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Infinite set

  1. Sep 20, 2011 #1
    infinite set

    1. The problem statement, all variables and given/known data
    [itex]f: \mathbb{R} \rightarrow \mathbb{Q} [/itex],

    show that there is a [itex] q \in \mathbb{Q}[/itex] st. [itex]f^{-1}(q)[/itex] is infinite set in [itex] \mathbb{R}[/itex].

    2. Relevant equations



    3. The attempt at a solution

    how can we show that is true?
     
  2. jcsd
  3. Sep 20, 2011 #2

    Dick

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    Re: infinite set

    Do you know R is uncountable? That would be a clue.
     
  4. Sep 20, 2011 #3
    Re: infinite set

    infinite set means ; we can't find 1-1 correspondence between the set {1,...,n} and [itex]f^{-1}(q)[/itex] , here [itex]f^{-1}(q)[/itex] has n elements.
    So, can we say [itex]f^{-1}(q)[/itex] is uncountable then we can't find 1-1 correspondence between the set {1,...,n}....????
     
  5. Sep 20, 2011 #4

    Dick

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    Re: infinite set

    Try to get the idea of your proof before you start writing symbols. Assume the opposite, that all of the f^(-1)(q) are finite. And you didn't answer my question. Do you know R is uncountable, and do you know what that means?
     
  6. Sep 20, 2011 #5
    Re: infinite set

    yeah R is uncountable but I can't find a relation to this question..
     
  7. Sep 20, 2011 #6

    Dick

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    Re: infinite set

    Every element if R is in one of the f^(-1)(q) sets, right? Suppose they are all finite?
     
  8. Sep 20, 2011 #7
    Re: infinite set

    I didn't understand clearly but ;

    for all [itex]q \in \mathbb{Q}[/itex], we suppose all the sets, [itex] f^{-1}(q)[/itex] are finite. That means, every element of [itex]\mathbb{R}[/itex] is one of these sets, [itex] f^{-1}(q)[/itex] ...right?
     
  9. Sep 20, 2011 #8

    Dick

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    Re: infinite set

    Every element of R is in one of the f^(-1)(q) whether those sets are finite or not, just because f:R->Q. Pick an element x of R, for which q in Q is x in f^(-1)(q)??
     
  10. Sep 20, 2011 #9
    Re: infinite set

    I think that is right because just now we said
    " every element of R is in one of the set f^{-1}(q) "
     
  11. Sep 20, 2011 #10

    Dick

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    Re: infinite set

    Yes, but can you tell me why?? Just because we said it doesn't prove it. You are going to need this as part of your proof.
     
    Last edited: Sep 20, 2011
  12. Sep 21, 2011 #11
    Re: infinite set

    Ok. I think; we have [itex]\mathbb{R}= \bigcup\limits_{q \in \mathbb{Q}} f^{-1}(q)[/itex], and we suppose that for all q in Q , [itex]f^{-1}(q)[/itex] finite, then union would be countable but R is uncountable so contradiction... right?
     
  13. Sep 21, 2011 #12

    Dick

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    Re: infinite set

    Right, if you are clear on why the union of all the f^(-1)(q) is R.
     
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