Symmetric difference of sets
From Encyclopedia of Mathematics
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An operation on sets. Given two sets 
and 
, their symmetric difference, denoted by 
, is given by
 |
where the symbols 
, 
, 
, 
denote the operations of union, intersection, difference, and complementation of sets, respectively.
Comments
The symmetric difference operation is associative, i.e. 
, and intersection is distributive over it, i.e. 
. Thus, 
and 
define a ring structure on the power set of a set 
(the set of subsets of 
), in contrast to union and intersection. This ring is the same as the ring of 
-valued functions on 
(with pointwise multiplication and addition). Cf. also Boolean algebra and Boolean ring for the symmetric difference operation in an arbitrary Boolean algebra.
References
| [a1] | C. Kuratowski, "Introduction to set theory and topology" , Pergamon (1961) pp. 34, 35 (Translated from French) |
How to Cite This Entry:
Symmetric difference of sets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetric_difference_of_sets&oldid=14687
Symmetric difference of sets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetric_difference_of_sets&oldid=14687
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article