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Math Help - Help With Cardinals

  1. #1
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    Help With Cardinals

    Hi, I've been asked to show the following assertion;

    {|A|}^2 = |A| > 1 \rightarrow 2 \cdot |A| = |A| \wedge {|A|}^{|A|} = 2^{|A|}

    I've managed to show the 2 \cdot |A| = |A| part, but I'm having trouble with the next part, the {|A|}^{|A|} = 2^{|A|}. Am I right in thinking that {|A|}^{|A|} is the number of functions A \rightarrow A, and is this on the right lines, or am I looking way off?

    Any help would be appreciated!
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  2. #2
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    Quote Originally Posted by jackprestonuk View Post
    Hi, I've been asked to show the following assertion;

    {|A|}^2 = |A| > 1 \rightarrow 2 \cdot |A| = |A| \wedge {|A|}^{|A|} = 2^{|A|}

    I've managed to show the 2 \cdot |A| = |A| part, but I'm having trouble with the next part, the {|A|}^{|A|} = 2^{|A|}. Am I right in thinking that {|A|}^{|A|} is the number of functions A \rightarrow A, and is this on the right lines, or am I looking way off?

    Any help would be appreciated!
    I suppose we're talking of cardinals here, so let us put \alpha:=|A|\,,\,\,so \,\,\,\alpha=\alpha^2\,\,\,and\,\,\,\alpha \neq 1\Longrightarrow \alpha\geq \aleph_0 , and thus by Cantor's Theorem:

    \alpha^\alpha\leq \left(2^\alpha\right)^\alpha=2^{\alpha\cdot\alpha}  =2^\alpha , and since \alpha^\alpha\geq 2^\alpha is straightforward, applying Cantor-Schroeder-Bernstein we're done.

    Tonio
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  3. #3
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    Ah, thanks very much, that's really helpful!
    In the meantime I'd come up with this argument; I wonder if it works just as well?

    Set |A| =: a \in \mathbb{N}. Then 2^a \subseteq a^a \subseteq \mathcal{P}(a \times a), and so 2^{|A|} = 2^a \leq a^a \leq |\mathcal{P}(a \times a)| = 2^{a^2} = 2^a. We have that a^a = |A|^{|A|} and so we have our equality by Cantor Bernstein.

    Apologies if sometimes I did or didn't put |...| in there, I still get a little confused when dealing with cardinals whether, say, a^a represents the set of functions from the natural number a to itself, or simply the cardinal number a^a (that is, effectively, the number of such functions), or can stand for both...
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