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.