# Symmetric difference of sets

From Encyclopedia of Mathematics

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. M.I. Voitsekhovskii (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Symmetric_difference_of_sets&oldid=14687

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098