A relationMathworldPlanetmath on A is antisymmetric iff x,yA, (xyyx)(x=y). For a finite setMathworldPlanetmath A with n elements, the number of possible antisymmetric relations is 2n3n2-n2 out of the 2n2 total possible relations.

Antisymmetric is not the same thing as “not symmetricPlanetmathPlanetmathPlanetmathPlanetmath”, as it is possible to have both at the same time. However, a relation that is both antisymmetric and symmetric has the condition that xyx=y. There are only 2n such possible relations on A.

An example of an antisymmetric relation on A={,×,} would be ={(,),(×,),(,),(,×)}. One relation that isn’t antisymmetric is ={(×,),(,),(,)} because we have both and , but

Title antisymmetric
Canonical name Antisymmetric
Date of creation 2013-03-22 12:15:50
Last modified on 2013-03-22 12:15:50
Owner aoh45 (5079)
Last modified by aoh45 (5079)
Numerical id 14
Author aoh45 (5079)
Entry type Definition
Classification msc 03E20
Synonym antisymmetry
Related topic ReflexiveMathworldPlanetmathPlanetmathPlanetmath
Related topic Symmetric
Related topic ExteriorAlgebra
Related topic SkewSymmetricMatrix