Transitive and Antitransitive Relation is Asymmetric

Theorem

Let $S$ be a set.

Let $\RR \subseteq S \times S$ be a relation in $S$.

Let $\RR$ be both transitive and antitransitive.

Then $\RR$ is asymmetric.

Proof

Let $\tuple {x, y} \in \RR$ for some $x, y \in S$.

Then as $\RR$ is antitransitive:

$\tuple {x, x} \notin \RR$

and so as $\RR$ is transitive and $\tuple {x, x} \notin \RR$:

$\tuple {y, x} \notin \RR$

That is, $\RR$ is asymmetric.

$\blacksquare$

Sources

• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $5$ Properties of Relations: Exercise $7 \ \text{(b)}$