# 交集

## 有限交集

${\displaystyle (\forall A)(\forall B)(\forall x)\left\{(x\in A\cap B)\Leftrightarrow \left[(x\in A)\wedge (x\in B)\right]\right\}}$

${\displaystyle A}$${\displaystyle B}$的交集写作「${\displaystyle A\cap B}$」，「對所有 ${\displaystyle x}$${\displaystyle x\in A\cap B}$ 等價於 ${\displaystyle x\in A}$${\displaystyle x\in B}$

${\displaystyle A\cap (B\cap C)=(A\cap B)\cap C}$

## 任意交集

${\displaystyle {\bar {\mathcal {M}}}:=\left\{A\,|\,(\exists M\in {\mathcal {M}})(A=M^{c})\right\}}$

${\displaystyle x\in \bigcup {\bar {\mathcal {M}}}\Leftrightarrow (\exists A)[(x\in A)\wedge (\exists M\in {\mathcal {M}})(A=M^{c})]}$

${\displaystyle x\in \bigcup {\bar {\mathcal {M}}}\Leftrightarrow (\exists M)[(M\in {\mathcal {M}})\wedge (x\notin M)\wedge (\exists A)(A=M^{c})]}$

${\displaystyle (\exists A)(A=M^{c})}$

${\displaystyle x\in \bigcup {\bar {\mathcal {M}}}\Leftrightarrow (\exists M\in {\mathcal {M}})(x\notin M)}$

${\displaystyle x\in {\left(\bigcup {\bar {\mathcal {M}}}\right)}^{c}\Leftrightarrow (\forall M\in {\mathcal {M}})(x\in M)}$

${\displaystyle \bigcap {\mathcal {M}}:={\left(\bigcup {\bar {\mathcal {M}}}\right)}^{c}}$

${\displaystyle A\cap B=\bigcap \{A,\,B\}}$

${\displaystyle \bigcap _{A\in {\mathcal {M}}}A}$

${\displaystyle I\,{\overset {A}{\cong }}\,{\mathcal {M}}}$${\displaystyle \bigcap _{i\in I}A(i):=\bigcap {\mathcal {M}}}$

${\displaystyle \mathbb {N} \,{\overset {A}{\cong }}\,{\mathcal {M}}}$${\displaystyle \bigcap _{i=1}^{\infty }A(i):=\bigcap {\mathcal {M}}}$