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Math Help - Proof of countable sets

  1. #1
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    Proof of countable sets

    In my class, we have this definition of countable: The statement that the set A is countable means that A is finite or else there is a one-to-one correspondence between C and A.

    Using this definition, prove that every subset of C is countable.

    To start, I've taken C to be {1, 2, 3, 4, 5, 6, 7, 8, 9, 10,...} and a subset, S, of C to be {2, 3, 5, 7, 9, 10,...} and I'm trying to show S is countable by making a one-to-one correspondence (map 1 to 2, 2 to 3, 3 to 5, 4 to 7, 5 to 9, 6 to 10, etc...). How do I show that this correspondence continues? I was thinking by induction, starting by letting x(sub 1) be the element of S that no other element of S precedes.

    Any more help proving this by induction, or any other method, would be helpful. Thanks.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by noles2188 View Post
    To start, I've taken C to be {1, 2, 3, 4, 5, 6, 7, 8, 9, 10,...} and a subset, S, of C to be {2, 3, 5, 7, 9, 10,...}
    This does not specify what set S is.

    CB
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