivak-seven-sketches_FO3702 | 316 | 0.586 | \iota_{B} \stackrel{\circ}{9}[f, g]=g | | |
| fong-spivak-seven-sketches_FO3703 | 316 | 0.546 | [f, g]: h=[f ; h, g ; h] | | |
| fong-spivak-seven-sketches_FO3704 | 316 | 1.000 | [f ; h, g ; h]: A+B \rightarrow D | | |
| fong-spivak-seven-sketches_FO3705 | 317 | 1.000 | +: \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C} | | |
| fong-spivak-seven-sketches_FO3706 | 317 | 0.984 | (A, B) | | |
| fong-spivak-seven-sketches_FO3707 | 317 | 1.000 | (f, g):(A, B) \rightarrow(C, D) | | |
| fong-spivak-seven-sketches_FO3708 | 317 | 0.510 | f+g=\left[f\right. | | |
| fong-spivak-seven-sketches_FO3709 | 317 | 0.510 | \iota_{C}, g | | |
| fong-spivak-seven-sketches_FO3710 | 317 | 0.510 | \left.\iota_{D}\right]: A+B \rightarrow C+D | | |
| fong-spivak-seven-sketches_FO3711 | 317 | 0.510 | \iota_{C}: C \rightarrow C+D | | |
| fong-spivak-seven-sketches_FO3712 | 317 | 0.510 | \iota_{D}: D \rightarrow C+D | | |
| fong-spivak-seven-sketches_FO3713 | 317 | 0.865 | (h, k):(C, D) \rightarrow(E, F) | | |
| fong-spivak-seven-sketches_FO3714 | 317 | 0.865 | (f+g) ;(h+k)=(f ; h)+(g ; k) | | |
| fong-spivak-seven-sketches_FO3715 | 317 | 1.000 | (f \circ h)+(g \circ k)= | | |
| fong-spivak-seven-sketches_FO3716 | 317 | 0.083 | \left[f ; h ; \iota_{E}, g \circ k ; \iota_{F}\right]=(f+g) ⿱ 宀 ⿱ 一 𧰨(h+k) | | |
| fong-spivak-seven-sketches_FO3717 | 317 | 1.000 | !_{A}: \varnothing \rightarrow A | | |
| fong-spivak-seven-sketches_FO3718 | 317 | 0.821 | \left[\operatorname{id}_{A},!_{A}\right] | | |
| fong-spivak-seven-sketches_FO3719 | 317 | 1.000 | A+\varnothing | | |
| fong-spivak-seven-sketches_FO3720 | 317 | 0.554 | \iota_{A} ~: ~\left[\mathrm{id}_{A},!{ }_{A}\right]=\mathrm{id}_{A} | | |
| fong-spivak-seven-sketches_FO3721 | 317 | 1.000 | \left[!_{A}, \mathrm{id}_{A}\right]: \varnothing+A \rightarrow A | | |
| fong-spivak-seven-sketches_FO3722 | 317 | 0.991 | \left[\iota_{A}, \iota_{B}+\mathrm{id}_{C}\right]: A+(B+C) \rightarrow(A+B)+C | | |
| fong-spivak-seven-sketches_FO3723 | 317 | 0.999 | \left[\iota_{A}, \iota_{B}\right]: A+B \rightarrow B+A | | |
| fong-spivak-seven-sketches_FO3724 | 317 | 0.990 | \iota_{A}: A \rightarrow A+B | | |
| fong-spivak-seven-sketches_FO3725 | 317 | 0.990 | \iota_{A}: A \rightarrow B+A | | |
| fong-spivak-seven-sketches_FO3726 | 317 | 0.991 | \left[\iota_{A}, \iota_{B}\right]: B+A \rightarrow A+B | | |
| fong-spivak-seven-sketches_FO3727 | 317 | 0.924 | \operatorname{Disc}_{S} | | |
| fong-spivak-seven-sketches_FO3728 | 317 | 0.924 | A=B=C | | |
| fong-spivak-seven-sketches_FO3729 | 317 | 1.000 | s^{\prime} | | |
| fong-spivak-seven-sketches_FO3730 | 317 | 1.000 | s \rightarrow s^{\prime} | | |
| fong-spivak-seven-sketches_FO3731 | 317 | 1.000 | s=s^{\prime} | | |
| fong-spivak-seven-sketches_FO3732 | 318 | 0.995 | \underline{5} \sqcup \underline{3} | | |
| fong-spivak-seven-sketches_FO3733 | 318 | 0.995 | \{f(a) \sim g(a) \mid a \in \underline{4}\} | | |
| fong-spivak-seven-sketches_FO3734 | 318 | 0.995 | \underline{5} | | |
| fong-spivak-seven-sketches_FO3735 | 318 | 1.000 | \{1, \ldots, 5\} | | |
| fong-spivak-seven-sketches_FO3736 | 318 | 1.000 | \underline{3} | | |
| fong-spivak-seven-sketches_FO3737 | 318 | 1.000 | \left\{1^{\prime}, 2^{\prime}, 3^{\prime}\right\} | | |
| fong-spivak-seven-sketches_FO3738 | 318 | 1.000 | \left\{1,1^{\prime}, 3\right\},\{2\},\{4\} | | |
| fong-spivak-seven-sketches_FO3739 | 318 | 1.000 | \left\{5,2^{\prime}, 3^{\prime}\right\} | | |
