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\title{Math 2001 \\ Quiz 11}
\date{\today}

\begin{document}
\maketitle

\begin{enumerate}
\item (1 pt) Write your name in the top right corner of the page.

\item (3 pts each) Complete each of the following definitions. (If you don't like how my sentences start, feel free to write your own, but your statement must be complete (define all your variables etc.).
\begin{enumerate}
\ifthenelse{\boolean{print_solutions}}{}{\setlength{\itemsep}{1in}}

\item Let $A$ be a set, then $R$ is a \emph{relation} on $A$ if

\begin{solutions}
$R \subseteq A \times A$.
\end{solutions}

\item Let $R$ be a relation on $A$, then $R$ is \emph{transitive} if

\begin{solutions}
whenever $(a,b), (b,c) \in R$, then $(a,c) \in R$.
\end{solutions}

\item If $R$ is an equivalence relation on $A$ and $a \in A$, then the \emph{equivalence class} of $a$ is

\begin{align*}
[a] = \ifthenelse{\boolean{print_solutions}}{\color{blue}\{b \in A : (a,b) \in R\}}{\hspace{2in}}
\end{align*}

\vspace{10mm}
(Set-builder notation is sufficient.)
\end{enumerate}

\item (3 pts each) Let $A = \{4,9,11\}$ and let $R$ be the relation on $A$ defined by 
\begin{align*}
R = \{(a,b) : 3 \mid (a-b) \text{ or } 3 \mid (a+b)\}. 
\end{align*}
\begin{enumerate}
\ifthenelse{\boolean{print_solutions}}{}{\setlength{\itemsep}{15mm}}
\item Write out the relation $R$ explicitly (list all of the elements of $R$ in set notation).

\begin{solutions}
$R = \{(4,4), (9,9), (11,11), (4,11), (11,4)\}$
\end{solutions}

\item Is $R$ an equivalence relation? If \textbf{yes}, write out all of the equivalence classes of $R$. If \textbf{no}, give a brief explanation of why not (e.g. give a counter example).

\begin{solutions}
Yes. 
\begin{align*}
[4] &= [11] = \{4,11\}\\
[9] &= \{9\}
\end{align*}
\end{solutions}
\end{enumerate}
\end{enumerate}


\end{document}