direct image

Let f:AB be a function, and let UA be a subset. The direct image of U is the set f(U)B consisting of all elements of B which equal f(u) for some uU.

Direct images satisfy the following properties:

  1. 1.

    Unions: For any collectionMathworldPlanetmath {Ui}iI of subsets of A,

  2. 2.

    IntersectionsMathworldPlanetmathPlanetmath: For any collection {Ui}iI of subsets of A,

  3. 3.

    Set differenceMathworldPlanetmath: For any U,VA,


    In particular, the complement of U satisfies f(U)f(A)f(U).

  4. 4.

    Subsets: If UVA, then f(U)f(V)B.

  5. 5.

    Inverse imagePlanetmathPlanetmath of a direct image: For any UA,


    with equality if f is injectivePlanetmathPlanetmath.

  6. 6.

    Direct image of an inverse image: For any VB,


    with equality if f is surjectivePlanetmathPlanetmath.

Title direct image
Canonical name DirectImage
Date of creation 2013-03-22 11:52:01
Last modified on 2013-03-22 11:52:01
Owner djao (24)
Last modified by djao (24)
Numerical id 10
Author djao (24)
Entry type Definition
Classification msc 03E20
Classification msc 81-00
Classification msc 18-00
Classification msc 17B37
Classification msc 18D10
Classification msc 18D35
Classification msc 16W30
Synonym image
Related topic InverseImage
Related topic Mapping