対称差集合の演算
対称差集合の演算
全体集合を\(X\)として、部分集合を\(A,B,C\subseteq X\)とする。
結合法則
交換法則
補集合
分配法則
空集合の対称差集合
全体集合の対称差集合
自分自身の対称差集合
補集合との対称差集合
対称差集合を含む式
全体集合を\(X\)として、部分集合を\(A,B,C\subseteq X\)とする。
(1)
\begin{align*} A\bigtriangleup B & =\left(A\cup B\right)\setminus\left(A\cap B\right)\\ & =\left(A\setminus B\right)\cup\left(B\setminus A\right) \end{align*}結合法則
(2)
\[ A\bigtriangleup\left(B\bigtriangleup C\right)=\left(A\bigtriangleup B\right)\bigtriangleup C \]交換法則
(3)
\[ A\bigtriangleup B=B\bigtriangleup A \]補集合
(4)
\begin{align*} \left(A\bigtriangleup B\right)^{c} & =A^{c}\bigtriangleup B\\ & =A\bigtriangleup B^{c} \end{align*}(5)
\[ A\bigtriangleup B=A^{c}\bigtriangleup B^{c} \]分配法則
(6)
\[ \left(A\bigtriangleup B\right)\cap C=\left(A\cap C\right)\bigtriangleup\left(B\cap C\right) \]空集合の対称差集合
(7)
\begin{align*} A\bigtriangleup\emptyset & =\emptyset\bigtriangleup A\\ & =A \end{align*}全体集合の対称差集合
(8)
\begin{align*} A\bigtriangleup X & =X\bigtriangleup A\\ & =A^{c} \end{align*}自分自身の対称差集合
(9)
\[ A\bigtriangleup A=\emptyset \]補集合との対称差集合
(10)
\[ A\bigtriangleup A^{c}=X \]対称差集合を含む式
(11)
\begin{align*} A\cup\left(B\bigtriangleup C\right) & =\left\{ \left(A\cup B^{c}\right)\bigtriangleup\left(A\cup C\right)\right\} ^{c}\\ & =\left(A\cup B\right)\bigtriangleup\left(C\setminus A\right) \end{align*}(12)
\begin{align*} A\cap\left(B\bigtriangleup C\right) & =\left(A^{c}\cup B\right)\bigtriangleup\left(A^{c}\cup C\right)\\ & =\left(A\cap B\right)\bigtriangleup\left(A\cap C\right) \end{align*}(13）
\begin{align*} A\bigtriangleup\left(B\cup C\right) & =\left\{ \left(A\bigtriangleup B\right)\cup\left(C\setminus A\right)\right\} \cap\left(A^{c}\cup C^{c}\right)\\ & =\left\{ \left(A\bigtriangleup B\right)\cap\left(A^{c}\cup C^{c}\right)\right\} \cup\left(C\setminus A\right) \end{align*}(14)
\begin{align*} A\bigtriangleup\left(B\cap C\right) & =\left\{ \left(A\bigtriangleup B\right)\cup\left(A\setminus C\right)\right\} \cap\left(A\cup C\right)\\ & =\left\{ \left(A\bigtriangleup B\right)\cap\left(A\cup C\right)\right\} \cup\left(A\setminus C\right) \end{align*}(15)
\[ \left(A\bigtriangleup B\right)\cup A=A\cup B \](16)
\[ \left(A\bigtriangleup B\right)\cup A^{c}=A^{c}\cup B^{c} \](17)
\[ \left(A\bigtriangleup B\right)\cap A=A\cap B^{c} \](18)
\[ \left(A\bigtriangleup B\right)\cap A^{c}=A^{c}\cap B \](19)
\[ \left(A\bigtriangleup B\right)=C\Leftrightarrow A=B\triangle C \](20)
\[ \left(A\bigtriangleup B\right)=\emptyset\Leftrightarrow A=B \](21)
\[ \left(A\bigtriangleup B\right)=X\Leftrightarrow A=B^{c} \](22)
\[ A_{1}\bigtriangleup A_{2}=B_{1}\bigtriangleup B_{2}\Leftrightarrow A_{1}\bigtriangleup B_{1}=A_{2}\bigtriangleup B_{2} \](23)
\[ \left(A\bigtriangleup B\right)\bigtriangleup A=B \](24)
\[ \left(A\bigtriangleup B\right)\bigtriangleup A^{c}=B^{c} \](25)
\[ A\cup B=\left(A\cup B\right)\cup\left(A\bigtriangleup B\right) \](26)
