# Antisymmetric Relation **A [[relation]] that does not relate a pair of distinct elements to each other.** > [!Example] Antisymmetric Relation > An *antisymmetric relation* is a binary relation $R$ on a [[set]] $X$ such that for every $x, y\in X$, if $xRy$ and $yRx$, then $x=y$. > > That is, $(x,x)\in R$ and either $(x,y)$ or $(y,x)\in R$, but not both. > > $\forall\;x,y\in X:(xRy\wedge yRx)\implies x=y$ > > > [!Visual definitions]- > > In a directed graph, a relation is antisymmetric if and only if *there are no double arrows*. > > > > ![[AntisymmetricRelationGraph.svg|400]] > > *** > > In a Boolean matrix, a relation is antisymmetric it *does not* have *reflection symmetry* along its major diagonal except for diagonal entries. > > > > ![[AntisymmetricRelationBooleanMatrix.svg]] > > *The left relation is antisymmetric; arrows point to corresponding entries which if both were true would make the relation not antisymmetric.* > > An antisymmetric relation can also be [[Symmetric Relation|symmetric]] if its ordered pairs contain only non-distinct elements - these properties are not mutually exclusive. > > If a relation is both symmetric and antisymmetric, then it must be [[Reflexive Relation|reflexive]].