Symmetric difference of sets

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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.


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.


[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. M.I. Voitsekhovskii (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098