# 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

- Null Relation is Antireflexive, Symmetric and Transitive for the case where $\mathcal R = \varnothing$.

## Sources

- 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): $\S 4.5$: Properties of Relations: Exercise $3$