(1)

\begin{align*} A\cup\left(B\cup C\right) & =A\cup\left\{ x;x\in B\lor x\in C\right\} \\ & =\left\{ x;x\in A\lor\left(x\in B\lor x\in C\right)\right\} \\ & =\left\{ x;\left(x\in A\lor x\in B\right)\lor x\in C\right\} \\ & =\left\{ x;\left(x\in A\lor x\in B\right)\right\} \cup C\\ & =\left(A\cup B\right)\cup C \end{align*}

(2)

\begin{align*} A\cap\left(B\cap C\right) & =A\cap\left\{ x;x\in B\land x\in C\right\} \\ & =\left\{ x;x\in A\land\left(x\in B\land x\in C\right)\right\} \\ & =\left\{ x;\left(x\in A\land x\in B\right)\land x\in C\right\} \\ & =\left\{ x;\left(x\in A\land x\in B\right)\right\} \cap C\\ & =\left(A\cap B\right)\cap C \end{align*}

(3)

\begin{align*} A\cup B & =\left\{ x;x\in A\lor x\in B\right\} \\ & =\left\{ x;x\in B\lor x\in A\right\} \\ & =B\cup A \end{align*}

(4)

\begin{align*} A\cap B & =\left\{ x;x\in A\land x\in B\right\} \\ & =\left\{ x;x\in B\land x\in A\right\} \\ & =B\cap A \end{align*}

(5)

\begin{align*} A^{cc} & =X\setminus\left(X\setminus A\right)\\ & =X\setminus\left\{ x;x\in X\nrightarrow x\in A\right\} \\ & =\left\{ x;x\in X\nrightarrow\left(x\in X\nrightarrow x\in A\right)\right\} \\ & =\left\{ x;x\in X\land\left(x\in X\rightarrow x\in A\right)\right\} \\ & =\left\{ x;x\in X\land\left(x\notin X\lor x\in A\right)\right\} \\ & =\left\{ x;\left(x\in X\land x\notin X\right)\lor\left(x\in X\land x\in A\right)\right\} \\ & =\left\{ x;x\in X\land x\in A\right\} \\ & =\left\{ x;x\in A\right\} \\ & =A \end{align*}

(6)

\begin{align*} A\cup A & =\left\{ x;x\in A\lor x\in A\right\} \\ & =\left\{ x;x\in A\right\} \\ & =A \end{align*}

(7)

\begin{align*} A\cap A & =\left\{ x;x\in A\land x\in A\right\} \\ & =\left\{ x;x\in A\right\} \\ & =A \end{align*}

(8)

\begin{align*} A\cup\left(A\cap B\right) & =\left\{ x;x\in A\lor\left(x\in A\land x\in B\right)\right\} \\ & =\left\{ x;x\in A\right\} \\ & =A \end{align*}

(9)

\begin{align*} A\cap\left(A\cup B\right) & =\left\{ x;x\in A\land\left(x\in A\lor x\in B\right)\right\} \\ & =\left\{ x;x\in A\right\} \\ & =A \end{align*}

(10)

\begin{align*} A\cup\left(A^{c}\cap B\right) & =\left\{ x;x\in A\lor\left(x\notin A\land x\in B\right)\right\} \\ & =\left\{ x;\left(x\in A\lor x\notin A\right)\land\left(x\in A\lor x\in B\right)\right\} \\ & =\left\{ x;x\in A\lor x\in B\right\} \\ & =A\cup B \end{align*}

(11)

\begin{align*} A\cap\left(A^{c}\cup B\right) & =\left\{ x;x\in A\land\left(x\notin A\lor x\in B\right)\right\} \\ & =\left\{ x;\left(x\in A\land x\notin A\right)\lor\left(x\in A\land x\in B\right)\right\} \\ & =\left\{ x;x\in A\land x\in B\right\} \\ & =A\cap B \end{align*}

(12)

\begin{align*} A\cup\left(B\cap C\right) & =\left\{ x;x\in A\lor\left(x\in B\land x\in C\right)\right\} \\ & =\left\{ x;\left(x\in A\lor x\in B\right)\land\left(x\in A\lor x\in C\right)\right\} \\ & =\left(A\cup B\right)\cap\left(A\cup C\right) \end{align*}

(13)

\begin{align*} A\cap\left(B\cup C\right) & =\left\{ x;x\in A\land\left(x\in B\lor x\in C\right)\right\} \\ & =\left\{ x;\left(x\in A\land x\in B\right)\lor\left(x\in A\land x\in C\right)\right\} \\ & =\left(A\cap B\right)\cup\left(A\cap C\right) \end{align*}

(14)

\begin{align*} \left(A\cup B\right)^{c} & =X\setminus\left(A\cup B\right)\\ & =\left\{ x;x\in X\nrightarrow\left(x\in A\lor x\in B\right)\right\} \\ & =\left\{ x;x\in X\land\left(x\notin A\land x\notin B\right)\right\} \\ & =\left\{ x;\left(x\in X\land x\notin A\right)\land\left(x\in X\land x\notin B\right)\right\} \\ & =\left\{ x;\left(x\in X\nrightarrow x\in A\right)\land\left(x\in X\nrightarrow x\in B\right)\right\} \\ & =A^{c}\cap B^{c} \end{align*}

