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\chapter[Equivalence relations and partitions.]{Prove the following correspondence between equivalence relations and partitions.}

Let $A$ be a set.
\begin{enumerate}
\item Prove that if $R$ is an equivalence relation on $A$, then $\{[a] : a \in A\}$ is a partition of $A$.
\item Prove that if $P$ is a partition of $A$, then $R = \{(x,y) : x,y \in X \text{ and } X \in P\}$ is an equivalence relation on $A$.
\end{enumerate}

\section{First draft}
Due Wednesday, April 20 at 6:00 PM.
\begin{proof}

\end{proof}