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Math Help - unlabeled subsets

  1. #1
    Senior Member Sampras's Avatar
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    unlabeled subsets

    1. How many unlabeled families of subsets of a  3 -set are there?

    It is just  4 ? Because you can have a subset with  0,1,2 or  3 elements?


    2. Prove that the number  F(n) of unlabeled families of subsets of an  n -set satisfies  \log_{2} F(n) = 2^n +O(n \log n) .

    So  F(n) = n+1 . Thus we want to show that  2^{2^{n}+O(n \log n)} = n+1 . Now  O(n \log n) means that there is a function  g(n) such that  |g(n)| < c(n \log n) for some constant  c . Maybe we can rewrite the LHS as  2^{2^n} \cdot 2^{O(n \log n)} ?
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  2. #2
    Senior Member Sampras's Avatar
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    actually these are families of subsets, so my answer is probably not correct.
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  3. #3
    Senior Member Sampras's Avatar
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    I know that there are  2^{2^3} labeled subsets. The power set  |\mathcal{P}(S)| = 2^n for example. Isn't this what is meant by families of subsets?
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