1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Bijection between sets of functions

  1. Oct 23, 2012 #1
    For two sets X and Y let X^Y be the set of functions from Y to X.

    Prove that there is a bijection between (X x Y)^Z and X^Z x Y^Z.

    Attempt: I could not get any further from that "there must be a function S with S(f)=g and S(f')=g for any g, f' in X^Z x Y^Z, and where f is in (X x Y)^Z."
  2. jcsd
  3. Oct 23, 2012 #2
    see attachment.

    Attached Files:

    • 001.jpg
      File size:
      13.5 KB
    Last edited: Oct 23, 2012
  4. Oct 23, 2012 #3
    see attachment
    Last edited: Oct 23, 2012
  5. Oct 24, 2012 #4
    Thank you, I think I'm done with proving that psi is an injection but I can't prove that it's also a surjection. Could you give me another hint?
  6. Oct 24, 2012 #5
    This is what I tried for surjectivity so far: psi is clearly a surjection, as for every ordered pair (h,g) there is an f such that psi(f)=(h,g), because when we look at the values of f, h and g at z we see that psi(x,y)=(x,y). But this is the identity function and therefore a surjection. Is that correct?
    Last edited: Oct 24, 2012
  7. Oct 24, 2012 #6
    define f(z)=(h(z),g(z)).verify that this is a counter image of (h,g)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook