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%\pagestyle{fancyplain} \chead{Math 3345, Autumn 2021, Kutler}


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\begin{center}
{\large{\textsc{Homework 21}}} \\
\textsc{Math 3345 -- Spring 2022 -- Kutler}
\end{center}

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 33(a)--(d)]}} Let $A$, $B$, $C$, and $D$ be sets.
\begin{enumerate}
\item Prove that $(A \times B) \cap (C \times D) = (A \cap C) \times (B \cap D)$
\item Prove that $(A \cup B) \times C = (A \times C) \cup (B \times C)$ and $A \times (B \cup C) = (A \times B) \cup (A \times C)$.
\item Prove that $(C \times D) \setminus (A \times B) = E \cup F$, where $E = (C \setminus A) \times D$ and $F = C \times (D \setminus B)$.
\item In the special case where $A = [1,3] = B$ and $C = [2,4] = D$, draw a picture to illustrate the result you proved for the general case in part (c).
\end{enumerate}

\item \textbf{{[Falkner Section 10 Exercise 34 -- modified]}}
\begin{enumerate}
\item Let $A$ be a set. Prove that $A \times \varnothing = \varnothing$.
\item Let $A$ and $B$ be sets. Deduce that $A \times \varnothing = B \times \varnothing$.
\item Let $A$, $B$, and $C$ be sets, and suppose that $C \neq \varnothing$. Prove that if $A \times C = B \times C$, then $A = B$.
\end{enumerate}
\end{enumerate}

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\begin{center}
\textbf{Practice Problems}
\end{center}

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.

\begin{enumerate}
\item \textbf{{[Falkner Section 10 Exercise 28]}} Find $\mathscr{P} \big( \{ 1, 2, 3 \} \big)$.

\item \textbf{{[Falkner Section 10 Exercise 32]}} Sketch the rectangle $[1,4] \times [2,3]$ in the coordinate plane. (Shade the set of points that belong to this rectangle.

\item \textbf{{[Falkner Section 10 Exercise 36]}} Let $A$, $B$, and $C$ be sets. Prove that
\[ A \setminus C \subseteq (A \setminus B) \cup (B \setminus C). \]
\end{enumerate}




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