Transitive Relation is Antireflexive iff Asymmetric
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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
- Null Relation is Antireflexive, Symmetric and Transitive for the case where $\RR = \O$.
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $5$ Properties of Relations: Exercise $3$