Prove or disprove if R is antisymmetric then R is asymmetric

Prove or disprove: if R is antisymmetric, then R is asymmetric.

Solution

An asymmetric relation must be antisymmetric, since the hypothesis of the condition for antisymmetry is false if the relation is asymmetric. The relation {(a,a)} on {a} is antisymmetric but not asymmetric, however, so the answer to the second question is no. In fact, it is easy to see that R is asymmetric if and only if R is antisymmetric and irreflexive.