# Definition:Equivalence Relation Induced by Mapping

## Definition

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

Let $\RR_f \subseteq S \times S$ be the relation defined as:

$\tuple {s_1, s_2} \in \RR_f \iff \map f {s_1} = \map f {s_2}$

Then $\RR_f$ 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 $f$.