\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}

\usepackage[margin=1in]{geometry}
\usepackage{enumitem}
\usepackage{amsmath, amssymb, amsthm}
\usepackage{xcolor}
\usepackage{verbatim}

% THEOREM ENVIRONMENTS
\theoremstyle{definition}
\newtheorem*{theorem}{{Theorem}}
\newtheorem*{lemma}{{Lemma}}
\newtheorem*{proposition}{{Proposition}}
\newtheorem*{definition}{{Definition}}
\newtheorem*{corollary}{{Corollary}}
\newtheorem*{example}{{Example}}
\newtheorem*{remark}{{Remark}}
\newtheorem*{claim}{{Claim}}
\newtheorem*{question}{{Question}}

\theoremstyle{plain}
\newtheorem*{reminder}{{\textit{Reminder}}}
\newtheorem*{hint}{{\textit{Hint}}}

% ENVIRONMENTS FOR CITING BOOK STATEMENTS
\newcommand{\prop}[1]{\paragraph{Proposition #1}}
\newcommand{\axm}[1]{\paragraph{Axiom #1}}
\newcommand{\thm}[1]{\paragraph{Theorem #1}}


% COMMANDS
\newcommand{\solution}{{\color{blue} \textsc{\ \\ Solution.\ \ }}}
\newcommand{\revisedsolution}{{\color{blue} \textsc{\ \\ Revised Solution.\ \ }}}
\newcommand{\Z}{\mathbb{Z}} % the integers
\newcommand{\N}{\mathbb{N}} % natural numbers
\newcommand{\divides}{\ \vert \ }
\newcommand{\st}{ \ : \ } % such that

\newcommand{\exheading}[1]{\section*{Exercise #1}}
\newcounter{exnum}
% \setcounter{exnum}{1} % default 0 start
\newcommand{\exercise}{
    \stepcounter{exnum} 
    \exheading{\theexnum}
    }



%%%%
\title{MAT 108: Problem Set 4}
\author{(ADD NAME)}
\date{Due 2/7/23 at 11:59 pm on Canvas}

\begin{document}

\maketitle


\paragraph{Reminders:}
\begin{itemize}
%\item Put your name at the top!
\item Your homework submission must be typed up in full sentences, with proper mathematical formatting. Handwritten homework submissions will receive a score of 0. Solutions containing incomplete sentences or poor formatting will lose points.
\item You will receive feedback on PS3 by next Tuesday, 2/7. PS3 revisions are due Friday, 2/10 at 11:59 pm. \textbf{New policy:} Keep your old solution for PS3. Underneath your old solution, type 
    \begin{verbatim}
        \paragraph{Revised solution.}
    \end{verbatim}
and then type your revised solution. This will help make the re-grading process go more smoothly. For future problem sets (such as PS4 revisions), you can just use the command 
    \begin{verbatim}
        \revisedsolution
    \end{verbatim}
to indicate the start of your revised solution.
\end{itemize}









%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise 
Let $A,B,C,D$ be sets. Decide whether each of the following statements is true or false; in each case, either prove the statement or give a counterexample.

\begin{enumerate}[label=(\alph*)]
    \item $A - (B \cup C) = (A - B) \cup (A-C)$
    \item $(A \times B) \cup (C \times D) = (A \cup C) \times (B\cup D)$
    \item $(A \times B) \cap (C \times D) = (A \cap C) \times (B\cap D)$
\end{enumerate}

\solution


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise 

For each of the following relations defined on $\Z$, determine whether it is an equivalence relation. If it is, determine the equivalence classes. 

\begin{enumerate}[label=(\alph*)]
    \item $x \sim y$ if $x \neq y$
    \item $x \sim y$ if $xy > 0$
    \item $x \sim y$ if $x \divides y$ or $y \divides x$
\end{enumerate}

\solution






\end{document}
