## 2016年5月13日金曜日

### 数学 – 群 - 群とその例(部分集合、対称差、可換群)

• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax

$\begin{array}{l}A,B,C\in P\left(S\right)\\ \left(A\Delta B\right)\Delta C\\ =\left(\left(A-B\right)\cup \left(B-A\right)\right)\Delta C\\ =\left(\left(\left(A-B\right)\cup \left(B-A\right)\right)-C\right)\cup \left(C-\left(\left(A-B\right)\cup \left(B-A\right)\right)\right)\\ =\left(\left(\left(A\cap {B}^{c}\right)\cup \left(B\cap {A}^{c}\right)\right)\cap {C}^{c}\right)\cup \left(C\cap {\left(\left(A\cap {B}^{c}\right)\cup \left(B\cap {A}^{c}\right)\right)}^{c}\right)\\ =\left(A\cap {B}^{c}\cap {C}^{c}\right)\cup \left({A}^{c}\cap B\cap {C}^{c}\right)\cup \left(C\cap \left(\left({A}^{c}\cup B\right)\cap \left({B}^{c}\cup A\right)\right)\right)\\ =\left(A\cap {B}^{c}\cap {C}^{c}\right)\cup \left({A}^{c}\cap B\cap {C}^{c}\right)\cup \left(C\cap \left(\left({A}^{c}\cap {B}^{c}\right)\cup \left(B\cap A\right)\right)\right)\\ =\left(A\cap {B}^{c}\cap {C}^{c}\right)\cup \left({A}^{c}\cap B\cap {C}^{c}\right)\cup \left({A}^{c}\cap {B}^{c}\cap C\right)\cup \left(A\cap B\cap C\right)\\ =\left(A\cap B\cap C\right)\cup \left(A\cap {B}^{c}\cap {C}^{c}\right)\cup \left({A}^{c}\cap B\cap {C}^{c}\right)\cup \left({A}^{c}\cap {B}^{c}\cap C\right)\\ =\left(A\cap \left(\left(B\cap C\right)\cup \left({B}^{c}\cap {C}^{c}\right)\right)\right)\cup \left({A}^{c}\cap B\cap {C}^{c}\right)\cup \left({A}^{c}\cap {B}^{c}\cap C\right)\\ =\left(A\cap \left(\left({B}^{c}\cup C\right)\cap \left(B\cup {C}^{c}\right)\right)\right)\cup \left({A}^{c}\cap B\cap {C}^{c}\right)\cup \left({A}^{c}\cap {B}^{c}\cap C\right)\\ =\left(A\cap {\left(\left(B\cap {C}^{c}\right)\cup \left(C\cap {B}^{c}\right)\right)}^{c}\right)\cup \left(\left(\left(B\cap {C}^{c}\right)\cup \left(C\cap {B}^{c}\right)\right)\cap {A}^{c}\right)\\ =\left(A-\left(\left(B-C\right)\cup \left(C-B\right)\right)\right)\cup \left(\left(\left(B-C\right)\cup \left(C-B\right)\right)-A\right)\\ =A\Delta \left(\left(B-C\right)\cup \left(C-B\right)\right)\\ =A\Delta \left(B\Delta C\right)\\ \\ \left(A\Delta B\right)C=A\Delta \left(B\Delta C\right)\\ \\ A\Delta \varphi =\left(A-\varphi \right)\cup \left(\varphi -A\right)=A\\ \varphi \Delta A=\left(\varphi -A\right)\cup \left(A-\varphi \right)=A\\ \\ A\Delta A=\left(A-A\right)\cup \left(A-A\right)=\varphi \\ \\ A\Delta B=\left(A-B\right)\cup \left(B-A\right)=\left(B-A\right)\cup \left(A-B\right)=B\Delta A\end{array}$