properties of symmetric difference

Recall that the symmetric differenceMathworldPlanetmathPlanetmath of two sets A,B is the set AB-(AB). In this entry, we list and prove some of the basic properties of .

  1. 1.

    (commutativity of ) AB=BA, because and are commutativePlanetmathPlanetmath.

  2. 2.

    If AB, then AB=B-A, because AB=B and AB=A.

  3. 3.

    A=A, because A, and A-=A.

  4. 4.

    AA=, because AA and A-A=.

  5. 5.

    AB=(A-B)(B-A) (hence the name symmetric difference).


    AB=(AB)-(AB)=(AB)(AB)=(AB)(AB)=((AB)A)((AB)B)=(BA)(AB)=(B-A)(A-B). ∎

  6. 6.

    AB=AB, because AB=(A-B)(B-A)=(AB)(BA)=(B-A)(A-B)=AB.

  7. 7.

    (distributivity of over ) A(BC)=(AB)(AC).


    A(BC)=A((BC)-(BC)), which is (A(BC))-(A(BC)), one of the properties of set difference (see proof here ( This in turns is equal to ((AB)(AC))-((AB)(AC))=(AB)(AC). ∎

  8. 8.

    (associativity of ) (AB)C=A(BC).


    Let U be a set containing A,B,C as subsets (take U=ABC if necessary). For a given B, let f:P(U)×P(U)P(U) be a function defined by f(A,C)=(AB)C. Associativity of is then then same as showing that f(A,C)=f(C,A), since A(BC)=(BC)A=(CB)A.

    By expanding f(A,C), we have

    (AB)C = ((AB)-C)(C-(AB))
    = (((A-B)(B-A))C)(C-((AB)-(AB)))
    = (((AB)(BA))C)((CAB)(C-(AB))
    = ((ABC)(BAC))((CAB)(CAB))
    = (BAC)(BAC)(BAC)(BAC).

    It is now easy to see that the last expression does not change if one exchanges A and C. Hence, f(A,C)=f(C,A) and this shows that is associative. ∎

Remark. All of the properties of on sets can be generalized to ( on Boolean algebrasMathworldPlanetmath.

Title properties of symmetric difference
Canonical name PropertiesOfSymmetricDifference
Date of creation 2013-03-22 14:36:56
Last modified on 2013-03-22 14:36:56
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 14
Author CWoo (3771)
Entry type Derivation
Classification msc 03E20