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\begin{document}
\begin{flushright}
Your Name Here\\
Math 220\\
HW \# 11\\
\today
\end{flushright}
%Please write your solutions in the designated solution area!
\begin{ex}
Let $A=\{0,1,2,3\}$ and let $R=\{(0,1),(0,2),(1,1),(1,3),(2,2),(3,0)\}$ be a relation on $A$. Find the transitive closure of $R$.
\end{ex}
\begin{proof}[Solution]
\end{proof}
\begin{ex}
Suppose $R$ and $S$ are binary relations on a set $A$.
\begin{enumerate}[$(a)$]
\item If $R$ and $S$ are reflexive, is $R\cap S$ reflexive? Why?
\item If $R$ and $S$ are symmetric, is $R\cap S$ symmetric? Why?
\item If $R$ and $S$ are transitive, is $R\cap S$ transitive? Why?
\end{enumerate}
\end{ex}
\begin{proof}
\end{proof}
\begin{ex}
Suppose $R$ and $S$ are binary relations on a set $A$.
\begin{enumerate}[$(a)$]
\item If $R$ and $S$ are reflexive, is $R\cup S$ reflexive? Why?
\item If $R$ and $S$ are symmetric, is $R\cup S$ symmetric? Why?
\item If $R$ and $S$ are transitive, is $R\cup S$ transitive? Why?
\end{enumerate}
\end{ex}
\begin{proof}
\end{proof}
\begin{ex}
Define the relation $S$ on $\mathbb{R}$ by $xSy$ if and only if $x-y$ is an integer. Show that $S$ is an equivalence relation.
\end{ex}
\begin{proof}
\end{proof}
\begin{ex}
Consider the following partition of the set $\{0,1,2,3,4\}$:
$$\mathcal{P}=\{\{0,2\},\{1\},\{3,4\}\}.$$
What is the relation $R$ that is induced by this partition? (Give $R$ as a set of ordered pairs.)
\end{ex}
\begin{proof}[Solution]
\end{proof}
\begin{ex}
Let $A=\{1,2,3,4,...,20\}$. An equivalence relation, $\sim$, is defined on $A$ by
\begin{center}
$x\sim y$ if and only if $4|(x-y)$.
\end{center}
Find the distinct equivalence classes of $\sim$.
\end{ex}
\begin{proof}[Solution]
\end{proof}
\begin{ex}
Let $\sim$ be the relation of congruence modulo $3$. Which of the following equivalence classes are equal?
$$[7],[-4],[-6],[17],[4],[27],[19].$$
Make sure to say why they are equal!
\end{ex}
\begin{proof}[Solution]
\end{proof}
\begin{ex}~
\begin{enumerate}[$(a)$]
\item Create an addition and multiplication table for $\mathbb{Z}_5$.
\item Create an addition and multiplication table for $\mathbb{Z}_6$.
\item In the multiplication tables, there is one big difference between $\mathbb{Z}_5$ and $\mathbb{Z}_6$, what is that difference?
\end{enumerate}
\end{ex}
\begin{proof}[Solution]
\end{proof}
\end{document}