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

Then $\mathcal R_f$ is an equivalence relation.

Proof
We need to show that $\mathcal R_f$ is an equivalence relation.

Checking in turn each of the criteria for equivalence:

Reflexive
$\mathcal R_f$ is reflexive:


 * $\forall x \in S: \map f x = \map f x \implies x \mathrel {\mathcal R_f} x$

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.