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 induced by (the mapping) $$f$$;
 * the (equivalence) relation defined by (the mapping) $$f$$;
 * the (equivalence) relation associated with (the mapping) $$f$$.