Serial Relation is not Null

Theorem

Let $S$ be a set such that $S \ne \O$.

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


Then $\RR$ is not a null relation.


Proof

As $S$ is non-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.

$\blacksquare$


Sources

  • 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $5$ Properties of Relations: Exercise $2 \ \text{(b)}$
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