$\text{1.Prove Theorem 1 (show that $x$ is in the left-hand set iff it is in the right-hand set).For example,for (d),}$
\begin{eqnarray}
x \in(A \cup B) \cap C &\Longleftrightarrow& [x \in(A \cup B) \text { and } x \in C] \\
&\Longleftrightarrow& [(x \in A \text { or } x \in B), \text { and } x \in C] \\
&\Longleftrightarrow& [(x \in A, x \in C) \text { or }(x \in B, x \in C)]
\end{eqnarray}
- $ A \cup A = A ; A \cap A = A$
\begin{eqnarray}
A \cup A &=& \{ x|x \in A \} \cup \{ x|x \in A \}\\
&=& \{ x|x \in A \}\\
&=&A\\
A \cap A &=& \{ x| x \in A \} \cup \{ x|x \in A \}\\
&=& \{ x|x \in A \}\\
&=&A\\
\end{eqnarray}
- $ A \cup B = B \cup A , A \cap B = B \cap A $
\begin{eqnarray}
A \cup B &=& \{ x| x \in A \} \cup \{ x|x \in B \} \\
&=& \{ x|x \in A\ or\ x \in B \}\\
&=& \{ x|x \in B \} \cup \{ x|x \in A \}\\
&=& B \cup A
\end{eqnarray}
