# Difference between revisions of "Symmetry (of a relation)"

From Encyclopedia of Mathematics

(Importing text file) |
(Category:Logic and foundations) |
||

(One intermediate revision by the same user not shown) | |||

Line 1: | Line 1: | ||

− | A property of a [[Binary relation|binary relation]]. A binary relation | + | A property of a [[Binary relation|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==== | ====Comments==== | ||

− | An anti-symmetric relation on a set | + | 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==== | ====References==== | ||

<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.M. Cohn, "Algebra" , '''1''' , Wiley (1982) pp. 17ff</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.M. Cohn, "Algebra" , '''1''' , Wiley (1982) pp. 17ff</TD></TR></table> | ||

+ | |||

+ | [[Category:Logic and foundations]] |

## Latest revision as of 18:49, 19 October 2014

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=16206

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