Transitive Relation is Antireflexive iff Asymmetric

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Theorem

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

Let $\mathcal R$ be transitive.


Then $\mathcal R$ is antireflexive iff $\mathcal R$ is asymmetric.


Proof

Necessary Condition

Let $\mathcal R \subseteq S \times S$ be antireflexive.

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

$\Box$


Sufficient Condition

Let $\mathcal R$ be asymmetric.

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

$\blacksquare$


Also see


Sources