Definition:Antisymmetric Relation/Definition 1

Definition
Let $S$ be a set.

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

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

Also see

 * Equivalence of Definitions of Antisymmetric Relation