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Thread: Intersection of Uncountable Sets Proof

  1. #1
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    Intersection of Uncountable Sets Proof

    I have a set theory question that I'm not really sure how to do it.

    Prove or disprove that there exists an uncountable set S\subseteq\mathbb{R} such that S\cap(\mathbb{R}\setminus\mathbb{Q}) is countable.

    Thanks!
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  2. #2
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    Re: Intersection of Uncountable Sets Proof

    'u' for binary union, and 'n' for binary intersection:

    Suppose S is an uncountable subset of R.

    S = S\Q u SnQ.

    SnQ is a subset of Q, and Q is countable, so SnQ is countable.

    If S\Q were countable, then S would be the union of two countable sets, so S would be countable.

    So S\Q is uncountable.

    Since S is a subset of R, we have S\Q = S n (R\Q).

    So S n (R\Q) is uncountable.

    /

    The key theorem used is that the union of two countable sets is a countable set.
    Last edited by MoeBlee; Oct 31st 2016 at 12:49 PM.
    Thanks from romsek and topsquark
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