Definition:Equivalence Relation Induced by Preordering

Definition
Let $\struct {S, \RR}$ be a relational structure such that $\RR$ is a preordering.

Let a relation $\sim_\RR$ be defined on $S$ by:
 * $x \sim_\RR y$ $x \mathrel \RR y$ and $y \mathrel \RR x$.

Then $\sim_\RR$ is known as the equivalence (relation) induced by $\RR$.

Also see

 * Preordering induces Equivalence Relation for a proof that $\sim_\RR$ is indeed an equivalence relation.