Definition:Antisymmetric Relation

Definition
Let $\mathcal R \subseteq S \times S$ be a relation in $S$. $\mathcal R$ is antisymmetric iff:


 * $\left({x, y}\right) \in \mathcal R \land \left({y, x}\right) \in \mathcal R \implies x = y$

Note the difference between:
 * asymmetric relation, in which the fact that $\left({x, y}\right) \in \mathcal R$ means that $\left({y, x}\right)$ is definitely not in $\mathcal R$

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

Also see

 * Symmetry


 * Symmetric Relation
 * Asymmetric Relation
 * Non-symmetric Relation