Definition:Equivalence Relation Induced by Mapping

Definition
Let $f: S \to T$ be a mapping.

The equivalence induced by $f$, $\mathcal R_f \subseteq S \times S$, is the relation defined as:
 * $\left({s_1, s_2}\right) \in \mathcal R_f \iff f \left({s_1}\right) = f \left({s_2}\right)$

The relation $\mathcal R_f$ is an equivalence relation, as demonstrated on Induced Equivalence is Equivalence Relation.

Also known as
The equivalence induced by $f$ is variously known as:


 * the (equivalence) relation (on $S$) induced by (the mapping) $f$
 * the (equivalence) relation (on $S$) defined by (the mapping) $f$
 * the (equivalence) relation (on $S$) associated with (the mapping) $f$
 * the equivalence kernel of $f$.

Also see

 * Induced Equivalence is Equivalence Relation for a demonstration that $\mathcal R_f$ is indeed an equivalence relation.