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Is there a way to prove equivalent cardinalites?

  1. Feb 5, 2012 #1
    1. The problem statement, all variables and given/known data
    I'm just a bit stupefied at trying to prove a cardinality. I'm not sure I have the skills, but I would like to show that the cardinalities of R (the real numbers) and the interval on (0,1) are equivalent.--to be clear, the interval here is all real numbers between (0,1).


    2. Relevant equations
    I suppose I could show some sort of bijection between the two, but I don't know how to begin it. I do realize that I will need a more general definition than an actual formula.


    3. The attempt at a solution
    I'm stuck here. I don't know how to begin exactly. This is not my homework problem. In fact, my homework is to show that the cardinalities of R and the interval of real numbers (1,infinity) are equivalent.

    Any tips would be very appreciated!
     
  2. jcsd
  3. Feb 5, 2012 #2

    Deveno

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    one thing that occurs to me is that if you suitably restricted the domain of the tangent function, you'd map an interval symmetric around 0 to all of R.

    then it's just a matter of tinkering around with the argument of the tan function to keep it within (0,1).

    finally, you want to show that this gives you a bijection of (0,1) with R, which is equivalent to showing the inverse function exists.
     
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