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Algebra question

  1. Jan 8, 2006 #1
    Hello,

    Im having a bit of trouble with this one question, a little help would be great.

    this is a mapping question involving surjectivity.

    S: A --> B T: B --> C

    T*: B --> C satisfies TS = T*S
    And S is surjective

    Show that T = T*

    Now i know the defenition of surjective. Im just having trouble showing what i have to show

    let x be in A and y be in B
    There exists an x in A, for every y in B such that S(x) = y
    --thats surjective--
    but where do i go now??
     
  2. jcsd
  3. Jan 8, 2006 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    If would phrase if differently: "For every y in B there exist x in A such that S(x)= y."
    Do you see the difference? Your phrasing implies you can just pick any x and y you like in advance. You can pick y as you like, but not x!

    Try proof by contradiction:
    If T and T* are not equal, then there exist some y in B such that [itex]Ty
    \ne T*y[/itex]. Since S is surjective, there exist x in A such that Sx= y. Then what is TSx?
    What is T*Sx
     
    Last edited: Jan 8, 2006
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