Reflexive Relation is Serial

Theorem

Every reflexive relation is also a serial relation.

Proof

Let $\RR \subseteq S \times S$ be a relation in $S$.

We have that $\RR$ is serial if and only if:

$\forall x \in S: \exists y \in S: \tuple {x, y} \in \RR$

That is, if and only if every element relates to at least one element.

We have that $\RR$ is reflexive if and only if:

$\forall x \in S: \tuple {x, x} \in \RR$

Hence if $\RR$ is reflexive, every $x$ is related to itself, thereby fulfilling the criterion for $\RR$ to be serial.

$\blacksquare$