\begin{definition}[Definition:Antisymmetric Relation/Definition 1]
Let $S$ be a set.
Let $\RR \subseteq S \times S$ be a relation in $S$.
$\RR$ is '''antisymmetric''' {{iff}}:
:$\tuple {x, y} \in \RR \land \tuple {y, x} \in \RR \implies x = y$
that is:
:$\set {\tuple {x, y}, \tuple {y, x} } \subseteq \RR \implies x = y$
\end{definition}