Antisymmetric and irreflexive relation which is not asymmetric

Question

Can anyone give me a counterexample for a relation $R\subset M\times M$ for the statement $$R\text{ antisymmetric} \wedge R\text{ not reflexive}\implies R\text{ asymmetric}$$

Answer 1

No, because a relation is asymmetric if and only if it is antisymmetric and not reflexive.

To see that your implication is always true, we could check the contrapositive statement: If R is symmetric then R is not antisymmetric or R is reflexive. This is easily seen to be true since if R is symmetric and anti-symmetric, it is a sub-relation of the equality relation, in which case it is obviously reflexive.