Reflexive Relation is Serial

Theorem
Every reflexive relation is also a serial relation.

Proof
Let $$\mathcal{R} \subseteq S \times S$$ be a relation in $S$.

We have that $$\mathcal{R}$$ is serial iff:
 * $$\forall x \in S: \exists y \in S: \left({x, y}\right) \in \mathcal{R}$$

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

We have that $$\mathcal{R}$$ is reflexive iff:
 * $$\forall x \in S: \left({x, x}\right) \in \mathcal{R}$$

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