Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Set Theory proofs

  1. Sep 7, 2004 #1
    Let f:X->Y be a function

    1) Given any subset B of Y, prove that f(f^-1(B)) is a subset of B

    2) Prove that f(f^-1(B))=B for all subsets B of Y if and only if f is surjective


    Help anybody?
     
  2. jcsd
  3. Sep 7, 2004 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    (If this is homework, you should post in the HW help section... lemme know and I'll move it)

    Sometimes, problems become more clear just by restating it.

    Note that your goal is to prove:

    If x is in f(f^-1(B)) then x is in B.

    So what is the criterion for x to be in f(f^-1(B))?

    Ask this question a few times, and I think it solves itself.


    I think the theorem and proof of (1) will provide some insight. Also, you might consider what happens if either of these conditions fails.

    In the end, I again think it will almost solve itself if you dig into more detail.
     
  4. Sep 7, 2004 #3
    It's not homework, just some problems in my topology book that I have been thinking about. My problem with 1) which I should have stated earlier is that I don't see why f(f^-1(B)) is a subset of B, and not simply equal to it.
     
  5. Sep 7, 2004 #4

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Well, I think your second question gives a strong clue as to how to find an example where f(f^1(B)) != B. :smile:

    It doesn't have to be complicated; try something very simple, like a function whose domain has only 1 or 2 elements, and is not surjective.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Set Theory proofs
  1. Set Theory Proof Help (Replies: 6)

  2. Set theory proof (Replies: 2)

Loading...