symmetric relation

A relation $\mathcal{R}$ on a set $A$ is symmetric if and only if whenever $x\mathcal{R}y$ for some $x,y\in A$ then also $y\mathcal{R}x$ .

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 is $\mathcal{R}=\{(b,b),(a,b),(b,a),(c,b)\}$ , because $(c,b)\in\mathcal{R}$ but $(b,c)\notin\mathcal{R}$ .

On a finite set with $n$ elements there are $2^{n^{2}}$ relations, of which $2^{\frac{n^{2}+n}{2}}$ are symmetric.

A relation $\mathcal{R}$ that is both symmetric and antisymmetric has the property that $x\mathcal{R}y$ implies $x=y$ . On a finite set with $n$ elements there are only $2^{n}$ 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	Reflexive
Related topic	Transitive3
Related topic	Antisymmetric
Defines	symmetry
Defines	symmetric