Definition:Equivalence Relation Induced by Mapping

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

Let $\mathcal R_f \subseteq S \times S$ be 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.

It is 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) $f$.

Also see

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