inverse image

Let f:AB be a function, and let UB be a subset. The inverse imagePlanetmathPlanetmath of U is the set f-1(U)A consisting of all elements aA such that f(a)U.

The inverse image commutes with all set operationsMathworldPlanetmath: For any collectionMathworldPlanetmath {Ui}iI of subsets of B, we have the following identitiesPlanetmathPlanetmathPlanetmath for

  1. 1.


  2. 2.



and for any subsets U and V of B, we have identities for

  1. 3.


  2. 4.

    Set differencesMathworldPlanetmath:

  3. 5.

    Symmetric differencesMathworldPlanetmathPlanetmath:


In addition, for XA and YB, the inverse image satisfies the miscellaneous identities

  1. 6.


  2. 7.


  3. 8.

    Xf-1(f(X)), with equality if f is injectivePlanetmathPlanetmath.

Title inverse image
Canonical name InverseImage
Date of creation 2013-03-22 11:51:58
Last modified on 2013-03-22 11:51:58
Owner djao (24)
Last modified by djao (24)
Numerical id 10
Author djao (24)
Entry type Definition
Classification msc 03E20
Classification msc 46L05
Classification msc 82-00
Classification msc 83-00
Classification msc 81-00
Synonym preimage
Related topic Mapping
Related topic DirectImage