Trivial Relation is Universally Congruent

Theorem
The trivial relation $\RR = S \times S$ on a set $S$ is universally congruent on $S$.

Proof
Let $\struct {S, \circ}$ be any algebraic structure which is closed for $\circ$.

By definition of trivial relation:
 * $x \in S \land y \in S \implies x \mathrel \RR y$

So: