Relation Induced by Mapping is Equivalence Relation

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

Let $$\mathcal{R}_f \subseteq S \times S$$ be the relation induced by $f$:
 * $$\left({s_1, s_2}\right) \in \mathcal{R}_f \iff f \left({s_1}\right) = f \left({s_2}\right)$$

Then $$\mathcal{R}_f$$ is an equivalence relation.

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

Checking in turn each of the criteria for equivalence:

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


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

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

$$ $$ $$

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

$$ $$ $$

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