Transitive Relation is Antireflexive iff Asymmetric
Let $\mathcal R$ be transitive.
Let $\mathcal R \subseteq S \times S$ be antireflexive.
Then by Antireflexive and Transitive Relation is Asymmetric it follows that $\mathcal R$ is asymmetric.
Let $\mathcal R$ be asymmetric.
- Null Relation is Antireflexive, Symmetric and Transitive for the case where $\mathcal R = \varnothing$.