Definition:Equivalence Relation Induced by Mapping

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

Then $$f$$ induces an equivalence $$\mathcal{R}_f$$ on its domain:

$$\left({s_1, s_2}\right) \in \mathcal{R}_f \iff f \left({s_1}\right) = f \left({s_2}\right)$$

$$\mathcal{R}_f$$ is known as the (equivalence) relation induced by $$f$$, or the relation defined by $$f$$.

Proof
We need to show that $$\mathcal{R}_f$$ is an equivalence.


 * $$\mathcal{R}_f$$ is reflexive:

$$\forall x \in S: f \left({x}\right) = f \left({x}\right) \Longrightarrow x \mathcal{R}_f x$$


 * $$\mathcal{R}_f$$ is symmetric:

$$ $$ $$


 * $$\mathcal{R}_f$$ is transitive:

$$ $$ $$

Thus $$\mathcal{R}_f$$ is reflexive, symmetric and transitive, and is therefore an equivalence relation.