## 2016年12月15日木曜日

### 数学 - 集合論 - 集合算(補集合、差集合、ド・モルガンの法則)

$\begin{array}{l}{A}^{c}\cup \left({B}^{c}\cap {C}^{c}\right)=\left({A}^{c}\cup {B}^{c}\right)\cap \left({A}^{c}\cup {C}^{c}\right)\\ {\left({A}^{c}\cup \left({B}^{c}\cap {C}^{c}\right)\right)}^{c}={\left(\left({A}^{c}\cup {B}^{c}\right)\cap \left({A}^{c}\cup {C}^{c}\right)\right)}^{c}\\ A\cap {\left({B}^{c}\cap {C}^{c}\right)}^{c}={\left({A}^{c}\cup {B}^{c}\right)}^{c}\cup {\left({A}^{c}\cup {C}^{c}\right)}^{c}\\ A\cap \left(B\cup C\right)=\left(A\cap B\right)\cup \left(A\cap C\right)\end{array}$

$\begin{array}{l}{A}^{c}\cap \left({B}^{c}\cup {C}^{c}\right)=\left({A}^{c}\cup {B}^{c}\right)\cap \left({A}^{c}\cup {C}^{c}\right)\\ {\left({A}^{c}\cap \left({B}^{c}\cup {C}^{c}\right)\right)}^{c}={\left(\left({A}^{c}\cup {B}^{c}\right)\cap \left({A}^{c}\cup {C}^{c}\right)\right)}^{c}\\ A\cup {\left({B}^{c}\cup {C}^{c}\right)}^{c}={\left({A}^{c}\cup {B}^{c}\right)}^{c}\cup {\left({A}^{c}\cup {C}^{c}\right)}^{c}\\ A\cup \left(B\cap C\right)=\left(A\cap B\right)\cup \left(A\cap C\right)\end{array}$