3 集合・クラスの最初の例

3.1 宇宙、空集合

word(,c) … \(\mathbb{V}\)
𝕍. … \(\mathbb{V} = \{ x \mid \top \}\)
\(x \in \mathbb{V}\) \(\blacktriangleleft\) O
\(X \subset \mathbb{V}\) \(\blacktriangleleft\) O

word(,s) … \(\emptyset\)
∅. … \(\emptyset = \{ x \mid {\perp} \}\)
\(x \notin \emptyset\) \(\blacktriangleleft\) \(\mathbb{W}.\)
\(\emptyset \subset X\) \(\blacktriangleleft\) \(\mathbb{W}.\)

3.2 有限集合

1以上の自然数 \({\tt n}\) に対して
word(s\(^{\tt n}\),s)F … \(\mathsf{set}_{\tt n}\)
abbr … \(\{ x_{1} , \cdots , x_{\tt n} \}\) ≈ \(\mathsf{set}_{\tt n} ( x_{1} , \cdots , x_{\tt n} )\)
set\({\tt n}\). … \(\{ x_{1} , \cdots , x_{\tt n} \} = \{ v \mid v = x_{1} \mathbin{\rm o\!r} \cdots \mathbin{\rm o\!r} v = x_{\tt n} \}\)

3.3 合併、共通部分、差

word(cc,c) … \(\cup\)  \(\cap\)  \(\setminus\)
∪_. … \(A \cup B = \{ x \mid x \in A \mathbin{\rm o\!r} x \in B \}\)
∩_. … \(A \cap B = \{ x \mid x \in A , x \in B \}\)
∖_. … \(A \setminus B = \{ x \mid x \in A , x \notin B \}\)
word(c,c) … \(\mathop\setminus\)
xs. … \(\mathop\setminus Y = \{ x \mid x \notin Y \}\)

word(ss,s) … \(\cup\)  \(\cap\)  \(\mathop\setminus\)
∪. … \([\cup] \stackrel{\star\!\star}{=} [\cup]\)
∩. … \([\cap] \stackrel{\star\!\star}{=} [\cap]\)
∖. … \([\mathop\setminus] \stackrel{\star\!\star}{=} [\setminus]\)
\(X \mathop\setminus Y = X \cap \mathop\setminus Y\) \(\blacktriangleleft\) \(\mathbb{W}.\)
\(X \subset Y \Longleftrightarrow X \cup Y = Y \Longleftrightarrow X \cap Y = X\) \(\blacktriangleleft\) \(\mathbb{W}.\)

3.4 総合併、総共通部分

word(c,c) … \(\bigcup\)  \(\bigcap\)
⋃_. … \(\bigcup \mathcal{X} = \{ x \mid \exists X \in \mathcal{X} . x \in X \}\)
⋂_. … \(\bigcap \mathcal{X} = \{ x \mid \forall X \in \mathcal{X} . x \in X \}\)
\(\bigcap \emptyset = \mathbb{V}\) \(\blacktriangleleft\) \(\mathbb{W}.\)

word(s,s)R … \(\bigcup\)  \(\bigcap\)
⋃. … \([\bigcup] \stackrel{\star}{=} [\bigcup]\)
⋂. … \(\mathcal{X} \neq \emptyset \Longrightarrow \bigcap \mathcal{X} = \bigcap \mathcal{X}\)
\(X = \bigcup \{ X \}\) \(\blacktriangleleft\) \(\mathbb{W}.\)
\(X = \bigcap \{ X \}\) \(\blacktriangleleft\) \(\mathbb{W}.\)
\(X \cup Y = \bigcup \{ X , Y \}\) \(\blacktriangleleft\) \(\mathbb{W}.\)
\(X \cap Y = \bigcap \{ X , Y \}\) \(\blacktriangleleft\) \(\mathbb{W}.\)