\[ A\cup B=\left(A\cap B\right)\cup\left(A\bigtriangleup B\right) \](27)
\[ A\cap B=\left(A\cup B\right)\setminus\left(A\bigtriangleup B\right) \](28)
\[ A\cap B=\left(A\cap B\right)\setminus\left(A\bigtriangleup B\right) \](29)
\[ \left(A\bigtriangleup B\right)\cap\left(A\cup B\right)=A\bigtriangleup B \](30)
\[ \left(A\bigtriangleup B\right)\cap\left(A\cap B\right)=\emptyset \](31)
\[ \left(A\bigtriangleup B\right)\bigtriangleup\left(C\bigtriangleup B\right)=A\bigtriangleup B \](1)
\begin{align*} A\bigtriangleup B & =\left\{ x;x\in A\nleftrightarrow x\in B\right\} \\ & =\left\{ x;\left(x\in A\land x\in B^{c}\right)\lor\left(x\in A^{c}\land x\in B\right)\right\} \\ & =\left\{ x;\left(x\in A\nrightarrow x\in B\right)\lor\left(x\in A\nleftarrow x\in B\right)\right\} \\ & =\left\{ x;\left(x\in A\setminus B\right)\lor\left(x\in B\setminus A\right)\right\} \\ & =\left(A\setminus B\right)\cup\left(B\setminus A\right) \end{align*} \begin{align*} A\bigtriangleup B & =\left\{ x;x\in A\nleftrightarrow x\in B\right\} \\ & =\left\{ x;\left(x\in A\land x\in B^{c}\right)\lor\left(x\in A^{c}\land x\in B\right)\right\} \\ & =\left\{ x;\left(x\in A\lor x\in B\right)\land\left(x\in A^{c}\lor x\in B^{c}\right)\right\} \\ & =\left\{ x;\left(x\in A\lor x\in B\right)\land\lnot\left(x\in A\land x\in B\right)\right\} \\ & =\left\{ x;\left(x\in A\cup B\right)\nrightarrow\left(x\in A\cap B\right)\right\} \\ & =\left(A\cup B\right)\setminus\left(A\cap B\right) \end{align*}(2)
\begin{align*} A\bigtriangleup\left(B\bigtriangleup C\right) & =\left\{ x;x\in A\nleftrightarrow\left(x\in B\nleftrightarrow x\in C\right)\right\} \\ & =\left\{ x;\left(x\in A\nleftrightarrow x\in B\right)\nleftrightarrow x\in C\right\} \\ & =\left(A\bigtriangleup B\right)\bigtriangleup C \end{align*}(3)
\begin{align*} A\bigtriangleup B & =\left(A\setminus B\right)\cup\left(B\setminus A\right)\\ & =\left(B\setminus A\right)\cup\left(A\setminus B\right)\\ & =B\bigtriangleup A \end{align*}(4)
\begin{align*} \left(A\bigtriangleup B\right)^{c} & =\left\{ \left(A\setminus B\right)\cup\left(B\setminus A\right)\right\} ^{c}\\ & =\left\{ \left(A\cap B^{c}\right)\cup\left(B\cap A^{c}\right)\right\} ^{c}\\ & =\left(A^{c}\cup B\right)\cap\left(B^{c}\cup A\right)\\ & =\left(A^{c}\cap B^{c}\right)\cup\left(B\cap A\right)\\ & =\left(A^{c}\cap B^{c}\right)\cup\left(B\cap A^{cc}\right)\\ & =\left(A^{c}\setminus B\right)\cup\left(B\setminus A^{c}\right)\\ & =A^{c}\bigtriangleup B \end{align*} \begin{align*} \left(A\bigtriangleup B\right)^{c} & =\left(B\bigtriangleup A\right)^{c}\\ & =B^{c}\bigtriangleup A\\ & =A\bigtriangleup B^{c} \end{align*}(5)
\begin{align*} A\bigtriangleup B & =\left(A\setminus B\right)\cup\left(B\setminus A\right)\\ & =\left(A\cap B^{c}\right)\cup\left(B\cap A^{c}\right)\\ & =\left(A^{c}\cap B^{cc}\right)\cup\left(B^{c}\cap A^{cc}\right)\\ & =\left(A^{c}\setminus B^{c}\right)\cup\left(B^{c}\setminus A^{c}\right)\\ & =A^{c}\bigtriangleup B^{c} \end{align*}(6)
