Serial Relation is not Null

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

Let $\mathcal R$ be a serial relation on $S$.

Then $\mathcal R$ is not a null relation.

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

As $\mathcal R$ be a serial relation on $S$:
 * $\exists y \in S: \left({x, y}\right) \in \mathcal R$

That is:
 * $\mathcal \ne \varnothing$

Hence the result by definition of null relation.