Definition:Antisymmetric Relation/Definition 1

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

that is:
 * $\set {\tuple {x, y}, \tuple {y, x} } \subseteq \mathcal R \implies x = y$

Also known as

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

Also see

 * Equivalence of Definitions of Antisymmetric Relation