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Math Help - A functions proof

  1. #1
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    A functions proof

    Hi, I have a question I am having trouble starting:

    Let A be a nonempty set. Prove there is no function A onto P(A) (power set of A).

    I am not sure if it just means a general function f:A -> P(A) or if we are talking about a function that is onto. Any help will be greatly appreciated.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by nzmathman View Post
    Hi, I have a question I am having trouble starting:

    Let A be a nonempty set. Prove there is no function A onto P(A) (power set of A).

    I am not sure if it just means a general function f:A -> P(A) or if we are talking about a function that is onto. Any help will be greatly appreciated.
    Yes, the question is asking about a surjection. Otherwise, you could take the function to be x\mapsto \{x\}.

    To prove this result, you can use arguments based on the cardinality of \mathcal{P}(A). Now, do you know about the rather handy relationship between |\mathcal{P}(A)| and |A| which states that |A| < |\mathcal{P}(A)|?
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  3. #3
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    It was just the wording I was slightly confused by, but your example x->{x} made it very clear what the question wanted me to do, and I think I have proved it o.k. now. Thanks!
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  4. #4
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    You may want to refer to this thread - I had the same question sometime back
    http://www.mathhelpforum.com/math-he...et-113875.html
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