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Math Help - How do I know if it is countable?

  1. #1
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    How do I know if it is countable?

    Hi there, I have to make some assignments(7), but I don't really get them. I have to 1) tell if a set is countable 2) tell if a set has the same cardinality as R (= all the real numbers).
    But I really don't get how to do this!
    For example:
    The intersection of Q and [3,4). How do I proof that this is countable (or not), and what about the cardinality?

    Can someone help me? Explain how to do this? I have to make 7 assignments exactly like this.
    Tnx,
    Mary
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  2. #2
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    1) A set is countable if it is either: 1) finite or 2) infinite and there exists a bijection from the set to the natural numbers.

    2) A set has the same cardinality as \mathbb{R} if there exists a bijection from the set to \mathbb{R}.

    In your example, you should know that \mathbb{Q} is countable. and also \mathbb{Q} \cap [3,4) \subseteq \mathbb{Q}, thus it is countable and does not have the same cardinality as \mathbb{R}.

    Basically, you have to 'see' if it is possible to make a mapping from the set to \mathbb{R} or \mathbb{N}..
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  3. #3
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    Quote Originally Posted by Defunkt View Post
    Basically, you have to 'see' if it is possible to make a mapping from the set to \mathbb{R} or \mathbb{N}..
    Okay I got this.
    I also have to proof both 1) and 2).
    I still have troubles with it...
    If I take a look at the second: [2,3] x [2,3] I have no idea what to do!
    1) is it countable?
    2) is this true: | [2,3] x [2,3] | = |R|?
    Please help!
    Last edited by MaryB; November 1st 2009 at 10:52 AM.
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  4. #4
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    I edited my last question, can someone please help me?
    By the way, I have a question: Can I also delete a reply or question? I'm sorry thought I was editing my old reply!
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  5. #5
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    Quote Originally Posted by MaryB View Post
    Okay I got this.
    I also have to proof both 1) and 2).
    I still have troubles with it...
    If I take a look at the second: [2,3] x [2,3] I have no idea what to do!
    1) is it countable?
    2) is this true: | [2,3] x [2,3] | = |R|?
    Please help!
    1) It is not countable. A way to prove this would be to assume by contradiction that it is -- that is, there exists a bijection f: \mathbb{N} \rightarrow [2,3]\times [2,3]. Then look at f(n) and f(n+1) for some n \in \mathbb{N} and conclude that it cannot happen.

    2) It is true. You need to first find a bijection from [2,3] to [0,1], and then find a bijection from [0,1]x[0,1] to R.

    From my experience, there is no 'method' to solve this type of questions. Simply solve as many as you can and it will be easier after you've done a few of them.
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