# Relation Partitions Set iff Equivalence

## Theorem

Let $\RR$ be a relation on a set $S$.

Then $S$ can be partitioned into subsets by $\RR$ if and only if $\RR$ is an equivalence relation on $S$.

The partition of $S$ defined by $\RR$ is the quotient set $S / \RR$.

## Proof

Let $\RR$ be an equivalence relation on $S$.

From the Fundamental Theorem on Equivalence Relations, we have that the equivalence classes of $\RR$ form a partition.

$\Box$

Let $S$ be partitioned into subsets by a relation $\RR$.

From Relation Induced by Partition is Equivalence, $\RR$ is an equivalence relation.

$\blacksquare$