Is an anti-symmetric Relation always Transitive?

Question

By the definition of anti-symmetry, there exist no two pairs $(a,b)$ and $(b,a)$ where $a \neq b$.

The pairs $ (a,b)$ and $(b,a)$ exists if and only if $a = b$.

Both the two cases fulfill the need of a transitive relation.

Answer 1

No, not at all. Antisymmetry is a restriction only; it doesn't force other relations in the way that transitivity does.

A relation that is both anti-symmetric and transitive would need to avoid cyclic relationships. You couldn't have $(a,b),(b,c),(c,d),(d,a)$ for example, since transitivity would give $(b,c),(c,d)\implies (b,d)$ and $(b,d),(d,a)\implies (b,a)$ which with $(a,b)$ breaks the antisymmetry.