(1)

\begin{align*} \left(A+B\right)/C & =\left\{ a+b;a\in A;b\in B\right\} \left\{ C\right\} \\ & =\left\{ a+b+C;a\in A;b\in B\right\} \\ & =\left\{ a;a\in A\right\} +\left\{ b+C;b\in B\right\} \\ & =A+\left(B/C\right) \end{align*} \begin{align*} \left(A+B\right)/C & =\left\{ a+b;a\in A;b\in B\right\} \left\{ C\right\} \\ & =\left\{ a+b+C;a\in A;b\in B\right\} \\ & =\left\{ a+C+b;a\in A;b\in B\right\} \\ & =\left\{ a+C;a\in A\right\} +\left\{ b;b\in B\right\} \\ & =\left(A/C\right)+B \end{align*} となるので題意は成り立つ。

(2)

\begin{align*} A+C & =\left\{ a;a\in A\right\} +\left\{ c;c\in C\right\} \\ & =\left\{ a+c;a\in A,c\in C\right\} \\ & =\left\{ c+a;c\in C,a\in A\right\} \\ & =\left\{ c;c\in C\right\} +\left\{ a;a\in A\right\} \\ & =C+A \end{align*}

(2)-2

\begin{align*} A+C & =\left\{ a;a\in A\right\} +\left\{ c;c\in C\right\} \\ & =\left\{ a+c;a\in A,c\in C\right\} \\ & =\bigcup_{a\in A}\left\{ a+c;c\in C\right\} \\ & =\bigcup_{a\in A}\left\{ a\right\} +\left\{ c;c\in C\right\} \\ & =\bigcup_{a\in A}a+C\\ & =\bigcup_{a\in A}C+a\\ & =\bigcup_{a\in A}\left\{ c;c\in C\right\} +\left\{ a\right\} \\ & =\bigcup_{a\in A}\left\{ c+a;c\in C\right\} \\ & =\left\{ c+a;c\in C,a\in A\right\} \\ & =\left\{ c;c\in C\right\} +\left\{ a;a\in A\right\} \\ & =C+A \end{align*}

(3)

\begin{align*} A+\left\{ C\right\} & =\left\{ a;a\in A\right\} +\left\{ C\right\} \\ & =\left\{ a+C;a\in A\right\} \\ & =\left\{ C+a;a\in A\right\} \\ & =\left\{ C\right\} +\left\{ a;a\in A\right\} \\ & =\left\{ C\right\} +A \end{align*}