Definition:Equivalence Relation

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


 * $(1): \quad$ reflexive
 * $(2): \quad$ symmetric
 * $(3): \quad$ transitive

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

Also denoted as
When discussing equivalence relations, various notations are used for $\left({x, y}\right) \in \mathcal R$.

Examples are:
 * $x \equiv y \left({\mathcal R}\right)$
 * $x \equiv y \mod \mathcal R$
 * $x \sim y$

and so on.

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

Also see

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


 * Relation Partitions Set iff Equivalence