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 the Fundamental Theorem on Equivalence Relations, 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.