# Trivial Relation is Universally Congruent

## Theorem

The trivial relation $\mathcal R = S \times S$ on a set $S$ is universally congruent with every closed operation on $S$.

## Proof

Let $\left({S, \circ}\right)$ be any algebraic structure which is closed for $\circ$.

By definition of trivial relation, $x \in S \land y \in S \implies x \mathcal R y$. So:

 $\ds x_1, x_2, y_1, y_2$ $\in$ $\ds S$ $\ds \implies \ \$ $\ds x_1 \circ y_1, x_2 \circ y_2$ $\in$ $\ds S$ Definition of closed algebraic structure $\ds \implies \ \$ $\ds \left({x_1 \circ y_1}\right)$ $\mathcal R$ $\ds \left({x_2 \circ y_2}\right)$ Definition of Trivial Relation

$\blacksquare$