Antisymmetric relation

In mathematics, a binary relation ${\displaystyle R}$ on a set ${\displaystyle X}$ is antisymmetric if there is no pair of distinct elements of ${\displaystyle X}$ each of which is related by ${\displaystyle R}$ to the other. More formally, ${\displaystyle R}$ is antisymmetric precisely if for all ${\displaystyle a,b\in X,}$

${\displaystyle {\text{if }}\,aRb\,{\text{ with }}\,a\neq b\,{\text{ then }}\,bRa\,{\text{ must not hold}},}$

or equivalently,

${\displaystyle {\text{if }}\,aRb\,{\text{ and }}\,bRa\,{\text{ then }}\,a=b.}$

The definition of antisymmetry says nothing about whether ${\displaystyle aRa}$ actually holds or not for any ${\displaystyle a}$. An antisymmetric relation ${\displaystyle R}$ on a set ${\displaystyle X}$ may be reflexive (that is, ${\displaystyle aRa}$ for all ${\displaystyle a\in X}$), irreflexive (that is, ${\displaystyle aRa}$ for no ${\displaystyle a\in X}$), or neither reflexive nor irreflexive. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.