Antisymmetric relation

Type of binary relation

In mathematics, a binary relation R {\displaystyle R} on a set X {\displaystyle X} is antisymmetric if there is no pair of distinct elements of X {\displaystyle X} each of which is related by R {\displaystyle R} to the other. More formally, R {\displaystyle R} is antisymmetric precisely if for all a , b ∈ X , {\displaystyle a,b\in X,} if a R b {\displaystyle aRb} with a ≠ b {\displaystyle a\neq b} then b R a {\displaystyle bRa} must not hold, or equivalently, if a R b {\displaystyle aRb} and b R a {\displaystyle bRa} then a = b {\displaystyle a=b} . The definition of antisymmetry says nothing about whether a R a {\displaystyle aRa} actually holds or not for any a {\displaystyle a} .

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