\begin{align*} \left(A\bigtriangleup B\right)\cap C & =\left\{ \left(A\setminus B\right)\cup\left(B\setminus A\right)\right\} \cap C\\ & =\left\{ \left(A\cap B^{c}\right)\cup\left(B\cap A^{c}\right)\right\} \cap C\\ & =\left(A\cap B^{c}\cap C\right)\cup\left(B\cap A^{c}\cap C\right)\\ & =\left\{ \left(A\cap C\right)\cap\left(B^{c}\cup C^{c}\right)\right\} \cup\left\{ \left(B\cap C\right)\cap\left(A^{c}\cup C^{c}\right)\right\} \\ & =\left\{ \left(A\cap C\right)\cap\left(B\cap C\right)^{c}\right\} \cup\left\{ \left(B\cap C\right)\cap\left(A\cap C\right)^{c}\right\} \\ & =\left(A\cap C\right)\bigtriangleup\left(B\cap C\right) \end{align*}(7)
\begin{align*} A\bigtriangleup\emptyset & =\left(A\setminus\emptyset\right)\cup\left(\emptyset\setminus A\right)\\ & =A \end{align*}(8)
\begin{align*} A\bigtriangleup\emptyset & =\left(A\setminus X\right)\cup\left(X\setminus A\right)\\ & =A^{c} \end{align*}(9)
\begin{align*} A\bigtriangleup A & =\left(A\setminus A\right)\cup\left(A\setminus A\right)\\ & =\emptyset \end{align*}(10)
\begin{align*} A\bigtriangleup A^{c} & =\left(A\setminus A^{c}\right)\cup\left(A^{c}\setminus A\right)\\ & =A\cup A^{c}\\ & =X \end{align*}(11)
\begin{align*} A\cup\left(B\bigtriangleup C\right) & =\left\{ x;x\in A\lor\left(x\in B\nleftrightarrow x\in C\right)\right\} \\ & =\left\{ x;\left(x\in A\lor x\in B^{c}\right)\leftrightarrow\left(x\in A\lor x\in C\right)\right\} \\ & =\left\{ \left(A\cup B^{c}\right)\bigtriangleup\left(A\cup C\right)\right\} ^{c} \end{align*} \begin{align*} A\cup\left(B\bigtriangleup C\right) & =A\cup\left(B^{c}\bigtriangleup C^{c}\right)\\ & =\left\{ \left(A\cup B\right)\bigtriangleup\left(A\cup C^{c}\right)\right\} ^{c}\\ & =\left(A\cup B\right)\bigtriangleup\left(A^{c}\cap C\right)\\ & =\left(A\cup B\right)\bigtriangleup\left(C\setminus A\right) \end{align*}(12)
\begin{align*} A\cap\left(B\bigtriangleup C\right) & =\left\{ x;x\in A\land\left(x\in B\nleftrightarrow x\in C\right)\right\} \\ & =\left\{ x;\left(x\in A^{c}\lor x\in B\right)\nleftrightarrow\left(x\in A^{c}\lor x\in C\right)\right\} \\ & =\left(A^{c}\cup B\right)\bigtriangleup\left(A^{c}\cup C\right) \end{align*} \begin{align*} A\cap\left(B\bigtriangleup C\right) & =A\cap\left(B^{c}\bigtriangleup C^{c}\right)\\ & =\left(A^{c}\cup B^{c}\right)\bigtriangleup\left(A^{c}\cup C^{c}\right)\\ & =\left(A\cap B\right)\bigtriangleup\left(A\cap C\right) \end{align*}(13）
