集合的运算
分配律
\(\text{(i)}A\cap(\displaystyle\bigcup_{\alpha\in I}B_\alpha)=\displaystyle\bigcup_{\alpha\in I}(A\cap B_\alpha)\);
\(\text{(ii)}A\cup(\displaystyle\bigcap_{\alpha\in I}B_\alpha)=\displaystyle\bigcap_{\alpha\in I}(A\cup B_\alpha)\).
De.Morgan法则
\(\text{(i)}(\displaystyle\bigcup_{\alpha\in I}A_\alpha)^c=\displaystyle\bigcap_{\alpha\in I}A_\alpha^c\);
\(\text{(ii)}(\displaystyle\bigcap_{\alpha\in I}A_\alpha)^c=\displaystyle\bigcup_{\alpha\in I}A_\alpha^c\).
集合列的极限
\(\varlimsup\limits_{k\to\infty}A_k=\displaystyle\bigcap_{j=1}^\infty\displaystyle\bigcup_{k=j}^\infty A_k=\{x:\forall j\in\mathbb N,\exists k\geqslant j,x\in A_k\}\);
\(\varliminf\limits_{k\to\infty}A_k=\displaystyle\bigcup_{j=1}^\infty\displaystyle\bigcap_{k=j}^\infty A_k=\{x:\exists j_0\in\mathbb N,\forall k\geqslant j_0,x\in A_k\}\).
映射
\(\text{(i)}f(\displaystyle\bigcup_{\alpha\in I}A_\alpha)=\displaystyle\bigcup_{\alpha\in I}f(A_\alpha)\);
\(\text{(ii)}f(\displaystyle\bigcap_{\alpha\in I}A_\alpha)\subset\displaystyle\bigcap_{\alpha\in I}f(A_\alpha)\);
\(\text{(iii)}f^{-1}(\displaystyle\bigcup_{\alpha\in I}B_\alpha)=\displaystyle\bigcup_{\alpha\in I}f^{-1}(B_\alpha)\);
\(\text{(iv)}f^{-1}(\displaystyle\bigcap_{\alpha\in I}B_\alpha)=\displaystyle\bigcap_{\alpha\in I}f^{-1}(B_\alpha)\).
