20 lines
900 B
TeX
20 lines
900 B
TeX
\section{Equivalence Relations}
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\begin{definition}[Equivalence Relation]
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An \textit{equivalence relation} on a set $S$ is a set $R$ of ordered pairs of elements of $S$ such that
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\begin{enumerate}
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\item $(a, a) \in R$ for all $a \in S$ (reflexive property).
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\item $(a, b) \in R$ implies $(b, a) \in R$ (symmetric property).
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\item $(a, b) \in R$ and $(b, c) \in R$ imply $(a, c) \in R$ (transitive property).
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\end{enumerate}
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\end{definition}
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\begin{definition}[Partition]
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A \textit{partition} of a set $S$ is a collection of nonempty disjoint subsets of $S$ whose union is $S$.
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\end{definition}
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\begin{theorem}[Equivalence Classes Partition]
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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 $P$.
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\end{theorem}
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