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Math Help - Help on proving equal cardinalities

  1. #1
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    Red face Help on proving equal cardinalities

    Hello, I am trying to proof the following claim.
    \left| P(A \times B) \right| = \left| P(B)^A \right|

    Where P is the power set and  P(B)^A := \left \{ \left f: ( f: A \to P(B)  \right ) \right \}.

    I am trying to construct a bijection or two injections and cantors theorem to proof they have the same cardinality, but I seem to be getting nowhere. Can anyone help me?
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  2. #2
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    Re: Help on proving equal cardinalities

    Quote Originally Posted by gordo151091 View Post
    Hello, I am trying to proof the following claim.
    \left| P(A \times B) \right| = \left| P(B)^A \right|

    Where P is the power set and  P(B)^A := \left \{ \left f: ( f: A \to P(B)  \right ) \right \}.

    I am trying to construct a bijection or two injections and cantors theorem to proof they have the same cardinality, but I seem to be getting nowhere. Can anyone help me?
    I would define a bijection \psi : P(A\times B) \to P(B)^A by \psi(S)(a) = \{b\in B:(a,b)\in S\}, for S\in P(A\times B) and a\in A.

    The inverse map \phi would be given by \phi(f) = \{(a,b):b\in f(a)\}, for f\in P(B)^A.

    Intuitively, a subset of the product space AxB can be identified with the set of "vertical slices" given by fixed elements of A, as in the picture.
    Attached Thumbnails Attached Thumbnails Help on proving equal cardinalities-powerset.png  
    Last edited by Opalg; November 18th 2011 at 05:09 AM.
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  3. #3
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    Re: Help on proving equal cardinalities

    So the b \in B in the red line from picture would be the resulting set by the function \psi for a particular S and a
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  4. #4
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    Re: Help on proving equal cardinalities

    Quote Originally Posted by gordo151091 View Post
    So the b \in B in the red line from picture would be the resulting set by the function \psi for a particular S and a
    Yes. The blobby area in the picture is supposed to represent a subset S of AxB, and \psi(S) is the function whose value at a point a is the set of second coordinates of all points in S whose first coordinate is a.
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