# 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 SymmetricRelation 2013-03-22 12:15:39 2013-03-22 12:15:39 yark (2760) yark (2760) 21 yark (2760) Definition msc 03E20 Reflexive Transitive3 Antisymmetric symmetry symmetric