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Math Help - Power Set of N-Intersection of finite subsets of N

  1. #1
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    Power Set of N-Intersection of infinitely many finite subsets of N

    Hello everyone,
    This is a problem I'm having a hard time solving:

    Let P(\mathbb{N})= the power set of \mathbb{N} and for every natural number, n, P(\mathbb{N}_n) be the power set of \mathbb{N}_n which is a finite subset of \mathbb{N}:  \mathbb{N}_n=\left \{ 1,2,3,...,n \right \}.

    Now, if for every n, A_n=P(\mathbb{N})-P(\mathbb{N}_n),

    what can be said about \bigcap_{n=1}^{\infty }A_n?

    Is that an infinite subset of \mathbb{N}?
    Thanks, in advance.
    Last edited by Neutriiino; July 21st 2012 at 07:36 AM.
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  2. #2
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    Re: Power Set of N-Intersection of infinitely many finite subsets of N

    Quote Originally Posted by Neutriiino View Post
    This is a problem I'm having a hard time solving:
    Let P(\mathbb{N})= the power set of \mathbb{N} and for every natural number, n, P(\mathbb{N}_n) be the power set of \mathbb{N}_n which is a finite subset of \mathbb{N}:  \mathbb{N}_n=\left \{ 1,2,3,...,n \right \}.
    Now, if for every n, A_n=P(\mathbb{N})-P(\mathbb{N}_n),
    what can be said about \bigcap_{n=1}^{\infty }A_n?
    Is that an infinite subset of \mathbb{N}?
    We realize that this is your first posting. Nonetheless, it is best if you show some work so that we know where to begin.
    If F is a finite subset of \mathbb{N} is it possible for F\in\bigcap_{n=1}^{\infty }A_n~?

    If F is an infinite subset of \mathbb{N} must for F\in\bigcap_{n=1}^{\infty }A_n~?

    Tell us what you think?
    Thanks from Neutriiino
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  3. #3
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    Re: Power Set of N-Intersection of finite subsets of N

    Note that N is an element of P(N), but cannot be an element of any P(Nn) (since these sets are all finite). Hence N is in every An, and thus in the intersection.

    The same exact logic hold for N - S, where S is any finite subset of N. Thus any co-finite subset of N is in the infinite intersection. Since we obtain a co-finite subset of N from any finite set S, simply by taking N - S, there are at least as many co-finite subsets of N, as there are finite subsets of N. Hence the intersection of all the An is indeed infinite.
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  4. #4
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    Re: Power Set of N-Intersection of finite subsets of N

    Thanks Deveno, I got it!
    Thank you too, Plato, for your rather Socratic method of instruction through questions.
    well, here we go:

    If F is a finite subset of \mathbb{N}, then F\subseteq \mathbb{N}_n_m_a_x, where nmax is the largest number in F. hence F\notin P(\mathbb{N})-P(\mathbb{N}_n_m_a_x), F\notin A_n (for any value of n)

    If G is an infinite subset of N, for every n, G\notin P(\mathbb{N}_n), hence G\in P(\mathbb{N})-P(\mathbb{N}_n).
    Right?
    I found this in a previously held exam in our department, and now I'm wondering where I can find more problems like this.
    Thanks, once again.
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