# Symmetric Relation **A [[relation]] that relates a pair of elements to each other.** > [!Example] Symmetric Relation > A *symmetric relation* is a binary relation $R$ on a [[set]] $X$ such that for every $x, y\in X$, if $xRy$, then $yRx$. That is, both $(x,y)$ and $(y,x) \in R$. > $\forall\; x, y\in X:xRy\implies yRx$ > > > [!Visual definitions]- > > In a directed graph, a relation is symmetric if and only if *every non-loop edge is a double arrow*. > > > > ![[SymmetricRelationGraph.svg|400]] > > *** > > In a Boolean matrix, a relation is symmetric if it has *reflection symmetry* along its major diagonal. > > > > ![[SymmetricRelationBooleanMatrix.svg]] > > *The left relation is symmetric; arrows point to corresponding matrix entries that make it symmetric.* > > A symmetric relation can also be [[Antisymmetric Relation|antisymmetric]] if the elements that form its ordered pairs are *not distinct* - these properties are not mutually exclusive. > > If a relation is both symmetric and antisymmetric, then it must be [[Reflexive Relation|reflexive]]. > [!Example] Asymmetric Relation > An *asymmetric relation* is a binary relation $R$ on a set $X$ such that for every $x, y\in X$, if $xRy$, then *not* $yRx$. > > That is, either $(x,y)$ or $(y,x) \in R$, but not both. > $\forall\; x, y\in X: xRy\implies\neg(yRx)$ > $\forall\; x, y\in X: \neg(xRy\wedge yRx)$ > > The [[complement]] of an asymmetric relation in the [[Cartesian product]] $X\times X$ is a symmetric relation. > > An asymmetric relation must also be [[Reflexive Relation#^4dc1ca|irreflexive]]. ^bd8e50