| fong-spivak-seven-sketches_FO3740 | 318 | 0.962 | \varnothing \rightarrow X+\Upsilon | | |
| fong-spivak-seven-sketches_FO3741 | 318 | 0.947 | f ; \iota_{X}=g{ }^{\circ} \iota_{Y} | | |
| fong-spivak-seven-sketches_FO3742 | 318 | 0.454 | t: X+\varnothing Y \rightarrow T | | |
| fong-spivak-seven-sketches_FO3743 | 318 | 0.454 | \iota_{X}: t=x | | |
| fong-spivak-seven-sketches_FO3744 | 318 | 0.454 | \iota_{Y}: t=y | | |
| fong-spivak-seven-sketches_FO3745 | 319 | 1.000 | A, B, X, Y | | |
| fong-spivak-seven-sketches_FO3746 | 319 | 1.000 | S \rightarrow T | | |
| fong-spivak-seven-sketches_FO3747 | 319 | 1.000 | X \leftarrow A \rightarrow Y | | |
| fong-spivak-seven-sketches_FO3748 | 319 | 1.000 | Q \rightarrow T | | |
| fong-spivak-seven-sketches_FO3749 | 319 | 1.000 | Y \leftarrow B \rightarrow Z | | |
| fong-spivak-seven-sketches_FO3750 | 319 | 1.000 | R \rightarrow T | | |
| fong-spivak-seven-sketches_FO3751 | 319 | 0.964 | (Y, Q, R, T) | | |
| fong-spivak-seven-sketches_FO3752 | 319 | 0.998 | X+{ }_{N} Y | | |
| fong-spivak-seven-sketches_FO3753 | 319 | 1.000 | X \sqcup N \sqcup Y | | |
| fong-spivak-seven-sketches_FO3754 | 319 | 1.000 | x \sim n | | |
| fong-spivak-seven-sketches_FO3755 | 319 | 1.000 | x=f(n) | | |
| fong-spivak-seven-sketches_FO3756 | 319 | 1.000 | y \sim n | | |
| fong-spivak-seven-sketches_FO3757 | 319 | 1.000 | y=g(n) | | |
| fong-spivak-seven-sketches_FO3758 | 319 | 1.000 | x \in X, y \in Y, n \in N | | |
| fong-spivak-seven-sketches_FO3759 | 319 | 1.000 | n \in N | | |
| fong-spivak-seven-sketches_FO3760 | 320 | 1.000 | B \rightarrow D \otimes D | | |
| fong-spivak-seven-sketches_FO3761 | 320 | 1.000 | g: X \rightarrow Y | | |
| fong-spivak-seven-sketches_FO3762 | 320 | 1.000 | \iota_{1}: X \rightarrow X+X | | |
| fong-spivak-seven-sketches_FO3763 | 320 | 1.000 | X+X | | |
| fong-spivak-seven-sketches_FO3764 | 320 | 0.748 | \iota_{1} | | |
| fong-spivak-seven-sketches_FO3765 | 320 | 0.748 | [\mathrm{id}, \mathrm{id}]=\mathrm{id} | | |
| fong-spivak-seven-sketches_FO3766 | 320 | 0.732 | f=\iota_{1} ;[\mathrm{id}, \mathrm{id}] ; f=\iota_{1} ;[\mathrm{id}, \mathrm{id}] ; g=g | | |
| fong-spivak-seven-sketches_FO3767 | 320 | 0.732 | f: X \rightarrow T | | |
| fong-spivak-seven-sketches_FO3768 | 321 | 0.384 | _{\mathbb{C}} | | |
| fong-spivak-seven-sketches_FO3769 | 321 | 0.809 | X \leftarrow N \rightarrow Y | | |
| fong-spivak-seven-sketches_FO3770 | 321 | 0.974 | F N=\{*\} | | |
| fong-spivak-seven-sketches_FO3771 | 321 | 0.974 | F N | | |
| fong-spivak-seven-sketches_FO3772 | 321 | 0.758 | \operatorname{Cospan}_{F}=\operatorname{Cospan}_{\mathcal{C}} | | |
| fong-spivak-seven-sketches_FO3773 | 321 | 0.484 | \operatorname{Cospan}_{\mathcal{C}} \rightarrow \operatorname{Cospan}_{F} | | |
| fong-spivak-seven-sketches_FO3774 | 321 | 0.983 | V=\{\mathrm{ul}, \mathrm{ur}, \mathrm{dl}, \mathrm{dr}\}, A= | | |
| fong-spivak-seven-sketches_FO3781 | 321 | 1.000 | \psi_{\underline{2}, \underline{2}}(b, s) | | |
| fong-spivak-seven-sketches_FO3782 | 321 | 0.220 | 1 \xrightarrow{f} \underline{2} \stackrel{g}{\leftarrow} \underline{1} | | |
| fong-spivak-seven-sketches_FO3783 | 321 | 0.220 | f(1)=1 | | |
| fong-spivak-seven-sketches_FO3784 | 321 | 0.220 | g(1)=2 | | |
| fong-spivak-seven-sketches_FO3785 | 321 | 0.954 | (\underline{2},\{a\}, s, t, \ell) | | |
| fong-spivak-seven-sketches_FO3786 | 321 | 0.954 | s(a)=1, t(a)=2 | | |
