# Diagonal Relation is Serial

## Theorem

Let $S$ be a set.

Let $\Delta_S$ be the diagonal relation on $S$.

Then $\Delta_S$ is a serial relation.

## Proof

By Diagonal Relation is Equivalence it follows a fortiori that $\Delta_S$ is reflexive.

The result follows from Reflexive Relation is Serial.

$\blacksquare$