\begin{align*} A\bigtriangleup\left(B\cup C\right) & =\left\{ x;x\in A\nleftrightarrow\left(x\in B\lor x\in C\right)\right\} \\ & =\left\{ x;\left(\left(x\in A\nleftrightarrow x\in B\right)\lor\left(x\in C\nrightarrow x\in A\right)\right)\land\left(x\in A^{c}\lor x\in C^{c}\right)\right\} \\ & =\left\{ \left(A\bigtriangleup B\right)\cup\left(C\setminus A\right)\right\} \cap\left(A^{c}\cup C^{c}\right) \end{align*} \begin{align*} A\bigtriangleup\left(B\cup C\right) & =\left\{ A^{c}\bigtriangleup\left(B\cup C\right)\right\} ^{c}\\ & =\left\{ \left(\left(A^{c}\bigtriangleup B\right)\cup\left(C\setminus A^{c}\right)\right)\cap\left(A\cup C^{c}\right)\right\} ^{c}\\ & =\left\{ \left(\left(A^{c}\bigtriangleup B\right)\cup\left(C\cap A\right)\right)\cap\left(A\cup C^{c}\right)\right\} ^{c}\\ & =\left\{ \left(A\bigtriangleup B\right)\cap\left(C^{c}\cap A^{c}\right)\right\} \cup\left(A^{c}\cap C\right)\\ & =\left\{ \left(A\bigtriangleup B\right)\cap\left(A^{c}\cup C^{c}\right)\right\} \cup\left(C\setminus A\right) \end{align*}(14)
\begin{align*} A\bigtriangleup\left(B\cap C\right) & =\left\{ x;x\in A\nleftrightarrow\left(x\in B\land x\in C\right)\right\} \\ & =\left\{ x;\left(\left(x\in A\nleftrightarrow x\in B\right)\lor\left(x\in A\nrightarrow x\in C\right)\right)\land\left(x\in A\lor x\in C\right)\right\} \\ & =\left\{ \left(A\bigtriangleup B\right)\cup\left(A\setminus C\right)\right\} \cap\left(A\cup C\right) \end{align*} \begin{align*} A\bigtriangleup\left(B\cap C\right) & =\left\{ A^{c}\bigtriangleup\left(B\cap C\right)\right\} ^{c}\\ & =\left\{ \left(\left(A^{c}\bigtriangleup B\right)\cup\left(A^{c}\cap C^{c}\right)\right)\cap\left(A^{c}\cup C\right)\right\} ^{c}\\ & =\left\{ \left(A\bigtriangleup B\right)\cap\left(A\cup C\right)\right\} \cup\left(A\cap C^{c}\right)\\ & =\left\{ \left(A\bigtriangleup B\right)\cap\left(A\cup C\right)\right\} \cup\left(A\setminus C\right) \end{align*}(15)
\begin{align*} \left(A\bigtriangleup B\right)\cup A & =\left\{ \left(A\cup B\right)\setminus\left(A\cap B\right)\right\} \cup A\\ & =\left\{ \left(A\cup B\right)\cap\left(A\cap B\right)^{c}\right\} \cup A\\ & =\left\{ \left(A\cup B\right)\cap\left(A^{c}\cup B^{c}\right)\right\} \cup A\\ & =\left\{ \left(A\cup B\cup A\right)\cap\left(A^{c}\cup B^{c}\cup A\right)\right\} \\ & =A\cup B \end{align*}(15)-2
\begin{align*} \left(A\bigtriangleup B\right)\cup A & =\left(A\setminus B\right)\cup\left(B\setminus A\right)\cup A\\ & =\left(A\cap B^{c}\right)\cup\left(B\cap A^{c}\right)\cup A\\ & =A\cup\left(B\cap A^{c}\right)\\ & =A\cup B \end{align*}(16)
\begin{align*} \left(A\bigtriangleup B\right)\cup A^{c} & =\left\{ \left(A\cup B\right)\setminus\left(A\cap B\right)\right\} \cup A^{c}\\ & =\left\{ \left(A\cup B\right)\cap\left(A\cap B\right)^{c}\right\} \cup A^{c}\\ & =\left\{ \left(A\cup B\right)\cap\left(A^{c}\cup B^{c}\right)\right\} \cup A^{c}\\ & =\left\{ \left(A\cup B\cup A^{c}\right)\cap\left(A^{c}\cup B^{c}\cup A^{c}\right)\right\} \\ & =A^{c}\cup B^{c} \end{align*}(16)-2
\begin{align*} \left(A\bigtriangleup B\right)\cup A^{c} & =\left(A\setminus B\right)\cup\left(B\setminus A\right)\cup A^{c}\\ & =\left(A\cap B^{c}\right)\cup\left(B\cap A^{c}\right)\cup A^{c}\\ & =\left(A\cap B^{c}\right)\cup A^{c}\\ & =A^{c}\cup B^{c} \end{align*}(17)
