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:
 * $\tuple {s_1, s_2} \in \mathcal R_f \iff f \paren {s_1} = f \paren {s_2}$

Then $\mathcal R_f$ is known as the equivalence (relation) induced by $f$.

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.