| fong-spivak-seven-sketches_FO3787 | 321 | 0.954 | \ell(a)= | | |
| fong-spivak-seven-sketches_FO3788 | 322 | 1.000 | C:=(V, A, s, t, \ell) | | |
| fong-spivak-seven-sketches_FO3789 | 322 | 0.462 | \underline{1} \xrightarrow{f} V \stackrel{g}{\leftarrow} \underline{2}, f(1)=\mathrm{ul}, g(1)=\mathrm{ur} | | |
| fong-spivak-seven-sketches_FO3790 | 322 | 0.462 | g(2)= | | |
| fong-spivak-seven-sketches_FO3791 | 322 | 0.953 | \underline{2} \xrightarrow{f^{\prime}} V^{\prime} \stackrel{g^{\prime}}{\leftarrow} \underline{2} | | |
| fong-spivak-seven-sketches_FO3792 | 322 | 0.953 | C^{\prime}:=\left(V^{\prime}, A^{\prime}, s^{\prime}, t^{\prime}, \ell^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO3793 | 322 | 1.000 | V^{\prime}=\{l, r, d\}, A^{\prime}=\left\{\mathrm{r} 1^{\prime}, \mathrm{r} 2^{\prime}\right\} | | |
| fong-spivak-seven-sketches_FO3798 | 322 | 0.971 | V \stackrel{g}{\leftarrow} \underline{2} \xrightarrow{f^{\prime}} V^{\prime} | | |
| fong-spivak-seven-sketches_FO3799 | 322 | 0.971 | V^{\prime \prime}= | | |
| fong-spivak-seven-sketches_FO3800 | 322 | 0.823 | \underline{1} \xrightarrow{h} V^{\prime \prime} \stackrel{k}{\leftarrow} \underline{2} | | |
| fong-spivak-seven-sketches_FO3801 | 322 | 0.823 | h(1)=\mathrm{ul}, k(1)=\mathrm{r} | | |
| fong-spivak-seven-sketches_FO3802 | 322 | 0.536 | k(2)=\mathrm{m} | | |
| fong-spivak-seven-sketches_FO3803 | 322 | 0.536 | \left(V^{\prime \prime} A+A^{\prime}, s,^{\prime \prime} t^{\prime \prime} \ell^{\prime \prime}\right) | | |
| fong-spivak-seven-sketches_FO3807 | 322 | 1.000 | \eta | | |
| fong-spivak-seven-sketches_FO3808 | 324 | 0.846 | (X, p, q) | | |
| fong-spivak-seven-sketches_FO3809 | 324 | 0.444 | q: f^{\prime} ; g^{\prime}=p ; h_{3} | | |
| fong-spivak-seven-sketches_FO3810 | 324 | 0.180 | r ; h_{2}=g | | |
| fong-spivak-seven-sketches_FO3811 | 324 | 0.180 | r ; g=p | | |
| fong-spivak-seven-sketches_FO3812 | 324 | 0.991 | r^{\prime}: X \rightarrow A | | |
| fong-spivak-seven-sketches_FO3813 | 324 | 0.991 | r^{\prime} ; f=r | | |
| fong-spivak-seven-sketches_FO3814 | 324 | 1.000 | r^{\prime} ; h_{1}=q | | |
| fong-spivak-seven-sketches_FO3815 | 324 | 1.000 | r^{\prime} ; f ; g=r ; g=p | | |
| fong-spivak-seven-sketches_FO3816 | 324 | 0.626 | r_{0} | | |
| fong-spivak-seven-sketches_FO3817 | 324 | 0.626 | f ; g=p | | |
| fong-spivak-seven-sketches_FO3818 | 324 | 0.626 | r_{0} ; h_{1}=q | | |
| fong-spivak-seven-sketches_FO3819 | 324 | 0.741 | r_{0}: f=r | | |
| fong-spivak-seven-sketches_FO3820 | 324 | 0.741 | r^{\prime} | | |
| fong-spivak-seven-sketches_FO3821 | 324 | 1.000 | r_{0}=r^{\prime} | | |
| fong-spivak-seven-sketches_FO3822 | 324 | 0.994 | r ; h_{2}=q ; f^{\prime} | | |
| fong-spivak-seven-sketches_FO3823 | 324 | 0.994 | p:=r ; g | | |
| fong-spivak-seven-sketches_FO3824 | 325 | 0.999 | r^{\prime} ; f ; g=p | | |
| fong-spivak-seven-sketches_FO3825 | 325 | 0.422 | r_{0}:=r^{\prime} ; f | | |
| fong-spivak-seven-sketches_FO3826 | 325 | 0.422 | r_{0} ; g=p | | |
| fong-spivak-seven-sketches_FO3827 | 325 | 0.422 | r_{0} ; h_{2}=q ; f^{\prime} | | |
| fong-spivak-seven-sketches_FO3828 | 325 | 0.554 | A, B, A^{\prime}, B^{\prime} | | |
| fong-spivak-seven-sketches_FO3829 | 325 | 1.000 | a_{1}, a_{2} \in A | | |
| fong-spivak-seven-sketches_FO3830 | 325 | 1.000 | f\left(a_{1}\right)=f\left(a_{2}\right) | | |
| fong-spivak-seven-sketches_FO3831 | 325 | 1.000 | a_{1}=a_{2} | | |
