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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.
     

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    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)
     
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