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{\large{\textsc{Homework 18}}} \\
\textsc{Math 3345 -- Spring 2022 -- Kutler}
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Please complete the following problems on your own paper. Solutions should be written clearly, legibly, and with appropriate style.

\begin{enumerate}

\item \textbf{{[Falkner Section 10 Exercise 7]}} Let $A = \big\{ 1, \{4, 7\}, 9 \big\}$ and $B = \big\{ \{1, 4\}, 7, 9 \big\}$. Find $A \cup B$, $A \cap B$, $A \setminus B$, and $B \setminus A$.

\item \textbf{{[Falkner Section 10 Exercise 8]}} Let $A$ and $B$ be sets. Show that $A \cap B \subseteq A$ and $A \cap B \subseteq B$.

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\textbf{Practice Problems}
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It is strongly recommended that you complete the following problems. There is no need to write up polished, final versions of your solutions (although you may find this a useful exercise). Please do not submit any work for these problems.

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\item Let $A$ and $B$ be sets. Prove that
\[ A \cup B = B \cup A \qquad \text{and} \qquad A \cap B = B \cap A. \]

\item Let $A$, $B$, and $C$ be sets. Prove that
\[ (A \cup B) \cup C = A \cup (B \cup C) \qquad \text{and} \qquad (A \cap B) \cap C = A \cap (B \cap C). \]

\item Give an example of sets $A$, $B$, and $C$ such that
\[ (A \cup B) \cap C \neq A \cup (B \cap C). \]
\end{enumerate}




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