| fong-spivak-seven-sketches_FO3832 | 325 | 0.402 | g_{1}, g_{2}: X \rightarrow A | | |
| fong-spivak-seven-sketches_FO3833 | 325 | 0.402 | f=g_{2} | | |
| fong-spivak-seven-sketches_FO3834 | 325 | 0.402 | g_{1}=g_{2} | | |
| fong-spivak-seven-sketches_FO3835 | 325 | 1.000 | X=\{*\} | | |
| fong-spivak-seven-sketches_FO3836 | 325 | 1.000 | g_{1}(*):=a_{1} | | |
| fong-spivak-seven-sketches_FO3837 | 325 | 1.000 | g_{2}(*):=a_{2} | | |
| fong-spivak-seven-sketches_FO3838 | 325 | 0.626 | =g_{2} | | |
| fong-spivak-seven-sketches_FO3839 | 325 | 1.000 | g_{1}(x)=g_{2}(x) | | |
| fong-spivak-seven-sketches_FO3840 | 325 | 1.000 | f\left(g_{1}(x)\right)=f\left(g_{2}(x)\right) | | |
| fong-spivak-seven-sketches_FO3841 | 325 | 0.974 | j:=i^{-1} | | |
| fong-spivak-seven-sketches_FO3842 | 325 | 0.974 | g:=(f: j): A \rightarrow B^{\prime} | | |
| fong-spivak-seven-sketches_FO3843 | 325 | 0.974 | g \circ i=f | | |
| fong-spivak-seven-sketches_FO3844 | 325 | 0.553 | j^{\prime}: A \rightarrow A^{\prime} | | |
| fong-spivak-seven-sketches_FO3845 | 325 | 0.553 | j^{\prime} ; i^{\prime}=\operatorname{id}_{A} | | |
| fong-spivak-seven-sketches_FO3846 | 325 | 0.392 | j^{\prime} ; f^{\prime}=f ; j | | |
| fong-spivak-seven-sketches_FO3847 | 325 | 0.392 | i^{\prime} ; j^{\prime}=\operatorname{id}_{A^{\prime}} | | |
| fong-spivak-seven-sketches_FO3848 | 325 | 0.392 | \left(i^{\prime} ; j^{\prime}\right) ; i^{\prime}=i^{\prime} | | |
| fong-spivak-seven-sketches_FO3849 | 325 | 0.024 | \left(i^{\prime}{ }_{9} j^{\prime}\right), f^{\prime}=i^{\prime} ; f ; j=f^{\prime}: i, j=f^{\prime} | | |
| fong-spivak-seven-sketches_FO3850 | 325 | 0.024 | \mathrm{id}_{A^{\prime}} | | |
| fong-spivak-seven-sketches_FO3851 | 325 | 0.024 | \mathrm{id}_{A^{\prime}}{ }^{\circ} i^{\prime}=i^{\prime} | | |
| fong-spivak-seven-sketches_FO3852 | 325 | 0.511 | f^{\prime}=f^{\prime} | | |
| fong-spivak-seven-sketches_FO3853 | 325 | 0.511 | \left(i^{\prime} ; j^{\prime}\right)=\mathrm{id}_{A^{\prime}} | | |
| fong-spivak-seven-sketches_FO3854 | 325 | 0.549 | g: X \rightarrow A | | |
| fong-spivak-seven-sketches_FO3855 | 325 | 0.549 | h: X \rightarrow B | | |
| fong-spivak-seven-sketches_FO3856 | 325 | 0.549 | g \circ f=h \circ \mathrm{id}_{B} | | |
| fong-spivak-seven-sketches_FO3857 | 325 | 0.554 | r: X \rightarrow A | | |
| fong-spivak-seven-sketches_FO3858 | 325 | 0.554 | r ; \operatorname{id}_{A}=g | | |
| fong-spivak-seven-sketches_FO3859 | 325 | 0.554 | r ; f=h | | |
| fong-spivak-seven-sketches_FO3860 | 325 | 1.000 | r=g | | |
| fong-spivak-seven-sketches_FO3861 | 326 | 0.666 | v \otimes w | | |
| fong-spivak-seven-sketches_FO3862 | 326 | 0.988 | v \leq(w \multimap x) | | |
| fong-spivak-seven-sketches_FO3863 | 326 | 0.988 | v \otimes W \leq x | | |
| fong-spivak-seven-sketches_FO3864 | 326 | 0.998 | \mathbb{N}^{\mathrm{op}} \times \mathbb{N}^{\mathrm{op}} | | |
| fong-spivak-seven-sketches_FO3865 | 326 | 0.998 | (a, b) \in \mathbb{N} \times \mathbb{N} | | |
| fong-spivak-seven-sketches_FO3866 | 326 | 0.998 | (a, b) \leq\left(a^{\prime}, b^{\prime}\right) | | |
| fong-spivak-seven-sketches_FO3867 | 326 | 1.000 | a^{\prime} \leq a | | |
| fong-spivak-seven-sketches_FO3868 | 326 | 1.000 | b^{\prime} \leq b | | |
| fong-spivak-seven-sketches_FO3869 | 326 | 0.658 | (0,0) | | |
| fong-spivak-seven-sketches_FO3870 | 326 | 0.658 | (a, b) \wedge\left(a^{\prime}, b^{\prime}\right)=\left(\max \left(a, a^{\prime}\right), \max \left(b, b^{\prime}\right)\right) | | |
