symmetric difference

The symmetric differenceMathworldPlanetmathPlanetmath between two sets A and B, written AB, is the set of all x such that either xA or xB but not both. In other words,


The Venn diagramMathworldPlanetmath for the symmetric difference of two sets A,B, represented by the two discs, is illustrated below, in light red:



Suppose that A, B, and C are sets.

  • AB=(AB)(BA).

  • AB=AcBc, where the superscript c denotes taking complementsPlanetmathPlanetmath.

  • Note that for any set A, the symmetric difference satisfies AA= and A=A.

  • The symmetric difference operator is commutativePlanetmathPlanetmathPlanetmath since AB=(AB)(BA)=(BA)(AB)=BA.

  • The symmetric difference operationMathworldPlanetmath is associative: (AB)C=A(BC). This means that we may drop the parentheses without any ambiguity, and we can talk about the symmetric difference of multipleMathworldPlanetmathPlanetmath sets.

  • Let A1,,An be sets. The symmetric difference of these sets is written


    In general, an element will be in the symmetric difference of several sets iff it is in an odd numberMathworldPlanetmathPlanetmath of the sets.

It is worth noting that these properties show that the symmetric difference operation can be used as a group law to define an abelian group on the power setMathworldPlanetmath of some set.

Finally, we note that intersectionMathworldPlanetmath distributes over the symmetric difference operator:


giving us that the power set of a given fixed set can be made into a Boolean ringMathworldPlanetmath using symmetric difference as additionPlanetmathPlanetmath, and intersection as multiplication.

Title symmetric difference
Canonical name SymmetricDifference
Date of creation 2013-03-22 11:59:41
Last modified on 2013-03-22 11:59:41
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 20
Author CWoo (3771)
Entry type Definition
Classification msc 03E20
Synonym set symmetric difference
Synonym symmetric set difference
Synonym symmetric difference between sets
Related topic SetDifference
Related topic ProofOfTheAssociativityOfTheSymmetricDifferenceOperator
Defines symmetric difference operator