View source for Symmetric difference of sets

An operation on sets. Given two sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s0916401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s0916402.png" />, their symmetric difference, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s0916403.png" />, is given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s0916404.png" /></td> </tr></table>

where the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s0916405.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s0916406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s0916407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s0916408.png" /> denote the operations of union, intersection, difference, and complementation of sets, respectively.



====Comments====
The symmetric difference operation is associative, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s0916409.png" />, and intersection is distributive over it, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s09164010.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s09164011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s09164012.png" /> define a ring structure on the power set of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s09164013.png" /> (the set of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s09164014.png" />), in contrast to union and intersection. This ring is the same as the ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s09164015.png" />-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091640/s09164016.png" /> (with [[pointwise multiplication]] and addition). Cf. also [[Boolean algebra]] and [[Boolean ring]] for the symmetric difference operation in an arbitrary Boolean algebra.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Kuratowski,   "Introduction to set theory and topology" , Pergamon  (1961)  pp. 34, 35  (Translated from French)</TD></TR></table>