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


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\begin{center}
{\large{\textsc{Homework 20}}} \\
\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 19 -- modified]}} Let $A$, $B$, and $X$ be sets.
\begin{enumerate}
\item Prove that if $A \subseteq B$, then $X \setminus B \subseteq X \setminus A$.
\item Prove that $A \subseteq X$ if and only if $A = X \setminus (X \setminus A)$. [\textsc{hint:} Use the result of Homework 19 Exercise 3 to express $X \setminus (X \setminus A)$ in a simpler form.]
\item Suppose $A \subseteq X$. Prove that if $X \setminus B \subseteq X \setminus A$, then $A \subseteq B$.
\item Show, by giving an example, that the implication
\[ \text{if $X \setminus B \subseteq X \setminus A$, then $A \subseteq B$} \]
may be \textbf{false} if $A \not \subseteq X$.

That is, give an example of sets $A$, $B$, and $X$ such that $X \setminus B \subseteq X \setminus A$ and $A \not \subseteq B$.
\end{enumerate}

\item \textbf{{[Falkner Section 10 Exercise 24]}} Prove Proposition 10.34(b): Let $\mathscr{A}$ be a nonempty set of sets and let $X$ be any object. Then
\[ \text{$x \notin \bigg( \bigcap_{A \in \mathscr{A}} A \bigg)$ if and only if there exists $A \in \mathscr{A}$ such that $x \notin A$.} \]
\end{enumerate}

\vspace{.05 in}

\begin{center}
\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.

\begin{enumerate}
\item \textbf{{[Falkner Section 10 Exercise 26]}} Prove Theorem 10.36(b): Let $S$ be a set and let $\mathscr{A}$ be a nonempty set of sets. Then
\[ S \cup \bigg( \bigcap_{A \in \mathscr{A}} A \bigg) = \bigcap_{A \in \mathscr{A}} (S \cup A). \]

\item \textbf{{[Falkner Section 10 Exercise 27]}} Let $A$ be a set and let $\mathscr{B}$ be a nonempty set of sets. Show that:
\begin{enumerate}
\item $\displaystyle A \cup \bigg( \bigcup_{B \in \mathscr{B}} B \bigg) = \bigcup_{B \in \mathscr{A}} (A \cup B)$
\item $\displaystyle A \cap \bigg( \bigcap_{B \in \mathscr{B}} B \bigg) = \bigcap_{B \in \mathscr{A}} (A \cap B)$
\end{enumerate}
\end{enumerate}




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