Antisymmetric relation

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In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X, if a is related to b and b is related to a, then a = b.

In mathematical notation, this is:

$\forall a, b \in X,\ a R b \and b R a \; \Rightarrow \; a = b$

or equally,

$\forall a, b \in X,\ a R b \and a \ne b \Rightarrow \lnot b R a$

Inequalities are antisymmetric, since for numbers a and b, a ≤ b and b ≤ a if and only if a = b.

Note that 'antisymmetric' is not the logical negative of 'symmetric' (whereby aRb implies bRa). (N.B.: Both are properties of relations expressed as universal statements about their members; their logical negations must be existential statements.) Thus, there are relations which are both symmetric and antisymmetric (e.g., the equality relation) and there are relations which are neither symmetric nor antisymmetric (e.g., the preys-on relation on biological species).

Antisymmetry is different from asymmetry. According to one definition of asymmetric, anything that fails to be symmetric is asymmetric. Another definition of asymmetric makes asymmetry equivalent to antisymmetry plus irreflexivity.