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Uncountable sets

  1. Sep 6, 2009 #1
    1. The problem statement

    Need to prove that the set of bijections from N to N is uncountable.


    2. The attempt at a solution

    I'm not really sure how to proceed here but what I did so far is this....

    f(2i) = { 2i+1, if fi(2i)=2i
    2i , if fi(2i)not equal to =2i }


    Not very sure what I'm going to do.

    Please help me.
     
  2. jcsd
  3. Sep 7, 2009 #2

    lanedance

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    Homework Helper

    you could try to start by representing each biejction as an infinite sequence of un-repeating integers (similar to a permutation)

    The problem then becomes showing the uncountability of the set of infinite sequences...

    if you have been shown the uncountabilty of the reals, you may be able to use a similar argument...
     
  4. Sep 7, 2009 #3

    HallsofIvy

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    Staff Emeritus
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    For each bijection, you can construct a real number by .0.f(1)f(2)f(3)... . Can you show that that set of real numbers is uncountable?
     
  5. Sep 7, 2009 #4
    Not really sure how to do that.

    Can you please help me out?
     
  6. Sep 7, 2009 #5
    Are you familiar with Cantor's argument to show the reals are uncountable?
     
  7. Sep 7, 2009 #6
    Is this proof on real numbers correct? Then how do I proceed from here to show the bijection from N to N is uncountable?

    Let A be a set of Real numbers between numbers 0 and 1.

    Suppose x0, x1, x2, ... is any sequence of elements of A, there is an element xεA that is not in the sequence for every i>=0 and x≠xi .

    X0 -> 0.x0,0 x0,1 x0,2 …
    X1 -> 0.x1,0 x1,1 x1,2 …
    |
    |
    |


    Assume the element not in A is X -> 0.x0 x1 x2…

    Then xi=xi,i and the sequence of x0 x1 x2 doesnot end with .99999…….
    The xi= 1 if xi,i≠1 and xi= 0 if xi,i=1

    As the real numbers between 0 and 1 are uncountable and it’s a subset of the total real numbers, the set of real numbers are uncountable.
     
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