\begin{align*} \left(A\bigtriangleup B\right)\cap A^{c} & =\left\{ \left(A\cup B\right)\setminus\left(A\cap B\right)\right\} \cap A\\ & =\left(A\cup B\right)\cap\left(A\cap B\right)^{c}\cap A\\ & =A\cap\left(A\cap B\right)^{c}\\ & =A\cap\left(A^{c}\cup B^{c}\right)\\ & =A\cap B^{c} \end{align*}(17)-2
\begin{align*} \left(A\bigtriangleup B\right)\cap A & =\left\{ \left(A\setminus B\right)\cup\left(B\setminus A\right)\right\} \cap A\\ & =\left\{ \left(A\cap B^{c}\right)\cup\left(B\cap A^{c}\right)\right\} \cap A\\ & =\left(A\cap B^{c}\cap A\right)\cup\left(B\cap A^{c}\cap A\right)\\ & =A\cap B^{c} \end{align*}(18)
\begin{align*} \left(A\bigtriangleup B\right)\cap A^{c} & =\left\{ \left(A\cup B\right)\setminus\left(A\cap B\right)\right\} \cap A^{c}\\ & =\left(A\cup B\right)\cap\left(A\cap B\right)^{c}\cap A^{c}\\ & =A^{c}\cap B\cap\left(A\cap B\right)^{c}\\ & =A^{c}\cap B\cap\left(A^{c}\cup B^{c}\right)\\ & =A^{c}\cap B \end{align*}(18)-2
\begin{align*} \left(A\bigtriangleup B\right)\cap A^{c} & =\left\{ \left(A\setminus B\right)\cup\left(B\setminus A\right)\right\} \cap A^{c}\\ & =\left\{ \left(A\cap B^{c}\right)\cup\left(B\cap A^{c}\right)\right\} \cap A^{c}\\ & =\left(A\cap B^{c}\cap A^{c}\right)\cup\left(B\cap A^{c}\cap A^{c}\right)\\ & =A^{c}\cap B \end{align*}(19)
\begin{align*} A\triangle B=C & \Leftrightarrow\left\{ x;\left(x\in A\nleftrightarrow x\in B\right)\leftrightarrow x\in C\right\} \\ & \Leftrightarrow\left\{ x;\left(x\in A\leftrightarrow x\notin B\right)\leftrightarrow x\in C\right\} \\ & \Leftrightarrow\left\{ x;x\in A\leftrightarrow\left(x\notin B\leftrightarrow x\in C\right)\right\} \\ & \Leftrightarrow\left\{ x;x\in A\leftrightarrow\left(x\in B\nleftrightarrow x\in C\right)\right\} \\ & \Leftrightarrow A=B\triangle C \end{align*}(20)
(19)より、\begin{align*} \left(A\bigtriangleup B\right)=\emptyset & \Leftrightarrow A=B\triangle\emptyset\\ & \Leftrightarrow A=\left(B\cup\emptyset\right)\setminus\left(B\cap\emptyset\right)\\ & \Leftrightarrow A=B \end{align*}
(20)-2
\begin{align*} \left(A\bigtriangleup B\right)=\emptyset & \Leftrightarrow\left\{ x;\left(x\in A\nleftrightarrow x\in B\right)\leftrightarrow x\in\emptyset\right\} \\ & \Leftrightarrow\left\{ x;\lnot\left(x\in A\nleftrightarrow x\in B\right)\right\} \\ & \Leftrightarrow\left\{ x;x\in A\leftrightarrow x\in B\right\} \\ & \Leftrightarrow A=B \end{align*}(21)
(19)より、\begin{align*} \left(A\bigtriangleup B\right)=X & \Leftrightarrow A=B\triangle X\\ & \Leftrightarrow A=\left(B\cup X\right)\setminus\left(B\cap X\right)\\ & \Leftrightarrow A=X\setminus B\\ & \Leftrightarrow A=B^{c} \end{align*}
(21)-2