(15)

\begin{align*} \left(A\cap B\right)^{c} & =\left(\left(A^{c}\cup B^{c}\right)^{c}\right)^{c}\\ & =\left(A^{c}\cup B^{c}\right)^{cc}\\ & =A^{c}\cup B^{c} \end{align*}

(16)

\begin{align*} A\cup\left(\bigcup_{\lambda\in\Lambda}B_{\lambda}\right) & =\left\{ x;x\in A\cup\bigcup_{\lambda\in\Lambda}B_{\lambda}\right\} \\ & =\left\{ x;x\in A\lor x\in\bigcup_{\lambda\in\Lambda}B_{\lambda}\right\} \\ & =\left\{ x;x\in A\lor\exists\lambda\in\Lambda,x\in B_{\lambda}\right\} \\ & =\left\{ x;\exists\lambda\in\Lambda,x\in A\lor x\in B_{\lambda}\right\} \\ & =\left\{ x;\exists\lambda\in\Lambda,x\in A\cup B_{\lambda}\right\} \\ & =\left\{ x;x\in\bigcup_{\lambda\in\Lambda}\left(A\cup B_{\lambda}\right)\right\} \\ & =\bigcup_{\lambda\in\Lambda}\left(A\cup B_{\lambda}\right) \end{align*}

(17)

\begin{align*} A\cap\left(\bigcap_{\lambda\in\Lambda}B_{\lambda}\right) & =\left\{ x;x\in A\cap\left(\bigcap_{\lambda\in\Lambda}B_{\lambda}\right)\right\} \\ & =\left\{ x;x\in A\land x\in\bigcap_{\lambda\in\Lambda}B_{\lambda}\right\} \\ & =\left\{ x;x\in A\land\forall\lambda\in\Lambda,x\in B_{\lambda}\right\} \\ & =\left\{ x;\forall\lambda\in\Lambda,x\in A\land x\in B_{\lambda}\right\} \\ & =\left\{ x;\forall\lambda\in\Lambda,x\in A\cap B_{\lambda}\right\} \\ & =\left\{ x;x\in\bigcap_{\lambda\in\Lambda}\left(A\cap B_{\lambda}\right)\right\} \\ & =\bigcap_{\lambda\in\Lambda}\left(A\cap B_{\lambda}\right) \end{align*}

(18)

\begin{align*} A\cup\left(\bigcap_{\lambda\in\Lambda}B_{\lambda}\right) & =\left\{ x;x\in A\cup\left(\bigcap_{\lambda\in\Lambda}B_{\lambda}\right)\right\} \\ & =\left\{ x;x\in A\lor x\in\bigcap_{\lambda\in\Lambda}B_{\lambda}\right\} \\ & =\left\{ x;x\in A\lor\forall\lambda\in\Lambda,x\in B_{\lambda}\right\} \\ & =\left\{ x;\forall\lambda\in\Lambda,x\in A\lor x\in B_{\lambda}\right\} \\ & =\left\{ x;\forall\lambda\in\Lambda,x\in A\cup B_{\lambda}\right\} \\ & =\left\{ x;x\in\bigcap_{\lambda\in\Lambda}\left(A\cup B_{\lambda}\right)\right\} \\ & =\bigcap_{\lambda\in\Lambda}\left(A\cup B_{\lambda}\right) \end{align*}

(19)

\begin{align*} A\cap\left(\bigcup_{\lambda\in\Lambda}B_{\lambda}\right) & =\left\{ x;x\in A\cap\left(\bigcup_{\lambda\in\Lambda}B_{\lambda}\right)\right\} \\ & =\left\{ x;x\in A\land x\in\bigcup_{\lambda\in\Lambda}B_{\lambda}\right\} \\ & =\left\{ x;x\in A\land\exists\lambda\in\Lambda,x\in B_{\lambda}\right\} \\ & =\left\{ x;\exists\lambda\in\Lambda,x\in A\land x\in B_{\lambda}\right\} \\ & =\left\{ x;\exists\lambda\in\Lambda,x\in A\cap B_{\lambda}\right\} \\ & =\left\{ x;x\in\bigcup_{\lambda\in\Lambda}\left(A\cap B_{\lambda}\right)\right\} \\ & =\bigcup_{\lambda\in\Lambda}\left(A\cap B_{\lambda}\right) \end{align*}

(20)

\begin{align*} \left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)^{c} & =\left\{ x;\exists\lambda\in\Lambda,x\in A_{\lambda}\right\} ^{c}\\ & =\left\{ x;\lnot\left(\exists\lambda\in\Lambda,x\in A_{\lambda}\right)\right\} \\ & =\left\{ x;\forall\lambda\in\Lambda,x\notin A_{\lambda}\right\} \\ & =\bigcap_{\lambda\in\Lambda}A_{\lambda}^{c} \end{align*}

(21)

\begin{align*} \left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)^{c} & =\left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)^{cc}\\ & =\bigcup_{\lambda\in\Lambda}A_{\lambda}^{\;c} \end{align*}

(21)-2

直接計算する。
\begin{align*} \left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)^{c} & =\left\{ x;\forall\lambda\in\Lambda,x\in A_{\lambda}\right\} ^{c}\\ & =\left\{ x;\lnot\left(\forall\lambda\in\Lambda,x\in A_{\lambda}\right)\right\} \\ & =\left\{ x;\exists\lambda\in\Lambda,x\notin A_{\lambda}\right\} \\ & =\bigcup_{\lambda\in\Lambda}A_{\lambda}^{c} \end{align*}