# Symmetry (of a relation)

From Encyclopedia of Mathematics

A property of a binary relation. A binary relation $R$ on a set $A$ is called symmetric if for any pair of elements $a,b \in A$, $aRb$ implies $b R a$, i.e. $R \subseteq R^{-1}$. An example of a symmetric relation is an Equivalence relation.

#### Comments

An anti-symmetric relation on a set $A$ is a reflexive relation $R$ such that $R \cap R^{-1} \subseteq \Delta = \{ (x,x) : \forall x \in A \}$.

#### References

[a1] | P.M. Cohn, "Algebra" , 1 , Wiley (1982) pp. 17ff |

**How to Cite This Entry:**

Symmetry (of a relation).

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Symmetry_(of_a_relation)&oldid=33962

This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article