set difference


Definition

Let A and B be sets. The set differenceMathworldPlanetmath (or simply difference) between A and B (in that order) is the set of all elements of A that are not in B. This set is denoted by AB or A-B (or occasionally AB). So we have

AB={xAxB}.
Venn diagramMathworldPlanetmath showing AB in blue

Properties

Here are some properties of the set difference operationMathworldPlanetmath:

  1. 1.

    If A is a set, then

    A=A

    and

    AA==A.
  2. 2.

    If A and B are sets, then

    B(AB)=BA.
  3. 3.

    If A and B are subsets of a set X, then

    AB=AB

    and

    (AB)=AB,

    where denotes complement in X.

  4. 4.

    If A, B, C and D are sets, then

    (AB)(CD)=(AC)(BD).

Remark

As noted above, the set difference is sometimes written as A-B. However, if A and B are sets in a vector space (or, more generally, a module (http://planetmath.org/Module)), then A-B is commonly used to denote the set

A-B={a-baA,bB}

rather than the set difference.

Title set difference
Canonical name SetDifference
Date of creation 2013-03-22 11:59:38
Last modified on 2013-03-22 11:59:38
Owner yark (2760)
Last modified by yark (2760)
Numerical id 33
Author yark (2760)
Entry type Definition
Classification msc 03E20
Synonym difference between sets
Synonym difference
Related topic SymmetricDifference
Related topic InverseImageCommutesWithSetOperations
Related topic Difference2