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\begin{document}

\begin{center}
{\large{\textsc{Homework 23}}} \\
\textsc{Math 3345 -- Autumn 2022 -- Kutler}
\end{center}

\thispagestyle{empty}

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 15(b) -- modified]}} Let $S$ and $T$ be sets. Prove that $S$ and $T$ are disjoint (i.e., $S \cap T = \varnothing)$ if and only if the following condition holds:
\hspace*{2em}
\begin{minipage}{.75\textwidth}
\vspace{6pt}
For all subsets $A_1, A_2 \subseteq S$ and $B_1, B_2 \subseteq T$, if $A_1 \cup B_1 = A_2 \cup B_2$, then $A_1 = A_2$ and $B_1 = B_2$.
\end{minipage}
\hfill $(\star)$

\vspace{6pt}
\textbf{Note:} The condition $(\star)$ is equivalent to the statement 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 injective. You proved on Homework 22 that this function is always surjective.

\item \textbf{{[Falkner Section 11 Exercise 17 -- modified]}} Let
\begin{align*}
f \colon [1,\infty) &\to \RR \\
x &\mapsto x-1.
\end{align*}
\begin{enumerate}
\item Show that $\Rng(f) \subseteq [0,\infty)$. That is, $f(x) \in [0,\infty)$ for every $x \in [1,\infty)$.
\item Prove that $\Rng(f) = [0,\infty)$. [\textsc{hint:} In light of part (a), you need only prove the other inclusion, $[0,\infty) \subseteq \Rng(f)$. That is, for each $y \in [0,\infty)$, you must find some $x \in \Dom(f) = [1,\infty)$ such that $f(x) = y$.] 
\item Prove that $f$ is an injection.
\item Conclude that $f$ is a bijection from $[1,\infty)$ to $[0,\infty)$, and give a formula for the inverse function $f^{-1} \colon [0,\infty) \to [1,\infty)$.
\item Sketch the graph of $f$.
\end{enumerate}

\item \textbf{{[Falkner Section 11 Exercise 26]}} Let $A$, $B$, and $C$ be sets. Prove that if $f \colon A \to B$ and $g \colon B \to C$ are bijections, then $g \circ f \colon A \to C$ is a bijection.





\item \textbf{{[Falkner Section 15 Exercise 1 -- modified]}} Show that the intervals $A = [1,\infty)$ and $B = (1,\infty)$ have the same cardinality by giving an example of a bijection $f \colon A \to B$.

[\textsc{hint:} Use one simple formula to define $f$ on $\NN$ and a different, even simpler formula to define $f$ on $A \setminus \NN$.]

Be sure to prove that $f$ is a bijection.
\end{enumerate}

\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 11 Exercise 20 -- modified]}} Let
\begin{align*}
g \colon [0,1) &\to \RR  &h \colon (-1,0) &\to \RR \\
x &\mapsto \frac{x}{1-x}. &x &\mapsto \frac{x}{1+x}.
\end{align*}
\begin{enumerate}
\item Prove that $\Rng(g) = [0,\infty)$ and $\Rng(h) = (-\infty,0)$.
\item Prove that both $g$ and $h$ are injections.
\item Conclude that $g$ is a bijection from $[0,1)$ to $[0,\infty)$ and that $h$ is a bijection from $(-1,0)$ to $(-\infty,0)$.
\item Find formulas for $g^{-1} \colon [0,\infty) \to [0,1)$ and $h^{-1} \colon (-\infty, 0) \to (-1,0)$.
\end{enumerate}

\item \textbf{{[Falkner Section 11 Exercise 23]}} Let
\begin{align*}
\varphi \colon (-1,1) &\to \RR \\
x &\mapsto \frac{x}{1-|x|}.
\end{align*}
\begin{enumerate}
\item Show that $\varphi$ is a bijection from $(-1,1)$ to $\RR$.
\item Find a formula for $\varphi^{-1} \colon \RR \to (-1,1)$.
\end{enumerate}
[\textsc{hint:} Use Practice Problem 1 above.]

\item \textbf{{[Falkner Section 15 Exercises 6 \& 7 -- modified]}}
\begin{enumerate}
\item Show that the intervals $[0,1)$ and $(0,1]$ have the same cardinality by giving an example of a bijection $f \colon [0,1) \to (0,1]$.
\item Show that the intervals $(0,1]$ and $(0,1)$ have the same cardinality by giving an example of a bijection $g \colon (0,1] \to (0,1)$.
\item Show that the intervals $[0,1]$ and $[0,1)$ have the same cardinality by giving an example of a bijection $h \colon [0,1] \to [0,1)$.
\item Conclude that the four intervals $[0,1]$, $[0,1)$, $(0,1]$, and $(0,1)$ all have the same cardinality.
\item Use the functions $f$, $g$, and $h$ to construct a bijection from $[0,1]$ to $(0,1)$. [\textsc{hint:} Use Exercise 3 above.]
\end{enumerate}

%\item Let
%\[ A = \{ n \in \NN \mid \, n \equiv 3 \!\!\! \mod 4 \}. \]
%Define a bijection
%\[ f \colon \NN \to A \]
%and prove that it is a bijection.

\end{enumerate}




\end{document}
