\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}

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

\begin{document}

\begin{flushright}
Your Name Here\\
Math 220\\
HW \# 7\\
\today
\end{flushright}

%Please write your solutions in the designated solution area!

\begin{ex}
A sequence $\{a_n\}$ is defined recursively by $a_1=1$, $a_2=2$, and $a_n=a_{n-1}+2a_{n-2}$ for $n\geq3$. Conjecture a formula for $a_n$ and verify that your conjecture is correct.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
A sequence $\{a_n\}$ is defined recursively by $a_1=1$, $a_2=4$, $a_3=9$, and $a_n=a_{n-1}-a_{n-2}+a_{n-3}+2(2n-3)$ for $n\geq4$. Conjecture a formula for $a_n$ and verify that your conjecture is correct.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Answer the following questions:
\begin{enumerate}[$(a)$]
\item Is $3\in\{1,2,3\}$?
\item Is $1\subset \{1\}$?
\item Is $\{2\}\in\{1,2\}$?
\item Is $\{3\}\in\{1,\{2\},\{3\}\}$?
\item Is $1\in\{1\}$?
\item Is $\{2\}\subset\{1,\{2\},\{3\}\}$?
\item Is $\{1\}\subset\{1,2\}$?
\item Is $1\in\{\{1\},2\}$?
\item Is $\{1\}\subset\{1,\{2\}\}$?
\item Is $\{1\}\subset\{1\}$?
\end{enumerate}
\end{ex}

\begin{proof}[Solution]

\end{proof}

\begin{ex}
Write each of the following sets by listing its elements within braces.
\begin{enumerate}[$(a)$]
\item $A=\{n\in\mathbb{Z}|-4<n\leq 4\}$
\item $B=\{n\in\mathbb{N}|n^3<100\}$
\item $C=\{x\in\mathbb{R}|x^2-x=0\}$
\item $D=\{4n|n\in\mathbb{Z}\}$
\end{enumerate}
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Let $U=\{1,2,3,4,5,6,7,8,9\}$ be the universal set and let $A=\{3,6,9\}$, $B=\{3,4,5,6,7\}$, and $C=\{4,5,7\}$. Find the following:
\begin{itemize}
\item $A\cup B$
\item $A\cap B$
\item $A\cap C$
\item $B^c$
\item $B\setminus C$
\item $U\cup A$
\item $B\cap U$
\end{itemize}
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Let $A=\{a,b,c\}$, $B=\{b,c,d\}$, and $C=\{b,c,e\}$.
\begin{enumerate}[$(a)$]
\item Find $A\cup(B\cap C)$, $(A\cup B)\cap C$, and $(A\cup B)\cap(A\cup C)$. Which of these sets are equal?
\item Find $A\cap(B\cup C)$, $(A\cap B)\cup C$, and $(A\cap B)\cup(A\cap C)$. Which of these sets are equal?
\item Find $A\setminus(B\setminus C)$ and $(A\setminus B)\setminus C$. Are these sets are equal?
\end{enumerate}
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Answer the following questions and give explanations for your answers.
\begin{enumerate}[$(a)$]
\item Is $\{\{a,d,e\},\{b,c\},\{d,f\}\}$ a partition of $\{a,b,c,d,e,f\}$?
\item Is $\{\{w,x,v\},\{u,y,q\},\{p,z\}\}$ a partition of $\{p,q,u,v,w,x,y,z\}$?
\item Is $\{\{3,7,8\},\{2,9\},\{1,4,5\}\}$ a partition of $\{1,2,3,4,5,6,7,8,9\}$?
\end{enumerate}
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Let $\mathbb{Z}$ be the set of all integers and let
\begin{eqnarray*}
A_0 &=& \{n\in\mathbb{Z}|n=4k, \mbox{for some integer }k\},\\
A_1 &=& \{n\in\mathbb{Z}|n=4k+1, \mbox{for some integer }k\},\\
A_2 &=& \{n\in\mathbb{Z}|n=4k+2, \mbox{for some integer }k\},\mbox{ and}\\
A_3 &=& \{n\in\mathbb{Z}|n=4k+3, \mbox{for some integer }k\}.
\end{eqnarray*}
Is $\{A_0,A_1,A_2,A_3\}$ a partition of $\mathbb{Z}$? Explain your answer.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Let $A=\{1,2,3,4\}$ and $B=\{a,c\}$. Compute the following:
\begin{enumerate}[$(a)$]
\item $A\times B$
\item $B\times A$
\item $A\times A$
\item $B\times B$
\item $\varnothing\times B$
\item $(A\times B)\times B$
\item $A\times(B\times B)$
\end{enumerate}
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Compute the following power sets. We will denote the power set of $A$ by $\PS(A)$. %\PS is not a native TeX/LaTeX command. I have defined it above for simplicity.
\begin{enumerate}[$(a)$]
\item $\PS(\{1,2,3,4\})$
\item $\PS(\varnothing)$
\item $\PS(\{\varnothing\})$
\end{enumerate}
\end{ex}

\begin{proof}

\end{proof}

\end{document}