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Bijection between (0,1) and [0,1) in R?

  1. Apr 6, 2015 #1
    1. The problem statement, all variables and given/known data
    I need to find a bijection between (0,1) and [0,1) in R. It can go in either direction since it is a bijection.

    2. Relevant equations
    I can't think of any equations at all!

    3. The attempt at a solution
    Something like f(x) = 1/[(1/x)+1] for x in A
    x for x not in A
    where A={1/2, 1/3, 1/4, ...}

    Having an issue with zero though.
     
  2. jcsd
  3. Apr 6, 2015 #2

    LCKurtz

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    Not sure which direction you are trying to do. For ##[0,1)\rightarrow (0,1)## try starting with ##0\to 1/2##, ##1/2\to 1/3## etc.
     
  4. Apr 7, 2015 #3

    HallsofIvy

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    The crucial point here is that the set of rational numbers, between 0 and 1, is countable. Write the rational numbers as "[itex]a_1, a_2, a_3, ...[/itex]" and map [itex]a_1[/itex] to 0 and [itex]a_{n+1}[/itex] to [itex]a_n[/itex]. Map each irrational number to itself.
     
  5. Apr 7, 2015 #4

    SammyS

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    Simplify ##\displaystyle\ \frac{1}{\displaystyle\frac{1}{x}+1}\ ## to get x out of the denominator.

    Although f(0) is undefined, ##\displaystyle\ \lim_{x\to 0}\,f(x)=0\ ##, so f(x) has a removable discontinuity at x=0 .
     
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