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Math Help - Proving Equal Cardinality

  1. #1
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    Proving Equal Cardinality

    Problem:

    Show that S = \{x \in \mathbb{R} | 0 < x < 1\} has the same cardinality as \mathbb{R}.

    -----------------

    From my understanding (and using the book's definition), the cardinality is just the number of elements in the set. So I think I need to prove that there exists a one-to-one function mapping the infinitely many numbers between 0 and 1 to all real numbers \mathbb{R}.

    I looked at cos(x) and sin(x) to maybe just try to say something like, function f(x) = cos(x) maps all real numbers into a number between 0 and 1, but this is not true. And even if I rigged that to work, it's not a one-to-one function since cos(x), sin(x) etc... can equal the same value with different inputs. Am I on the right track, though?

    I don't want the answer, just a little guidance.

    Thanks.
    Last edited by Plato; September 12th 2011 at 11:25 AM. Reason: LaTeX fix
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  2. #2
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    Re: Proving Equal Cardinality

    Quote Originally Posted by tangibleLime View Post
    Problem:
    Show that S = \{x \in \mathbb{R} | 0 < x < 1\} has the same cardinality as \mathbb{R}.
    Is the function \tan\left[\left(-\frac{1}{2}+x\right){\pi}\right] one-to-one and onto S\to\mathbb{R}~?
    Last edited by Plato; September 12th 2011 at 11:43 AM.
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  3. #3
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    Re: Proving Equal Cardinality

    Thanks for the response! I was working with tan for a bit, but was discouraged about the facts of when it was undefined... I was thinking about it the wrong way.

    It seems to me that it would be one-to-one since for each value of x between 0 and 1, it would give a distinct output in \mathbb{R}. I think it is also onto since, looking at the graph of this function, it spans the entire codomain off into infinity in both vertical directions.
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