View source for Symmetry (of a relation)

A property of a [[Binary relation|binary relation]]. A binary relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091770/s0917701.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091770/s0917702.png" /> is called symmetric if for any pair of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091770/s0917703.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091770/s0917704.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091770/s0917705.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091770/s0917706.png" />. An example of a symmetric relation is an [[Equivalence|equivalence]].



====Comments====
An anti-symmetric relation on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091770/s0917707.png" /> is a reflexive relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091770/s0917708.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091770/s0917709.png" />.

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