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Antisymmetric relation
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{{short description|Binary relation such that if A is related to B and is different from it then B is not related to A}} {{Refimprove|date=January 2010}} {{distinguish|Asymmetric relation}} {{stack|{{Binary relations}}}} In [[mathematics]], a [[binary relation]] <math>R</math> on a [[Set (mathematics)|set]] <math>X</math> is '''antisymmetric''' if there is no pair of ''distinct'' elements of <math>X</math> each of which is related by <math>R</math> to the other. More formally, <math>R</math> is antisymmetric precisely if for all <math>a, b \in X,</math> <math display=block>\text{if } \,aRb\, \text{ with } \,a \neq b\, \text{ then } \,bRa\, \text{ must not hold},</math> or equivalently, <math display=block>\text{if } \,aRb\, \text{ and } \,bRa\, \text{ then } \,a = b.</math> The definition of antisymmetry says nothing about whether <math>aRa</math> actually holds or not for any <math>a</math>. An antisymmetric relation <math>R</math> on a set <math>X</math> may be [[Reflexive relation|reflexive]] (that is, <math>aRa</math> for all <math>a \in X</math>), [[Irreflexive relation|irreflexive]] (that is, <math>aRa</math> for no <math>a \in X</math>), or neither reflexive nor irreflexive. A relation is [[Asymmetric relation|asymmetric]] if and only if it is both antisymmetric and irreflexive.
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