Serial Relation is not Null

Theorem
Let $S$ be a set such that $S \ne \varnothing$.

Let $\RR$ be a serial relation on $S$.

Then $\RR$ is not a null relation.

Proof
As $S$ is not the empty set:
 * $\exists x: x \in S$

As $\RR$ be a serial relation on $S$:
 * $\exists y \in S: \tuple {x, y} \in \RR$

That is:
 * $\RR \ne \O$

Hence the result by definition of null relation.