Definition:Equivalence Relation/Definition 2

Definition
Let $\mathcal R \subseteq S \times S$ be a relation on a set $S$.

$\mathcal R$ is an equivalence relation :


 * $\Delta_S \cup \mathcal R^{-1} \cup \mathcal R \circ \mathcal R \subseteq \mathcal R$

where:
 * $\Delta_S$ denotes the diagonal relation on $S$
 * $\mathcal R^{-1}$ denotes the inverse relation
 * $\circ$ denotes composition of relations

Also known as
An equivalence relation is frequently referred to just as an equivalence.

Also see

 * Equivalence of Definitions of Equivalence Relation