1 内包記法

1.1 集合とクラスの基本word

word(sc,p) … \(\in\)
word(ss,p) … \(\in\)
=_. … \(X = Y \Longleftrightarrow \forall x \, ( x \in X \Leftrightarrow x \in Y )\)
word(vp,c)F … \(\mathsf{cls}\)
abbr … \(\{ x \mid P \}\) ≈ \(\mathsf{cls} ( x , P )\)
abbr … \(\{ x \mathop{{\sf A}} \mid P \}\) ≈ \(\{ x \mid x \mathop{{\sf A}} , ( P ) \}\)
abbr … \(\{ T \mid P \}\) ≈ \(\{ x \mid \exists \mathop{{\sf X}} ( ( P ) , x = T ) \}\)
abbr … \(\{ T \mathop{{\sf A}} \mid P \}\) ≈ \(\{ x \mid x \mathop{{\sf A}} , \exists \mathop{{\sf X}} ( ( P ) , x = T ) \}\)
abbr … \(_{ P } T\) ≈ \(\{ T \mid P \}\)

2 一意量化子

abbr … \(! x P\) ≈ \(\forall y , z \in \{ x \mid P \} . y = z\)
abbr … \(\exists! x P\) ≈ \(\exists x P , ! x P\)
abbr … \(! x \mathop{{\sf A}} . P\) ≈ \(! x ( x \mathop{{\sf A}} , ( P ) )\)
abbr … \(\exists! x \mathop{{\sf A}} . P\) ≈ \(\exists! x ( x \mathop{{\sf A}} , ( P ) )\)

3 関数の像

word+(s,s→s,c) … \(\text{^}\)
lower … \(\stackrel{\text{^}}{\mathbin{{\sf f}}} A\) ≃ \(\{ \mathbin{{\sf f}} a \mid a \in A \}\)
word+(ss,s→ss,c) … \(\text{^*}\)  \(\text{*^}\)  \(\text{^^}\)
lower … \(A \stackrel{\text{^*}}{\mathbin{{\sf f}}} b\) ≃ \(\{ a \mathbin{{\sf f}} b \mid a \in A \}\)
lower … \(a \stackrel{\text{*^}}{\mathbin{{\sf f}}} B\) ≃ \(\{ a \mathbin{{\sf f}} b \mid b \in B \}\)
lower … \(A \stackrel{\text{^^}}{\mathbin{{\sf f}}} B\) ≃ \(\{ a \mathbin{{\sf f}} b \mid a \in A , b \in B \}\)

4 アロー関数

word(vs,(s,s))F … \(\mathsf{fn}\)
abbr … \(\lceil x \mapsto X \rfloor\) ≈ \(\mathsf{fn} ( x , X )\)