\documentclass[12pt,reqno]{amsart} 

\usepackage{amsmath,amssymb,amsfonts,amsthm,mathrsfs}
\usepackage{graphicx}
\usepackage{enumerate}
\usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry}
\usepackage{circuitikz}

\renewcommand{\baselinestretch}{1.25}

\newtheorem{thm}{Theorem}
\newtheorem{ex}{Exercise}
\newtheorem*{defn}{Definition}

\newcommand{\PS}{\mathscr{P}}

\begin{document}

\begin{flushright}
Your Name Here\\
Math 220\\
HW \# 9\\
\today
\end{flushright}

%Please write your solutions in the designated solution area!

\begin{ex}
A function $f:\mathbb{Z}\to\mathbb{Z}$ is defined by $f(n)=2n+1$. Determine whether $f$ is (a) injective, (b) surjective. Give proof or a counterexample for your claims.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
A function $f:\mathbb{Z}\to\mathbb{Z}$ is defined by $f(n)=n-3$. Determine whether $f$ is (a) injective, (b) surjective. Give proof or a counterexample for your claims.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
How many integers must you pick in order to be sure that at least two of them have the same remainder when divided by $15$?
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Let $X=\{1,2,3,4\}$ and consider the functions $f,g,h:X\to X$ given by
\begin{eqnarray*}
f&=&\{(1,2),(2,1),(3,1),(4,4)\}\\
g&=&\{(1,2),(2,2),(3,1),(4,3)\}\\
h&=&\{(1,1),(2,3),(3,1),(4,3)\}
\end{eqnarray*}
\begin{enumerate}[$(a)$]
\item Find the following compositions: $f\circ g$, $h\circ f$, and $g^2 = g\circ g$.
\item Find the compositions: $h\circ g\circ f$ and $f\circ g\circ h$.
\end{enumerate}
\end{ex}

\begin{proof}[Solution]

\end{proof}

\begin{ex}
Consider the functions $f,g:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}\times\mathbb{Z}$ given by $f(m,n)=(3m-4n,2m+n)$ and $g(m,n)=(5m+n,m)$. Find formulas for $f\circ g$ and $g\circ f$.
\end{ex}

\begin{proof}[Solution]

\end{proof}

\begin{ex}
Let $X,Y,Z$ be sets and suppose $f:X\to Y$ and $g:Y\to Z$. Suppose that $g\circ f:X\to Z$ is a bijection. Prove that $f$ is an injection and $g$ is a surjection.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Let $X$ and $Y$ be sets and let $A\subset X$. Is $A=f^{-1}(f(A))$? If so, give a proof, if not, give a counter example and prove whichever of $A\subset f^{-1}(f(A))$ or $A\supset f^{-1}(f(A))$ are true.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Prove that the function $f:\mathbb{R}\setminus\{2\}\to\mathbb{R}\setminus\{5\}$ given by $f(x)=\dfrac{5x+1}{x-2}$ is bijective. What is the inverse function $f^{-1}:\mathbb{R}\setminus\{5\}\to\mathbb{R}\setminus{2}$?
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Recall that for an integer $c\in\mathbb{Z}$, the set $c\mathbb{Z}=\{cn|n\in\mathbb{Z}\}$. Show that $3\mathbb{Z}$ and $5\mathbb{Z}$ have the same cardinality.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Prove that the rational numbers, $\mathbb{Q}$, are countable. (You may use the fact that the union of two countable sets is countable.)
\end{ex}

\begin{proof}

\end{proof}

\end{document}