Definition:Antisymmetric Relation

Definition
Let $\mathcal R \subseteq S \times S$ be a relation in $S$. $\mathcal R$ is antisymmetric iff:
 * $\left({x, y}\right) \in \mathcal R \land \left({y, x}\right) \in \mathcal R \implies x = y$

that is:
 * $\left\{{\left({x, y}\right), \left({y, x}\right)}\right\} \subseteq \mathcal R \implies x = y$

Also see

 * Symmetry of Relations


 * Symmetric Relation
 * Asymmetric Relation
 * Non-symmetric Relation