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:

:<math>\forall a, b \in X,\ a R b \and b R a \; \Rightarrow \; a = b</math>

or equally,

<!-- this is the same formula as above, but due to the addition of the negation, it is more clear where the term anti-symmetric comes from -->
:<math>\forall a, b \in X,\ a R b \and a \ne b \Rightarrow \lnot b R a</math>

[[inequality|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 relation|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., [[equality (mathematics)|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 [[Asymmetric relation|asymmetry]]. According to one definition of '''asymmetric''', anything that fails to be symmetric is asymmetric; the definition of antisymmetry is more specific than this. Another definition of '''asymmetric''' makes asymmetry equivalent to antisymmetry plus [[reflexive relation|irreflexivity]].

==Examples==
* [[Equality (mathematics)|Equality]]
* "... is even, ... is odd"
::::::[[Image:Evenandodd.PNG]]

==Properties containing antisymmetry==
* [[Partial order]] - An antisymmetric relation that is also [[transitive relation|transitive]] and [[reflexive relation|reflexive]].

* [[total order]] - An antisymmetric relation that is also [[transitive relation|transitive]] and [[total relation|total]].

==See also==
* [[Symmetry in mathematics]]
* [[Symmetric relation]]
* [[antisymmetry]] in linguistics
* [[nonsymmetric relation]]
* [[asymmetric relation]]

[[Category:Mathematical relations]]

[[cs:Slabě antisymetrická relace]]
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[[ko:반대칭관계]]
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[[pl:Relacja antysymetryczna]]
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