## You are here

Homesymmetric relation

## Primary tabs

# 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.

Defines:

symmetry, symmetric

Related:

Reflexive, Transitive3, Antisymmetric

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

03E20*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections