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Math Help - Help with set theory proof

  1. #1
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    Help with set theory proof

    I've been given a question on my problem set which I'm a bit stuck with. I'm getting the sense that the first part is easier than the second part, but I can't even seem to get the first part!

    a) Let A, X, Y be sets such that X\preceq A. Show that X^Y\preceq A^Y. Therefore show that, for cardinals \kappa, \lambda, \mu , if \kappa \leq \lambda then \kappa^\mu \leq \lambda^\mu.

    b) Let A, B, X, Y be sets with X\preceq A and Y\preceq B. Show that, apart from some exceptional cases, X^Y\preceq A^B. What are the exceptional cases?

    I'd really appreciate any help you can provide!
    Last edited by longhorn22; March 8th 2011 at 07:04 AM. Reason: fixed latex
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  2. #2
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    Quote Originally Posted by longhorn22 View Post
    a) Let A, X, Y be sets such that X\preceq A. Show that X^Y\preceq A^Y. Therefore show that, for cardinals \kappa, \lambda, \mu , if \kappa \leq \lambda then \kappa^\mu \leq  \lambda^\mu.
    From X\preceq A we know that there is an injection f:X\to A.
    Using f find an injection \Phi:X^Y\to A^Y.
    Last edited by Plato; March 8th 2011 at 08:11 AM.
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  3. #3
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    Thank you - I was able to find an injection for (a), and then deduce the result about cardinals. However, I'm still stuck on (b), which seems to be a fair bit more difficult.
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  4. #4
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    Quote Originally Posted by longhorn22 View Post
    I've been given a question on my problem set which I'm a bit stuck with. I'm getting the sense that the first part is easier than the second part, but I can't even seem to get the first part!

    a) Let A, X, Y be sets such that X\preceq A. Show that X^Y\preceq A^Y. Therefore show that, for cardinals \kappa, \lambda, \mu , if \kappa \leq \lambda then \kappa^\mu \leq \lambda^\mu.

    b) Let A, B, X, Y be sets with X\preceq A and Y\preceq B. Show that, apart from some exceptional cases, X^Y\preceq A^B. What are the exceptional cases?

    I'd really appreciate any help you can provide!
    Here's something to get you started for the second one. Let f be an injection from X to A, and g an injection from Y to B. Now, let h be an arbitrary function from Y to X. The most natural thing to do is let F(h) be the function k from B to A where to compute k(b), you take g^{-1}(b) if possible (this gives an element of Y), then apply h (this gives an element of X), and finally apply f (this gives an element of A). Now think about how you would define k on elements not in the range of g. Going through the argument carefully will probably shed some light on the exceptional cases.
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