Definition:Antisymmetric Relation/Definition 1

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 known as

 * Some sources render this concept as anti-symmetric relation.

Also see

 * Equivalence of Definitions of Antisymmetric Relation