Relation Partitions Set iff Equivalence

Theorem
Let $$\mathcal{R}$$ be a relation on a set $$S$$.

Then $$S$$ can be partitioned into subsets by $$\mathcal{R}$$ iff $$\mathcal{R}$$ is an equivalence relation on $$S$$.

The partition of $$S$$ defined by $$\mathcal{R}$$ is the quotient set $$S / \mathcal R$$.

Proof

 * Let $$\mathcal{R}$$ be an equivalence relation on $$S$$.

From Quotient Set forms a Partition, we have shown that the equivalence classes of $$\mathcal{R}$$ form a partition.


 * Let $$S$$ be partitioned into subsets by a relation $$\mathcal{R}$$.

From Equivalence Relation Defined by a Partition, $$\mathcal{R}$$ must be an equivalence.