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Math Help - Cardinal Exponentials

  1. #1
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    Cardinal Exponentials

    Show that if κ ≤ λ then κ^μ ≤ λ^μ for any cardinals μ, κ and λ .

    I was thinking setting the sets A, B and C to be sets with cardinals κ, λ and μ respectively. Then I'd just need to prove that A^C ≤ B^C gives an injection. If someone could help out with a sketch of the proof that would be great.

    (I think it's also helpful to note that A^B is defined to be the set of all functions from B to A and κ^λ = card(A^B), but I'm not sure how to apply these).
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  2. #2
    Senior Member
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    Using your notations, what we assume is that A\prec B.
    You want to send a map from C to A on a map from C to B.

    A\prec B means there is a map f:A\rightarrow B such that f is a bijection from A into f(A)\subseteq B.

    Prove that A^C\rightarrow B^C:g\rightarrow fg is an injection.
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