Definition:Symmetry (Relation)/Antisymmetric and Asymmetric

Antisymmetric and Asymmetric Relations
Note the difference between:
 * An asymmetric relation, in which the fact that $\tuple {x, y} \in \mathcal R$ means that $\tuple {y, x}$ is definitely not in $\mathcal R$

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