| fong-spivak-seven-sketches_FO3871 | 327 | 0.958 | x \multimap y | | |
| fong-spivak-seven-sketches_FO3872 | 327 | 0.958 | x \multimap y=\vee\{w \mid w \wedge x \leq y\} | | |
| fong-spivak-seven-sketches_FO3873 | 327 | 0.997 | y \wedge x \leq y | | |
| fong-spivak-seven-sketches_FO3874 | 327 | 0.997 | y \leq x \multimap y | | |
| fong-spivak-seven-sketches_FO3875 | 327 | 0.431 | x \multimap y=\vee\{w \mid y \leq w | | |
| fong-spivak-seven-sketches_FO3876 | 327 | 0.431 | w \wedge x \leq y\} | | |
| fong-spivak-seven-sketches_FO3877 | 327 | 0.431 | \mathbb{N}^{\text {op }} \times \mathbb{N}^{\text {op }} | | |
| fong-spivak-seven-sketches_FO3878 | 327 | 1.000 | m: \mathbb{Z} \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO3879 | 327 | 1.000 | \mathbb{N} \subseteq \mathbb{Z} | | |
| fong-spivak-seven-sketches_FO3880 | 327 | 1.000 | \ulcorner m\urcorner(-5)= | | |
| fong-spivak-seven-sketches_FO3881 | 327 | 1.000 | \ulcorner m\urcorner(0)= | | |
| fong-spivak-seven-sketches_FO3882 | 327 | 1.000 | A \subseteq \mathbb{B} | | |
| fong-spivak-seven-sketches_FO3883 | 327 | 0.993 | \ulcorner\Rightarrow\urcorner: \mathbb{B} \times \mathbb{B} \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO3884 | 327 | 0.991 | \subseteq \mathbb{B} \times \mathbb{B} | | |
| fong-spivak-seven-sketches_FO3885 | 327 | 1.000 | \ulcorner E\urcorner,\ulcorner P\urcorner,\ulcorner T\urcorner: \mathbb{N} \rightarrow \mathbb{B} | | |
| fong-spivak-seven-sketches_FO3886 | 327 | 1.000 | \}, P:=\{n \in \mathbb{N} \mid | | |
| fong-spivak-seven-sketches_FO3887 | 327 | 0.994 | \ulcorner E\urcorner(17)= | | |
| fong-spivak-seven-sketches_FO3888 | 327 | 1.000 | \ulcorner P\urcorner(17)= | | |
| fong-spivak-seven-sketches_FO3889 | 327 | 0.857 | \ulcorner T\urcorner(17)= | | |
| fong-spivak-seven-sketches_FO3890 | 327 | 0.857 | 17 \geq 10 | | |
| fong-spivak-seven-sketches_FO3891 | 327 | 0.679 | B(x, \epsilon):=\left\{x^{\prime} \in \mathbb{R}| | x-x^{\prime} \mid<\epsilon\right\} | | |
| fong-spivak-seven-sketches_FO3892 | 327 | 1.000 | x \in U | | |
| fong-spivak-seven-sketches_FO3893 | 327 | 1.000 | B(x, \epsilon) \subseteq U | | |
| fong-spivak-seven-sketches_FO3894 | 327 | 1.000 | U_{1}:=\{x \in \mathbb{R} \mid 0<x<2\} | | |
| fong-spivak-seven-sketches_FO3895 | 327 | 1.000 | U_{2}:=\{x \in \mathbb{R} \mid 1<x<3\} | | |
| fong-spivak-seven-sketches_FO3896 | 327 | 1.000 | U:=U_{1} \cup U_{2}=\{x \in \mathbb{R} \mid 0< | | |
| fong-spivak-seven-sketches_FO3897 | 327 | 1.000 | x<3\} | | |
| fong-spivak-seven-sketches_FO3898 | 327 | 1.000 | y=(a, b) | | |
| fong-spivak-seven-sketches_FO3899 | 327 | 1.000 | (a+1) *(b+1) | | |
| fong-spivak-seven-sketches_FO3900 | 328 | 1.000 | I=\{1,2,3,4, \ldots\} | | |
| fong-spivak-seven-sketches_FO3901 | 328 | 1.000 | U_{i}:=\left\{x \in \mathbb{R} \left\lvert\, \frac{1}{i}<x<1\right.\right\} | | |
| fong-spivak-seven-sketches_FO3902 | 328 | 1.000 | U_{1} \subseteq U_{2} \subseteq | | |
| fong-spivak-seven-sketches_FO3903 | 328 | 1.000 | U_{3} \subseteq \cdots | | |
| fong-spivak-seven-sketches_FO3904 | 328 | 1.000 | U:=\bigcup_{i \in I} U_{i}=\{x \in \mathbb{R} \mid 0<x<1\} | | |
| fong-spivak-seven-sketches_FO3905 | 328 | 1.000 | \varnothing \subseteq X | | |
| fong-spivak-seven-sketches_FO3906 | 328 | 0.999 | \bigcup_{i \in I} A_{i} | | |
| fong-spivak-seven-sketches_FO3907 | 328 | 0.999 | A_{i}=X | | |
| fong-spivak-seven-sketches_FO3908 | 328 | 1.000 | A \subseteq X | | |
