Definition:Symmetry (Relation)/Antisymmetric and Asymmetric

Antisymmetric and Asymmetric Relations

Note the difference between:

An asymmetric relation, in which the fact that $\tuple {x, y} \in \RR$ means that $\tuple {y, x}$ is definitely not in $\RR$

and:

An antisymmetric relation, in which there may be instances of both $\tuple {x, y} \in \RR$ and $\tuple {y, x} \in \RR$ but if there are, then it means that $x$ and $y$ have to be the same object.