Definition:Equivalence Relation

A relation on a set $$S$$ which is:


 * reflexive,
 * symmetric and
 * transitive

is called an equivalence relation, or an equivalence, on $$S$$.

When discussing equivalence relations, we also use the symbolism $$x \equiv y \left({\mathcal{R}}\right)$$ for $$\left({x, y}\right) \in \mathcal{R}$$.