# Symmetric Preordering is Equivalence Relation

## Theorem

Let $\RR \subseteq S \times S$ be a preordering on a set $S$.

Let $\RR$ also be symmetric.

Then $\RR$ is an equivalence relation on $S$.

## Proof

By definition, a preordering on $S$ is a relation on $S$ which is:

$(1): \quad$ reflexive

and:

$(2): \quad$ transitive.

Thus $\RR$ is a relation on $S$ which is reflexive, transitive and symmetric.

Thus by definition $\RR$ is an equivalence relation on $S$.

$\blacksquare$