# Reflexive Relation **A [[relation]] that relates every element to itself.** > [!Example] Reflexive Relation > A *reflexive relation* is a binary relation $R$ on a [[set]] $X$ such that for every $x\in X$, there is $xRx$ and $(x,x)\in R$. > $\forall\;x\in X: xRx$ > > > [!Visual definitions]- > > In a directed graph, a relation is reflexive if and only if *every vertex has a loop* to itself. > > > > ![[ReflexiveRelationGraph.svg|400]] > > *** > > In a Boolean matrix, a relation is reflexive if and only if its *major diagonal is all true*. > > > > ![[ReflexiveRelationBooleanMatrix.svg|400]] > > *The translucent squares indicate related elements which are not necessary for the relation to be reflexive.* > [!Example] Irreflexive Relation > An *irreflexive relation* is a binary relation $R$ on a set $X$ such that for every $x\in X$, there is *not* $xRx$ and $(x,x)\notin R$. > $\forall\;x\in X: \neg(xRx)$ > > The [[complement]] of an irreflexive relation in the [[Cartesian product]] $X\times X$ is a reflexive relation. ^4dc1ca