Symmetric and Antisymmetric Relation is Transitive

Theorem

Let $S$ be a set.

Let $\RR \subseteq S \times S$ be a relation in $S$ which is both symmetric and antisymmetric.

Then $\RR$ is transitive.

Proof

Let $\tuple {x, y}, \tuple {y, z} \in \RR$.

By Relation is Symmetric and Antisymmetric iff Coreflexive:

$x = y, y = z$

and so trivially:

$\tuple {x, z} = \tuple {x, x} \in \RR$

Thus $\RR$ is transitive.

$\blacksquare$

Sources

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