最初の写像
具体的な写像を作るときには\(\lambda\)計算の記号が便利です。abbr …【\([ {\sf x} \mid {\sf X} ]\)】≈【\(\{ {\sf z} \mid \exists {\sf x} \, {\sf z} = \langle {\sf x} , {\sf X} \rangle \}\)】
abbr …【\([ {\sf x} {\sf A} \mid {\sf X} ]\)】≈【\(\{ \langle {\sf x} , {\sf X} \rangle \mid {\sf x} {\sf A} \}\)】
【\(f \in \text{mapon} ( X ) \Longrightarrow f = [ x \in X \mid f ( x ) ]\)】\(\,{\blacktriangleleft}\,\mathbb{W}.\)
一点集合への写像
word(ss,s)! … on 【\({\sf Y} _{ \mid {\sf X} }\)】
\(\text{on.}\)【\(y _{ \mid X } = X \times \{ y \}\)】
\(\text{on..}\)【\(y _{ \mid X } = [ x \in X \mid y ]\)】\(\,{\blacktriangleleft}\,\mathbb{W}.\)
\(\text{on0}\)【\(X \to \{ y \} = \{ y _{ \mid X } \}\)】\(\,{\blacktriangleleft}\,\mathbb{W}.\)
互換
word(ss,s) … \(\leftrightharpoons\)
\({\leftrightharpoons}.\)【\(x \leftrightharpoons y = \{ \langle x , y \rangle , \langle y , x \rangle \}\)】
入力補助を？
恒等写像
word(s,s)! … id 【\(\text{id} _{ {\sf X} }\)】\(\text{id}.\)【\(\text{id} _{ X } = [ x \in X \mid x ]\)】
\(\text{id0}\)【\(\text{id} _{ X } \in X \stackrel{\rm IS}\to X\)】\(\,{\blacktriangleleft}\,\mathbb{W}.\)
\({\circ}単\)【\(R \subset X \times Y \Longrightarrow \text{id} _{ Y } \circ R = R = R \circ \text{id} _{ X }\)】\(\,{\blacktriangleleft}\,\mathbb{W}.\)
\(|\text{id}\)【\(R \in \text{Rel} \Longrightarrow R | _{ X } = R \circ \text{id} _{ X }\)】\(\,{\blacktriangleleft}\,\mathbb{W}.\)
\({\stackrel{\rm I}\to}{.}{.}\)【\(X \stackrel{\rm I}\to Y = \{ f \in X \to Y \mid f ^\leftrightarrow \circ f = \text{id} _{ X } \}\)】\(\,{\blacktriangleleft}\,\mathbb{W}.\)
\({\stackrel{\rm S}\to}{.}{.}\)【\(X \stackrel{\rm S}\to Y = \{ f \in X \to Y \mid f \circ f ^\leftrightarrow = \text{id} _{ Y } \}\)】\(\,{\blacktriangleleft}\,\mathbb{W}.\)