Definition:Equivalence Relation

Definition
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, various symbols are used for $$\left({x, y}\right) \in \mathcal R$$. Examples are: and so on.
 * $$x \equiv y \left({\mathcal R}\right)$$
 * $$x \sim y$$

Specialised equivalence relations generally have their own symbols, which can be defined as they are needed.

Also see

 * Equivalence Class
 * Quotient Set
 * Quotient Mapping, also known as the Canonical Surjection


 * Relation Partitions a Set iff Equivalence