\begin{align*} \left(A\bigtriangleup B\right)=X & \Leftrightarrow\left\{ x;\left(x\in A\nleftrightarrow x\in B\right)\leftrightarrow x\in X\right\} \\ & \Leftrightarrow\left\{ x;x\in A\nleftrightarrow x\in B\right\} \\ & \Leftrightarrow\left\{ x;x\in A\leftrightarrow x\in B^{c}\right\} \\ & \Leftrightarrow A=B^{c} \end{align*}(22)
\begin{align*} A_{1}\bigtriangleup A_{2}=B_{1}\bigtriangleup B_{2} & \Leftrightarrow A_{1}=A_{2}\bigtriangleup B_{1}\bigtriangleup B_{2}\cmt{\because\left(A\bigtriangleup B\right)=C\Leftrightarrow A=B\triangle C}\\ & \Leftrightarrow A_{1}=B_{1}\bigtriangleup A_{2}\bigtriangleup B_{2}\\ & \Leftrightarrow A_{1}\bigtriangleup B_{1}=A_{2}\bigtriangleup B_{2} \end{align*}(23)
\begin{align*} \left(A\bigtriangleup B\right)\bigtriangleup A & =\left(B\bigtriangleup A\right)\bigtriangleup A\\ & =B\bigtriangleup\left(A\bigtriangleup A\right)\\ & =B\bigtriangleup\emptyset\\ & =B \end{align*}(24)
\begin{align*} \left(A\bigtriangleup B\right)\bigtriangleup A^{c} & =\left(B\bigtriangleup A\right)\bigtriangleup A^{c}\\ & =B\bigtriangleup\left(A\bigtriangleup A^{c}\right)\\ & =B\bigtriangleup X\\ & =B^{c} \end{align*}(25)
\begin{align*} \left(A\bigtriangleup B\right)\cup\left(A\cup B\right) & =\left\{ \left(A\cup B\right)\setminus\left(A\cap B\right)\right\} \cup\left(A\cup B\right)\\ & =\left\{ \left(A\cup B\right)\cap\left(A\cap B\right)^{c}\right\} \cup\left(A\cup B\right)\\ & =A\cup B \end{align*}(25)-2
\begin{align*} \left(A\bigtriangleup B\right)\cup\left(A\cup B\right) & =\left(A\setminus B\right)\cup\left(B\setminus A\right)\cup A\cup B\\ & =\left(A\cap B^{c}\right)\cup\left(B\cap A^{c}\right)\cup A\cup B\\ & =\left(B\cap A^{c}\right)\cup A\cup B\\ & =A\cup B \end{align*}(26)
\begin{align*} A\cup B & =\left(A\cup B\right)\cup\left\{ \left(A\cup B\right)\cap\left(A\cap B\right)^{c}\right\} \\ & =\left(A\cup B\right)\cup\left\{ \left(A\cup B\right)\setminus\left(A\cap B\right)\right\} \\ & =\left(A\cap B\right)\cup\left(A\bigtriangleup B\right) \end{align*}(26)-2
\begin{align*} \left(A\bigtriangleup B\right)\cup\left(A\cap B\right) & =\left(A\setminus B\right)\cup\left(B\setminus A\right)\cup\left(A\cap B\right)\\ & =\left(A\cap B^{c}\right)\cup\left(B\cap A^{c}\right)\cup\left(A\cap B\right)\\ & =\left\{ A\cap\left(B\cup B^{c}\right)\right\} \cup\left(B\cap A^{c}\right)\\ & =A\cup\left(B\cap A^{c}\right)\\ & =A\cup B \end{align*}(27)
\begin{align*} A\cap B & =\left(A\cup B\right)\cap\left(A\cap B\right)\\ & =\left(A\cup B\right)\cap\left\{ \left(A\cup B\right)^{c}\cup\left(A\cap B\right)\right\} \\ & =\left(A\cup B\right)\cap\left\{ \left(A\cup B\right)\setminus\left(A\cap B\right)\right\} ^{c}\\ & =\left(A\cup B\right)\cap\left(A\bigtriangleup B\right)^{c}\\ & =\left(A\cup B\right)\setminus\left(A\bigtriangleup B\right) \end{align*}(28)
