対
word(ss,s)! …【\(\langle {\sf X} , {\sf Y} \rangle\)】対の定義はされません。次のものは対の成分を取り出します。
word(s,s) … \(\triangleleft\) \(\triangleright\)
\({\triangleleft.}\)【\(\triangleleft \langle x , y \rangle = x\)】
\({\triangleright.}\)【\(\triangleright \langle x , y \rangle = y\)】
\(\text{%0}\)【\(\langle x_{0} , y_{0} \rangle = \langle x_{1} , y_{1} \rangle \Longleftrightarrow x_{0} = x_{1} , y_{0} = y_{1}\)】\(\,{\blacktriangleleft}\,\mathbb{D}.\)
直積
word(cc,c) … \(\times\)def …【\({\sf X} \times {\sf Y}\)】≃【\(\{ \langle {\sf x} , {\sf y} \rangle \mid {\sf x} \in {\sf X} , {\sf y} \in {\sf Y} \}\)】
\(\text{Set}^2\)【\(\widehat{\times} \text{Set} = \{ x \mid x = \langle \triangleleft x , \triangleright x \rangle \} \)】\(\,{\blacktriangleleft}\,\mathbb{D}.\)
word(ss,s) … \(\times\)
\(\times.\)【\(X \times Y = X \times Y\)】
対の集合は関係と呼ばれます。
word(,c) … \(\text{Rel}\)
def …【\(\text{Rel}\)】≃【\(\{ R \mid R \subset \widehat{\times} \text{Set} \} \)】
定義域、値域
word(c,c)F … \(\text{dom}\) \(\text{im}\)def …【\(\text{dom} ( {\sf R} )\)】≃【\(\{ {\sf x} \mid \exists {\sf y} \, \langle {\sf x} , {\sf y} \rangle \in {\sf R} \} \)】
def …【\(\text{im} ( {\sf R} )\)】≃【\(\{ {\sf y} \mid \exists {\sf x} \, \langle {\sf x} , {\sf y} \rangle \in {\sf R} \} \)】
word(s,s)F … \(\text{dom}\) \(\text{im}\)
\(\text{dom.}\)【\(\text{dom} ( R ) = \text{dom} ( R )\)】
\(\text{im.}\)【\(\text{im} ( R ) = \text{im} ( R )\)】
\(\text{Rel0}\)【\(R \in \text{Rel} \Longrightarrow R \subset \text{dom} ( R ) \times \text{im} ( R )\)】\(\,{\blacktriangleleft}\,\mathbb{D}.\)
定義域の制限、像
word(cc,c)! …【\({\sf R} | _{ {\sf X} }\)】def …【\({\sf R} | _{ {\sf X} }\)】≃【\(\{ \langle {\sf x} , {\sf y} \rangle \in {\sf R} \mid {\sf x} \in {\sf X} \}\)】
word(ss,s)! …【\({\sf R} | _{ {\sf X} }\)】
\(|.\)【\(R | _{ X } = R | _{ X }\)】
word(ss,s)! …【\({\sf R} ( {\sf A} )\)】
\(\text{Pap}.\)【\(R ( A ) = \{ y \mid \exists a \in A . \langle a , y \rangle \in R \} \)】
成分の入れ替え
word(s,s) … \(^\leftrightarrow\)\(\text{sw.}\)【\(\langle x , y \rangle ^\leftrightarrow = \langle y , x \rangle\)】
\(\text{sw}巾\)【\(\langle x , y \rangle ^\leftrightarrow \, \! ^\leftrightarrow = \langle x , y \rangle\)】\(\,{\blacktriangleleft}\,\mathbb{D}.\)
word(c,c) … \(^\leftrightarrow\)
def …【\({\sf R} ^\leftrightarrow\)】≃【\(\{ {\sf p} ^\leftrightarrow \mid {\sf p} \in {\sf R} \}\)】
word(s,s) … \(^\leftrightarrow\)
\(\text{Psw.}\)【\(R ^\leftrightarrow = R ^\leftrightarrow\)】
\(\text{Psw}巾\)【\(R \in \text{Rel} \Longrightarrow R ^\leftrightarrow \, \! ^\leftrightarrow = R\)】\(\,{\blacktriangleleft}\,\mathbb{D}.\)
合成
word(cc,c) … \(\circ\)def …【\({\sf S} \circ {\sf R}\)】≃【\(\{ \langle {\sf x} , {\sf z} \rangle \mid \exists {\sf y} \, ( \langle {\sf x} , {\sf y} \rangle \in {\sf R} , \langle {\sf y} , {\sf z} \rangle \in {\sf S} ) \}\)】
word(ss,s) … \(\circ\)
\({\circ}.\)【\(S \circ R = S \circ R\)】
\({\circ}\text{Rel}\)【\(R \subset X \times Y , S \subset Y \times Z \Longrightarrow S \circ R \subset X \times Z\)】\(\,{\blacktriangleleft}\,\mathbb{D}.\)
\({\circ}結 \)【\(R , S , T \in \text{Rel} \Longrightarrow ( T \circ S ) \circ R = T \circ ( S \circ R )\)】\(\,{\blacktriangleleft}\,\mathbb{D}.\)
\({\circ}逆\)【\(R , S \in \text{Rel} \Longrightarrow ( R \circ S ) ^\leftrightarrow = S ^\leftrightarrow \circ R ^\leftrightarrow\)】\(\,{\blacktriangleleft}\,\mathbb{D}.\)