symmetric relation (original) (raw)

A relation ℛ on a set A is symmetric if and only if whenever x⁢ℛ⁢y for some x,y∈A then also y⁢ℛ⁢x.

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 isℛ={(b,b),(a,b),(b,a),(c,b)}, because (c,b)∈ℛ but (b,c)∉ℛ.

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

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