Relation Induced by Mapping is Equivalence Relation

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

Let $\RR_f \subseteq S \times S$ be the relation induced by $f$:
 * $\tuple {s_1, s_2} \in \RR_f \iff \map f {s_1} = \map f {s_2}$

Then $\RR_f$ is an equivalence relation.

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

Checking in turn each of the criteria for equivalence:

Reflexive
$\RR_f$ is reflexive:


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

Symmetric
$\RR_f$ is symmetric:

Transitive
$\RR_f$ is transitive:

Thus $\RR_f$ is reflexive, symmetric and transitive, and is therefore an equivalence relation.