symmetric relation

A relationMathworldPlanetmathPlanetmath on a set A is symmetricPlanetmathPlanetmathPlanetmathPlanetmath if and only if whenever xy for some x,yA then also yx.

An example of a symmetric relation on {a,b,c} is {(a,a),(c,b),(b,c),(a,c),(c,a)}. One relation that is not symmetric is ={(b,b),(a,b),(b,a),(c,b)}, because (c,b) but (b,c).

On a finite setMathworldPlanetmath with n elements there are 2n2 relations, of which 2n2+n2 are symmetric.

A relation that is both symmetric and antisymmetric has the property that xy implies x=y. On a finite set with n elements there are only 2n such relations.

Title symmetric relation
Canonical name SymmetricRelation
Date of creation 2013-03-22 12:15:39
Last modified on 2013-03-22 12:15:39
Owner yark (2760)
Last modified by yark (2760)
Numerical id 21
Author yark (2760)
Entry type Definition
Classification msc 03E20
Related topic ReflexiveMathworldPlanetmathPlanetmathPlanetmath
Related topic Transitive3
Related topic Antisymmetric
Defines symmetry
Defines symmetric