| fong-spivak-seven-sketches_FO3909 | 328 | 0.405 | (X, \mathbf{P}(X)) | | |
| fong-spivak-seven-sketches_FO3910 | 328 | 1.000 | f^{-1}(U) \subseteq X | | |
| fong-spivak-seven-sketches_FO3911 | 328 | 1.000 | \varnothing \rightarrow\{1\} \rightarrow\{1,2\} | | |
| fong-spivak-seven-sketches_FO3912 | 328 | 0.994 | U_{i}=U | | |
| fong-spivak-seven-sketches_FO3913 | 328 | 1.000 | Y=B \cap Y | | |
| fong-spivak-seven-sketches_FO3914 | 328 | 1.000 | B=Y | | |
| fong-spivak-seven-sketches_FO3915 | 328 | 1.000 | B=X | | |
| fong-spivak-seven-sketches_FO3916 | 328 | 1.000 | \varnothing=\varnothing \cap Y | | |
| fong-spivak-seven-sketches_FO3917 | 328 | 1.000 | \varnothing \in \mathbf{O p}_{? \cap Y} | | |
| fong-spivak-seven-sketches_FO3918 | 328 | 1.000 | A_{1}, A_{2} \in \mathbf{O p}_{? \cap Y} | | |
| fong-spivak-seven-sketches_FO3919 | 328 | 0.999 | B_{1}, B_{2} \in \mathbf{O p} | | |
| fong-spivak-seven-sketches_FO3920 | 328 | 0.999 | A_{1}=B_{1} \cap Y | | |
| fong-spivak-seven-sketches_FO3921 | 328 | 0.999 | A_{2}=B_{2} \cap Y | | |
| fong-spivak-seven-sketches_FO3922 | 328 | 0.999 | A_{1} \cap A_{2}=\left(B_{1} \cap Y\right) \cap\left(B_{2} \cap Y\right)= | | |
| fong-spivak-seven-sketches_FO3923 | 328 | 0.995 | \left(B_{1} \cap B_{2}\right) \cap Y | | |
| fong-spivak-seven-sketches_FO3924 | 328 | 0.995 | B_{1} \cap B_{2} \in \mathbf{O p} | | |
| fong-spivak-seven-sketches_FO3925 | 328 | 1.000 | A_{i} | | |
| fong-spivak-seven-sketches_FO3926 | 328 | 1.000 | A_{i}=B_{i} \cap Y | | |
| fong-spivak-seven-sketches_FO3927 | 328 | 1.000 | B_{i} \in \mathbf{O p} | | |
| fong-spivak-seven-sketches_FO3928 | 328 | 1.000 | U \subseteq \mathbf{O p} | | |
| fong-spivak-seven-sketches_FO3929 | 328 | 1.000 | \mathcal{C}(B, C)= | | |
| fong-spivak-seven-sketches_FO3930 | 329 | 1.000 | \left(U_{3} \wedge U_{1}\right) \vee\left(U_{4} \wedge U_{2}\right) | | |
| fong-spivak-seven-sketches_FO3931 | 329 | 1.000 | \mathcal{C}(a, b) | | |
| fong-spivak-seven-sketches_FO3932 | 329 | 1.000 | \left\{a_{1}, a_{2}\right\} | | |
| fong-spivak-seven-sketches_FO3933 | 329 | 1.000 | \left\{c_{1}\right\} | | |
| fong-spivak-seven-sketches_FO3934 | 329 | 1.000 | \operatorname{Sec}_{f}\left(V_{2}\right)=\varnothing | | |
| fong-spivak-seven-sketches_FO3935 | 329 | 0.949 | \left.\operatorname{Sec}_{f}\left(V_{3}\right)\right) | | |
| fong-spivak-seven-sketches_FO3936 | 329 | 0.949 | 2 * 3 * 1 * 2=12 | | |
| fong-spivak-seven-sketches_FO3937 | 330 | 1.000 | g_{1}:=\left(a_{1}, b_{1}\right) | | |
| fong-spivak-seven-sketches_FO3938 | 330 | 1.000 | g_{2}:=\left(b_{2}, e_{1}\right) | | |
| fong-spivak-seven-sketches_FO3939 | 330 | 1.000 | g \in \operatorname{Sec}_{f}\left(U_{1} \cup U_{2}\right) | | |
| fong-spivak-seven-sketches_FO3940 | 330 | 1.000 | \left.g\right|_{U_{1}}=g_{1} | | |
| fong-spivak-seven-sketches_FO3941 | 330 | 1.000 | \left.g\right|_{U_{2}}=g_{2} | | |
| fong-spivak-seven-sketches_FO3942 | 330 | 0.997 | F(\{1,2\}) \rightarrow F(\{1\}) \rightarrow F(\varnothing) | | |
| fong-spivak-seven-sketches_FO3943 | 330 | 1.000 | F(\varnothing) \cong\{1\} | | |
| fong-spivak-seven-sketches_FO3944 | 330 | 1.000 | F(\{1,2\}) \rightarrow F(\{1\}) | | |
| fong-spivak-seven-sketches_FO3945 | 330 | 1.000 | S(\varnothing)=\{()\} | | |
| fong-spivak-seven-sketches_FO3946 | 330 | 1.000 | S(\{1\}) | | |
| fong-spivak-seven-sketches_FO3947 | 330 | 0.751 | \Omega: \mathbf{O p}(X)^{\mathrm{op}} \rightarrow | | |
| fong-spivak-seven-sketches_FO3948 | 330 | 0.947 | \Omega(\{1\}) | | |
