Created the Abstract Algebra theorems and definitions cheat sheet

part-1/chapters/chapter-0/equivalence-relations.tex

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+\section{Equivalence Relations}
+
+\begin{definition}[Equivalence Relation]
+	An \textit{equivalence relation} on a set $S$ is a set $R$ of ordered pairs of elements of $S$ such that
+
+	\begin{enumerate}
+		\item $(a, a) \in R$ for all $a \in S$  (reflexive property).
+		\item $(a, b) \in R$ implies $(b, a) \in R$  (symmetric property).
+		\item $(a, b) \in R$ and $(b, c) \in R$ imply $(a, c) \in R$  (transitive property).
+	\end{enumerate}
+\end{definition}
+
+\begin{definition}[Partition]
+	A \textit{partition} of a set $S$ is a collection of nonempty disjoint subsets of $S$ whose union is $S$.
+\end{definition}
+
+\begin{theorem}[Equivalence Classes Partition]
+	The equivalence classes of an equivalence relation on a set $S$ constitute a partition of $S$. Conversely, for any partition $P$ of $S$, there is an equivalence relation on $S$ whose equivalence classes are the elements of $P$.
+\end{theorem}