\begin{align*} \left(A\cap B\right)\setminus\left(A\bigtriangleup B\right) & =\left(A\cap B\right)\setminus\left\{ \left(A\cup B\right)\setminus\left(A\cap B\right)\right\} \\ & =\left(A\cap B\right)\cap\left\{ \left(A\cup B\right)\setminus\left(A\cap B\right)\right\} ^{c}\\ & =\left(A\cap B\right)\cap\left\{ \left(A\cup B\right)^{c}\cup\left(A\cap B\right)\right\} \\ & =A\cap B \end{align*}(29)
\begin{align*} \left(A\bigtriangleup B\right)\cap\left(A\cup B\right) & =\left\{ \left(A\cup B\right)\setminus\left(A\cap B\right)\right\} \cap\left(A\cup B\right)\\ & =\left(A\cup B\right)\cap\left(A\cap B\right)^{c}\cap\left(A\cup B\right)\\ & =\left(A\cup B\right)\cap\left(A\cap B\right)^{c}\\ & =\left(A\cup B\right)\setminus\left(A\cap B\right)\\ & =A\bigtriangleup B \end{align*}(29)-2
\begin{align*} \left(A\bigtriangleup B\right)\cap\left(A\cup B\right) & =\left\{ \left(A\setminus B\right)\cup\left(B\setminus A\right)\right\} \cap\left(A\cup B\right)\\ & =\left\{ \left(A\cap B^{c}\right)\cup\left(B\cap A^{c}\right)\right\} \cap\left(A\cup B\right)\\ & =\left\{ A\cap B^{c}\cap\left(A\cup B\right)\right\} \cup\left\{ B\cap A^{c}\cap\left(A\cup B\right)\right\} \\ & =\left(A\cap B^{c}\right)\cup\left(B\cap A^{c}\right)\\ & =\left(A\setminus B\right)\cup\left(B\setminus A\right)\\ & =A\bigtriangleup B \end{align*}(30)
\begin{align*} \left(A\bigtriangleup B\right)\cap\left(A\cap B\right) & =\left\{ \left(A\cup B\right)\setminus\left(A\cap B\right)\right\} \cap\left(A\cap B\right)\\ & =\left(A\cup B\right)\cap\left(A\cap B\right)^{c}\cap\left(A\cap B\right)\\ & =\emptyset \end{align*}(30)-2
\begin{align*} \left(A\bigtriangleup B\right)\cap\left(A\cap B\right) & =\left\{ \left(A\setminus B\right)\cup\left(B\setminus A\right)\right\} \cap A\cap B\\ & =\left\{ \left(A\cap B^{c}\right)\cup\left(B\cap A^{c}\right)\right\} \cap A\cap B\\ & =\left(A\cap B^{c}\cap A\cap B\right)\cup\left(B\cap A^{c}\cap A\cap B\right)\\ & =\emptyset\cup\emptyset\\ & =\emptyset \end{align*}(31)
\begin{align*} \left(A\bigtriangleup B\right)\bigtriangleup\left(C\bigtriangleup B\right) & =A\bigtriangleup\left\{ B\bigtriangleup\left(C\bigtriangleup B\right)\right\} \\ & =A\bigtriangleup\left\{ B\bigtriangleup\left(B\bigtriangleup C\right)\right\} \\ & =A\bigtriangleup\left\{ \left(B\bigtriangleup B\right)\bigtriangleup C\right\} \\ & =A\bigtriangleup\left\{ \emptyset\bigtriangleup C\right\} \\ & =A\bigtriangleup C \end{align*}
ページ情報
| タイトル | 対称差集合の演算 |
| URL | https://www.nomuramath.com/m8l2t8if/ |
| SNSボタン |
集合の演算その他
\[
A\cup B=A\cap B\Leftrightarrow A=B
\]
差集合の演算
\[
\left(A\setminus B\right)\setminus C=A\setminus\left(B\cup C\right)
\]
否定包含関係を含む式
\begin{align*}
A\nsubseteq B & \Leftrightarrow B^{c}\nsubseteq A^{c}\\
& \Leftrightarrow A\cap B^{c}\ne\emptyset\\
& \Leftrightarrow B\subsetneq B\cup A\\
& \Leftrightarrow A\cap B\subsetneq A\\
& \Leftrightarrow A\setminus B\ne\emptyset
\end{align*}
等号なし包含関係を含む式
\[
A\subsetneq B\Rightarrow A\subseteq B
\]