恒等写像
word(s,s)! …【\(\text{id} _{ {\sf X} }\)】\(\text{id}.\)【\(\text{id} _{ X } = [ x \in X \mid x ]\)】
\(\text{id0}\)【\(\text{id} _{ X } \in X \stackrel{\rm IS}\to X\)】\(\,{\blacktriangleleft}\,\mathbb{D}.\)
\({\circ}単\)【\(R \subset X \times Y \Longrightarrow \text{id} _{ Y } \circ R = R = R \circ \text{id} _{ X }\)】\(\,{\blacktriangleleft}\,\mathbb{D}.\)
\(|\text{id}\)【\(R \in \text{Rel} \Longrightarrow R | _{ X } = R \circ \text{id} _{ X }\)】\(\,{\blacktriangleleft}\,\mathbb{D}.\)
\({\stackrel{\rm I}\to}{.}{.}\)【\(X \stackrel{\rm I}\to Y = \{ f \in X \to Y \mid f ^\leftrightarrow \circ f = \text{id} _{ X } \}\)】\(\,{\blacktriangleleft}\,\mathbb{D}.\)
\({\stackrel{\rm S}\to}{.}{.}\)【\(X \stackrel{\rm S}\to Y = \{ f \in X \to Y \mid f \circ f ^\leftrightarrow = \text{id} _{ Y } \}\)】\(\,{\blacktriangleleft}\,\mathbb{D}.\)
to:I.. / d. ＜ %0 ， sw. と to:S.. / d. ＜ %0 ， sw. もOK
濃度
word(ss,p) … \(\stackrel{\#}=\) \(\stackrel{\#}\le\) \(\stackrel{\#}<\)\(\stackrel{\#}=.\)【\(X \stackrel{\#}= Y \Longleftrightarrow X \stackrel{\rm IS}\to Y \neq \emptyset\)】\({\bf /}\emptyset{.}{.}\)
\(\stackrel{\#}\le.\)【\(X \stackrel{\#}\le Y \Longleftrightarrow X \stackrel{\rm I}\to Y \neq \emptyset\)】\({\bf /}\emptyset{.}{.}\)
\(\stackrel{\#}<.\)【\(X \stackrel{\#}< Y \Longleftrightarrow X \stackrel{\#}\le Y , X \cancel{\stackrel{\#}=} Y\)】
\({\stackrel{\#}=}自推\)【\(\stackrel{\#}= \cdots \text{自推}\)】\(\,{\blacktriangleleft}\,\mathbb{D}.\)
\({\stackrel{\#}=}対\)【\(X \stackrel{\#}= Y \Longrightarrow Y \stackrel{\#}= X\)】\(\,{\blacktriangleleft}\,\mathbb{D}.\)
\({\stackrel{\#}\le}自推\)【\(\stackrel{\#}\le \cdots \text{自推}\)】\(\,{\blacktriangleleft}\,\mathbb{D}.\)
\({\stackrel{\#}<}推\)【\(( X \stackrel{\#}< Y \stackrel{\#}\le Z ) \mathbin{\rm o\!r} ( X \stackrel{\#}\le Y \stackrel{\#}< Z ) \Longrightarrow X \stackrel{\#}< Z\)】\(\,{\blacktriangleleft}\,\mathbb{D}.\)
\({\stackrel{\#}=}0\)【\(X \stackrel{\#}= 0 \Longleftrightarrow X = \emptyset\)】
\({\stackrel{\#}=}1\)【\(X \stackrel{\#}= 1 \Longleftrightarrow \exists x \, X = \{ x \}\)】