Serial Relation is not Null
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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 $2 \ \text{(b)}$