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Math Help - Constructing a bijection between a unit interval and a unit square.

  1. #1
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    Constructing a bijection between a unit interval and a unit square.

    Hi thanks for helping in advance,

    Here is the question....

    Construct a Bijection between the closed unit interval [0,1] and the closed unit square.

    [0,1] x [0,1] = {(x,y) : 0 <= x <= 1 and 0 <= y <= 1}.





    Any ideas would be appreciated

    Thanks again.
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  2. #2
    Super Member Matt Westwood's Avatar
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    The definitive answer is to take each (real) number in the interval [0, 1]. Take the even-indexed digits of its decimal expansion to form the x co-ordinate of a point in the square, and the odd-indexed digits to form the y co-ordinate.

    That's the overall policy - are you in a position to be able to tighten it up, i.e. prove that this is indeed a bijection?
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  3. #3
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    Thanks Matt,

    I can certainly see that it is a Bijection. May main problem seems to be how I write that explicitly. Actually that seems to be most of the problem I have in the class. Describing what is in my head.

    Probably just takes practice.

    An example would be great.
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  4. #4
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    Actually it turns out that as stated, it is not a bijection. Since it is not injective.

    Consider cases: .71707070707....

    and .707979797979797.....

    Clearly different numbers but both map to the same place. namely (.7777777...., .1 )

    So you have to tweak it to make it work.
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