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\begin{document}

\begin{center}
{\large{\textsc{Homework 22}}} \\
\textsc{Math 3345 -- Autumn 2022 -- Kutler}
\end{center}

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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 11 Exercise 6]}} Let $f(x) = x^2 + 1$ for all $x \in \RR$, let $g(y) = \sqrt{y-1}$ for all $y \in [1, \infty)$, and let $h(u) = 1 - u$ for all $u \in [2,3)$. Find the range of $f$, the range of $g$, and the range of $h$.

\vspace{.5in}

\item \textbf{{[Falkner Section 11 Exercise 15(a) -- modified]}} Recall that for a set $X$, the \textbf{power set} $\mathscr{P}(X)$ is the set of all subsets of $X$.

Let $S$ and $T$ be sets.

\begin{enumerate}
\item Prove that if $A \subseteq S$ and $B \subseteq T$, then $A \cup B \subseteq S \cup T$.

\item Prove that \emph{every} subset of $S \cup T$ is of the form $A \cup B$, where $A \subseteq S$ and $B \subseteq T$. That is, if $Y \subseteq S \cup T$, then there exist subsets $A \subseteq S$ and $B \subseteq T$ such that $Y = A \cup B$.
\end{enumerate}
We may understand the result of part (a) as saying that the function
\begin{align*}
f \colon \mathscr{P}(S) \times \mathscr{P}(T) &\to \mathscr{P}(S \cup T) \\
(A,B) &\mapsto A \cup B
\end{align*}
is well-defined. That is, if $A \subseteq S$ and $B \subseteq T$, then $f(A,B) = A \cup B$ is a well-defined subset of $S \cup T$.

The result of part (b) then shows that the range of $f$ is all of $\mathscr{P}(S \cup T)$. That is, if $Y \in \mathscr{P}(S \cup T)$, then there exist $A \in \mathscr{P}(S)$ and $B \in \mathscr{P}(T)$ such that $f(A,B) = Y$.
\begin{enumerate}
\item[(c)] Illustrate this line of thinking in the case where $S = \{1,2\}$ and $T = \{2,3\}$. The eight subsets of $S \cup T = \{1,2,3\}$ are
\[ \varnothing, \ \{1\}, \ \{2\}, \ \{3\}, \ \{1,2\}, \ \{1,3\}, \ \{2,3\}, \ \{1,2,3\}. \]
For each set $Y$ in this list, find $A \subseteq \{1,2\}$ and $B \subseteq \{2,3\}$ such that $A \cup B = Y$.

\item[(d)] Continuing with the example from part (c), for which of the sets $Y$ is there a \textbf{unique} choice of $A \subseteq \{1,2\}$ and $B \subseteq \{2,3\}$ such that $A \cup B = Y$?

\end{enumerate}
\end{enumerate}

\newpage
\vspace{.05 in}

\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 35]}} Let $A$, $B$, $C$, and $D$ be sets. Suppose that $A \times B = C \times D \neq \varnothing$. Prove that $A = C$ and $B = D$.

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

%\item \textbf{{[Falkner Section 11 Exercise 7]}} Let $A$ and $B$ be sets, and let $\pi_A$ and $\pi_B$ be the projections from $A \times B$ to $A$ and $B$ respectively. That is,
%\begin{align*}
%\pi_A \colon A \times B &\to A  &\pi_B \colon A \times B &\to B \\
%(x,y) &\mapsto x. &(x,y) &\mapsto y.
%\end{align*}
%Show that if $B \neq \varnothing$, then $\Rng(\pi_A) = A$, and that if $A \neq \varnothing$, then $\Rng(\pi_B) = B$.
\end{enumerate}




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