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Math Help - Power set of N

  1. #1
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    Power set of N

    Is there an uncountable family such that is finite for all distinct ,?
    (here denotes a power set for N).
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Pythagorean12 View Post
    Is there an uncountable family such that is finite for all distinct ,?
    (here denotes a power set for N).
    Before you even try to construct an example or craft a proof, what do you think the answer is?
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  3. #3
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    I don't know if it can be useful in your case, but there is a general theorem that states that, given a cardinal \kappa \geq \omega such that 2^{< \kappa} = \kappa, then there exists an almost disjoint family F \subset P(\kappa) with |F| = 2^\kappa.
    This theorem, with \kappa = \omega answers your question.
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  4. #4
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    Come on, do not take a sledgehammer to crack a nut. There is a simple example:

    Let the P_i be such infinite set that p_i^k \in P_i, where p_i is the i-th prime number and  k \in \mathbb{N}.

    Then infinite family  \{ P_1, P_2, P_3, ... \} is an example satisfying the requirements. Because for every distinct P_i \cap P_j = \{1\} and P_i \subseteq \mathcal{P}(\mathbb{N}) and there are infinitely many primes.
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  5. #5
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    Quote Originally Posted by sidor View Post
    Come on, do not take a sledgehammer to crack a nut. There is a simple example:

    Let the P_i be such infinite set that p_i^k \in P_i, where p_i is the i-th prime number and  k \in \mathbb{N}.

    Then infinite family  \{ P_1, P_2, P_3, ... \} is an example satisfying the requirements. Because for every distinct P_i \cap P_j = \{1\} and P_i \subseteq \mathcal{P}(\mathbb{N}) and there are infinitely many primes.

    The family \{P_1,\ldots\}\subset P(\mathbb{N}) is not uncountable, as requested in the OP.

    Tonio
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  6. #6
    Senior Member Shanks's Avatar
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    I read a problem like this somewhere.
    since there is a bijective mapping between N and Q, let A be the collection of all rational number cauchy sequence converging to 0, B be the collection of all rational number cauchy sequence converging to 1.
    A union B is a uncountable set. for each a in A, and b in B, there are only finite common elements.
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  7. #7
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Shanks View Post
    I read a problem like this somewhere.
    since there is a bijective mapping between N and Q, let A be the collection of all rational number cauchy sequence converging to 0, B be the collection of all rational number cauchy sequence converging to 1.
    A union B is a uncountable set. for each a in A, and b in B, there are only finite common elements.
    I actually did a similar problem to this (as you described). The question was "Does there exists an uncountable family of subsets of a countable set such that the interesction of any two is finite?" The answer is yes. I gave two examples. I actually did it on the forum if you are interested. But! The OP said that the sets need to be subsets of \mathbb{N} and as your sets are not this doesn't work. I refuse to actually answer the question upon the grounds that the OP has NEVER shown any effort.
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  8. #8
    Senior Member Shanks's Avatar
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    That is Why I said there is a bijective mapping between N and Q.
    This bijective mapping play a role in this example.
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