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单词 DirectImage
释义

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,

    f(iIUi)=iIf(Ui).
  2. 2.

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

    f(iIUi)iIf(Ui).
  3. 3.

    Set differenceMathworldPlanetmath: For any U,VA,

    f(VU)f(V)f(U).

    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,

    f-1(f(U))U

    with equality if f is injectivePlanetmathPlanetmath.

  6. 6.

    Direct image of an inverse image: For any VB,

    f(f-1(V))V

    with equality if f is surjectivePlanetmathPlanetmath.

Titledirect image
Canonical nameDirectImage
Date of creation2013-03-22 11:52:01
Last modified on2013-03-22 11:52:01
Ownerdjao (24)
Last modified bydjao (24)
Numerical id10
Authordjao (24)
Entry typeDefinition
Classificationmsc 03E20
Classificationmsc 81-00
Classificationmsc 18-00
Classificationmsc 17B37
Classificationmsc 18D10
Classificationmsc 18D35
Classificationmsc 16W30
Synonymimage
Related topicInverseImage
Related topicMapping
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更新时间:2025/5/3 12:47:52