According to the definition of an Anti-Symmetric Relation if xRy and yRx then x = y
Which means, effectively, x is in relation with itself. Does this mean that anti-symmetry implies reflexive property as well?
Set Theory – Is an Anti-Symmetric Relation Also Reflexive?
elementary-set-theoryrelations
Best Answer
Your definition is wrong. The relation $R$ is antisymmetric if, whenever $x\mathrel{R}y$ and $y\mathrel R x$ it holds that $x=y$.
An example of a relation that is antisymmetric but not reflexive is $>$ on the set of integers.