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Countable and Uncountable Sets

  1. Jan 24, 2008 #1
    Show that (0,1) is uncountable if and only if [tex]\Re[/tex] is uncountable.

    I have a nice little proof showing (0,1) is uncountable, however I'm wondering how I can make implications that [tex]\Re[/tex] and vice versa.
     
  2. jcsd
  3. Jan 24, 2008 #2
    Well, i think that you might want to see if you can construct a 1-1 function (correspondence) with the naturals( positive integers).
    f:(0,1)-->Z (integers) . Well, you might also use the property that if a set A is uncountable, and further if this set A is a subset of B, then also B is uncountable.
    So basically if you manage to show that (0,1) is uncountable, then automatically you have shown that R is uncountable, since even if we managed to put all other elements of R in an order and count them, we defenitely could not count the elements of R that are within the interval (0,1).
     
    Last edited: Jan 24, 2008
  4. Jan 24, 2008 #3

    morphism

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    You can setup a bijection between (0,1) and R - try playing around with tan(x).
     
  5. Jan 24, 2008 #4
    I basically just wanted to say that I honestly think you should coin the term proberty because it makes this feel like a property of probabilistic outcomes!
     
  6. Jan 24, 2008 #5
    Well, if you really like, you can start using it from now on, i will not suit u for plagiarism!!:cool:
     
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