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Question

# Explain the relationship between equivalence relations on a set and partitions of this set.

Solution

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A relation $R$ is an $\textbf{equivalence relation}$ if the relation $R$ is transitive, symmetric and reflexive.

A $\textbf{partition}$ of a set $S$ is a collection of $\textit{disjoint subsets}$ whose union is equal to the set $S$.

The collection of ordered pairs produced by a partition will form an equivalence relation (thus if $a$ and $b$ are elements of the same set in the partition, then $(a,b)$ is in the equivalence relation).

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