| fong-spivak-seven-sketches_FO3949 | 330 | 1.000 | \mathbf{O p}(\{1\}) | | |
| fong-spivak-seven-sketches_FO3950 | 330 | 0.600 | \Omega(U):=\left\{U^{\prime} \in \mathbf{O p} \mid U^{\prime} \subseteq U\right\} | | |
| fong-spivak-seven-sketches_FO3951 | 330 | 1.000 | U^{\prime} \mapsto U^{\prime} \cap V | | |
| fong-spivak-seven-sketches_FO3952 | 331 | 0.998 | U^{\prime} \subseteq U | | |
| fong-spivak-seven-sketches_FO3953 | 331 | 0.998 | \left(U^{\prime} \cap V\right) \cap W=U^{\prime} \cap W | | |
| fong-spivak-seven-sketches_FO3954 | 331 | 1.000 | W \subseteq V | | |
| fong-spivak-seven-sketches_FO3955 | 331 | 1.000 | U^{\prime} \cap U=U^{\prime} | | |
| fong-spivak-seven-sketches_FO3956 | 331 | 0.947 | G^{\prime} | | |
| fong-spivak-seven-sketches_FO3957 | 331 | 0.947 | \gamma:=\left\ulcorner G^{\prime}\right. | | |
| fong-spivak-seven-sketches_FO3958 | 331 | 1.000 | \gamma(D)=0 | | |
| fong-spivak-seven-sketches_FO3959 | 331 | 1.000 | A, B, C | | |
| fong-spivak-seven-sketches_FO3960 | 331 | 1.000 | \gamma(A)=\gamma(B)=\gamma(C)=V | | |
| fong-spivak-seven-sketches_FO3961 | 331 | 1.000 | \gamma(i)=(V, 0 ; 0) | | |
| fong-spivak-seven-sketches_FO3962 | 331 | 1.000 | \gamma(f)=(V, V ; A) | | |
| fong-spivak-seven-sketches_FO3963 | 331 | 1.000 | \gamma(g)=\gamma(h)=(V, V ; 0) | | |
| fong-spivak-seven-sketches_FO3964 | 331 | 1.000 | U=\mathbb{R}-\{0\} \subseteq \mathbb{R} | | |
| fong-spivak-seven-sketches_FO3965 | 331 | 1.000 | \mathbb{R}-U=\{0\} | | |
| fong-spivak-seven-sketches_FO3966 | 331 | 1.000 | \neg U=\varnothing | | |
| fong-spivak-seven-sketches_FO3967 | 331 | 1.000 | \mathbb{R}-\varnothing=\mathbb{R} | | |
| fong-spivak-seven-sketches_FO3968 | 331 | 1.000 | \neg \neg U=\mathbb{R} | | |
| fong-spivak-seven-sketches_FO3969 | 331 | 0.943 | \neg \neg U \subseteq | | |
| fong-spivak-seven-sketches_FO3970 | 331 | 0.995 | V \in \mathbf{O p} | | |
| fong-spivak-seven-sketches_FO3971 | 331 | 0.995 | \top \wedge V=V | | |
| fong-spivak-seven-sketches_FO3972 | 331 | 0.995 | V=X | | |
| fong-spivak-seven-sketches_FO3973 | 331 | 0.995 | \top \wedge X:=\top \cap X=X | | |
| fong-spivak-seven-sketches_FO3974 | 331 | 0.953 | \top=\top \cap X=X | | |
| fong-spivak-seven-sketches_FO3975 | 331 | 0.477 | (\top \vee V):=(X \cup V)=X | | |
| fong-spivak-seven-sketches_FO3976 | 331 | 0.477 | (V \Rightarrow X)=\bigcup_{\{R \in \mathbf{O p} \mid R \cap V \subseteq X\}} R=X | | |
| fong-spivak-seven-sketches_FO3977 | 331 | 0.477 | (X \cap V) \subseteq X | | |
| fong-spivak-seven-sketches_FO3978 | 331 | 0.885 | (\perp \vee V)=V | | |
| fong-spivak-seven-sketches_FO3979 | 331 | 0.885 | V=\emptyset | | |
| fong-spivak-seven-sketches_FO3980 | 331 | 0.885 | (\perp \vee \varnothing)=(\perp \cup \varnothing)=\varnothing | | |
| fong-spivak-seven-sketches_FO3981 | 331 | 0.807 | (\perp \wedge V)=(\varnothing \cap V)=\varnothing | | |
| fong-spivak-seven-sketches_FO3982 | 331 | 0.807 | (\perp \Rightarrow V)=\cup_{\{R \in \mathbf{O p} \mid R \cap \varnothing \subseteq V\}} R=X | | |
| fong-spivak-seven-sketches_FO3983 | 331 | 0.807 | (X \cap \varnothing) \subseteq V | | |
| fong-spivak-seven-sketches_FO3984 | 331 | 1.000 | \{S \mid p\} | | |
| fong-spivak-seven-sketches_FO3985 | 331 | 0.983 | S=\mathbb{N} | | |
| fong-spivak-seven-sketches_FO3986 | 331 | 0.983 | 24 \leq s \leq 28 | | |
| fong-spivak-seven-sketches_FO3987 | 331 | 0.983 | q(s) | | |
