Transitive Relation is Antireflexive iff Asymmetric

Theorem

Let $\RR \subseteq S \times S$ be a relation which is not null.

Let $\RR$ be transitive.


Then $\RR$ is antireflexive if and only if $\RR$ is asymmetric.


Proof

Necessary Condition

Let $\RR \subseteq S \times S$ be antireflexive.

Then by Antireflexive and Transitive Relation is Asymmetric it follows that $\RR$ is asymmetric.

$\Box$


Sufficient Condition

Let $\RR$ be asymmetric.

Then from Asymmetric Relation is Antireflexive it follows directly that $\RR$ is antireflexive.

$\blacksquare$


Also see


Sources

  • 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $5$ Properties of Relations: Exercise $3$
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