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Math Help - Bijection

  1. #1
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    Bijection

    Give an explicit example of a bijection from [0,1] to the set of real numbers.

    I'm confused on how to do this one because it is a closed set.
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  2. #2
    Senior Member MacstersUndead's Avatar
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    Re: Bijection

    Recall that a finite composition of bijections is also a bijection. [I don't remember the infinite case]

    Let f be a bijection between [0,1] and (0,1) and let g be a bijection between (0,1) and R. then g o f is a bijection between [0,1] and R

    f(x): [0, 1] -> (0, 1) by

    f(0) = 1/2
    If n is a positive integer, then f(1/n) = 1/(n + 2)
    Otherwise (that is, if x is not zero and cannot be written in the form 1/n for n a positive integer), then f(x) = x.

    g(x) = tan( \pix - \pi/2)
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  3. #3
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    Re: Bijection

    Quote Originally Posted by lovesmath View Post
    Give an explicit example of a bijection from [0,1] to the set of real numbers. I'm confused on how to do this one because it is a closed set.
    You should not be confused: the real number set is closed as a topological space.

    There is a bijection \left[ {0,1} \right]\mathop  \Leftrightarrow \limits^f \left( {0,1} \right), this done by shifting a countable set.

    g(x)=\tan\left(x\pi-\frac{\pi}{2}\right) is a bijection \mathbb{R} \Leftrightarrow \left( {0,1} \right).

    What about a composition of those two?
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  4. #4
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    Re: Bijection

    See also this thread for a bijection between [0, 1] and (0, 1).
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