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Math Help - Cantor's Theorem

  1. #1
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    Cantor's Theorem

    Prove by induction on n: If a set A has n elements, its power set has 2^n elements.

    *Given a set A, the power set of A is the set of all subsets of A
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  2. #2
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    Quote Originally Posted by thamathkid1729 View Post
    Prove by induction on n: If a set A has n elements, its power set has 2^n elements.

    *Given a set A, the power set of A is the set of all subsets of A

    P(k)

    The power of a set with k elements is 2^k


    P(k+1)

    The power of a set with k+1 elements is 2^{k+1}


    Show that P(k+1) must be true if P(k) is true.

    If a set of k elements has a power of 2^k

    then an added element will form 2^k new subsets as it can be placed with all 2^k existing subsets.

    Therefore a set of k+1 elements has a power of (2)2^k if a set of k elements has power 2^k.

    Finally prove for the base case.
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  3. #3
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    This proof turns on a simple fact: 2^n+2^n=2(2^n)=2^{n+1}.

    Any subset of \{1,2,\cdots,n\} is a subset of \{1,2,\cdots,n,n+1\}.

    So by uniting any subset of \{1,2,\cdots,n\} with \{n+1}\} we get a subset of \{1,2,\cdots,n+1\}.
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