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Math Help - Nondenumerable - Denumerable = Nondenumerable

  1. #1
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    Lightbulb Nondenumerable - Denumerable = Nondenumerable

    Given: S is a nondenumerable set (uncountable)
    T is a denumerable subset of S

    Prove: (S - T) ~ S. (S - T is nondenumerable)
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  2. #2
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    Is this true T\cup (S\setminus T)=S?

    What if (S\setminus T) were denumerable?
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  3. #3
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    isn't that just another way of wording it?
    it would be a denumerable subset of a nondenumerable set, which i believe is acceptable...just not the other way around
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  4. #4
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    Quote Originally Posted by DavyHilbert View Post
    isn't that just another way of wording it?
    it would be a denumerable subset of a nondenumerable set, which i believe is acceptable...just not the other way around
    No it in not a reword.
    The union of two denumerable sets is a denumerable set.
    But S is not denumerable.
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  5. #5
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    ahhh, thanks. understood.

    Because they are both nondenumerable, we can still say (S - T) ~ S, correct? Even though there are different "levels" of nondenumerable?
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  6. #6
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    Because they are both nondenumerable, we can still say (S - T) ~ S, correct?
    Well, no, precisely because there are many levels.

    To show that S - T ~ S, you can consider a denumerable subset T0 of S - T and construct a bijection between S - T and S as follows. On S - T - T0 the bijection is identity, and on T0 it is a bijection between T0 and (T0 union T).
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