| fong-spivak-seven-sketches_FO3988 | 332 | 1.000 | p(n, z) | | |
| fong-spivak-seven-sketches_FO3989 | 332 | 1.000 | n \leq|z| | | |
| fong-spivak-seven-sketches_FO3990 | 332 | 1.000 | \{0\} \subseteq \mathbb{N} | | |
| fong-spivak-seven-sketches_FO3991 | 332 | 1.000 | \mathbb{N} \subseteq \mathbb{N} | | |
| fong-spivak-seven-sketches_FO3992 | 332 | 0.988 | \varnothing \subseteq \mathbb{Z} | | |
| fong-spivak-seven-sketches_FO3993 | 332 | 1.000 | \mathbb{Z} \subseteq \mathbb{Z} | | |
| fong-spivak-seven-sketches_FO3994 | 332 | 0.786 | \forall(t: T) | | |
| fong-spivak-seven-sketches_FO3995 | 332 | 1.000 | t_{i} | | |
| fong-spivak-seven-sketches_FO3996 | 332 | 0.990 | V= | | |
| fong-spivak-seven-sketches_FO3997 | 332 | 0.695 | s^{\prime \prime} | | |
| fong-spivak-seven-sketches_FO3998 | 332 | 1.000 | p:=j(q) | | |
| fong-spivak-seven-sketches_FO3999 | 332 | 0.999 | j(p) \leq p | | |
| fong-spivak-seven-sketches_FO4000 | 332 | 0.999 | p \cong j(p) | | |
| fong-spivak-seven-sketches_FO4001 | 332 | 0.999 | p=j(p) | | |
| fong-spivak-seven-sketches_FO4002 | 333 | 1.000 | s \in S(U), p(s) | | |
| fong-spivak-seven-sketches_FO4003 | 333 | 1.000 | j(j(p(s))=j(p(s) | | |
| fong-spivak-seven-sketches_FO4004 | 333 | 1.000 | 0 \leq 2 \leq 6 \leq 8 | | |
| fong-spivak-seven-sketches_FO4005 | 333 | 0.977 | [2,6] \in^{?} o_{[0,5]} \cup o_{[4,8]} | | |
| fong-spivak-seven-sketches_FO4006 | 333 | 0.977 | [2,6] \in^{?} o_{[0,5]} | | |
| fong-spivak-seven-sketches_FO4007 | 333 | 0.977 | [2,6] \in^{?} o_{[4,8]} | | |
| fong-spivak-seven-sketches_FO4008 | 333 | 0.510 | U^{\prime} \subseteq \mathbb{R} | | |
| fong-spivak-seven-sketches_FO4009 | 333 | 1.000 | U=U^{\prime} \cap \mathbb{R} | | |
| fong-spivak-seven-sketches_FO4010 | 333 | 0.999 | U^{\prime}=\bigcup_{i \in I} o_{\left[a_{i}, b_{i}\right]} | | |
| fong-spivak-seven-sketches_FO4011 | 333 | 0.999 | o_{\left[a_{i}, b_{i}\right]}=\left\{[d, u] \in \mathbb{R} \mid a_{i}<d<u<b_{i}\right\} | | |
| fong-spivak-seven-sketches_FO4012 | 333 | 0.999 | \mathbb{R}=\{[x, x] \in \mathbb{R}\} | | |
| fong-spivak-seven-sketches_FO4013 | 333 | 1.000 | U^{\prime} \cap \mathbb{R} | | |
| fong-spivak-seven-sketches_FO4014 | 333 | 1.000 | B\left(m_{i}, r_{i}\right) | | |
| fong-spivak-seven-sketches_FO4015 | 333 | 1.000 | m_{i}:=\frac{a_{i}+b_{i}}{2} | | |
| fong-spivak-seven-sketches_FO4016 | 333 | 1.000 | r_{i}:=\frac{b_{i}-a_{i}}{2} | | |
| fong-spivak-seven-sketches_FO4017 | 333 | 0.877 | \left(a_{i}, b_{i}\right) | | |
| fong-spivak-seven-sketches_FO4018 | 333 | 1.000 | U=\bigcup_{j \in J} B\left(m_{j}, \epsilon_{j}\right) | | |
| fong-spivak-seven-sketches_FO4019 | 333 | 1.000 | a_{j}:=m_{j}-\epsilon_{j} | | |
| fong-spivak-seven-sketches_FO4020 | 333 | 1.000 | b_{j}:=m_{j}+\epsilon_{j} | | |
| fong-spivak-seven-sketches_FO4021 | 333 | 0.950 | H_{X}(U):= | | |
| fong-spivak-seven-sketches_FO4022 | 333 | 1.000 | \{f: U \cap R \rightarrow X \mid f | | |
| fong-spivak-seven-sketches_FO4023 | 333 | 1.000 | f: U \cap R \rightarrow X | | |
| fong-spivak-seven-sketches_FO4024 | 333 | 1.000 | V \cap R \subseteq U \cap R | | |
| fong-spivak-seven-sketches_FO4025 | 333 | 1.000 | U=\bigcup_{i} U_{i} | | |
| fong-spivak-seven-sketches_FO4026 | 333 | 1.000 | f_{i}: U_{i} \cap R \rightarrow X | | |
| fong-spivak-seven-sketches_FO4027 | 333 | 0.913 | U \cap R | | |
| fong-spivak-seven-sketches_FO4028 | 333 | 0.913 | U \cap R=\left(\bigcup_{i} U_{i}\right) \cap R=\bigcup_{i}\left(U_{i} \cap R\right) | |