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

That is, every element relates to at least one element.

We have that $\RR$ is reflexive :
 * $\forall x \in S: \tuple {x, x} \in \RR$

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