Difference between revisions of "Symmetry (of a relation)"
From Encyclopedia of Mathematics
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| − | 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> | ||
Revision as of 19:45, 12 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
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