Antisymmetric:
if R(a,b) and R(b,a), then a = b, or, equivalently, if R(a,b) with a ≠ b, then R(b,a) must not hold.
If a relation is irreflexive as well as transitive then it should be asymmetric and antisymmetric
for example:
A={1,2,3}
R={(1,2)(2,3),(1,3)} is irreflexive as well as transitive
and it is asymmetric and antisymmetric
if a relation is asymmetric then it must be antisymmetric also