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Math Help - infinite prove

  1. #1
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    infinite prove

    I need to prove this question: 2 the set of all even natural numbers. Prove that 2 is countably infinite.

    I know that I can combine each even number with an number so that 2=1, 4=2, 6=3,....
    ={1,2,3,4,5,6,.....,n} (n∈ )
    2={2,4,6,8,10,12,...2n} (n∈ )

    I know how to write it down but not how to prove it proper.
    And ik know I need to prove its bijective but I dont know how.
    Last edited by Burcin; September 22nd 2012 at 03:05 AM.
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  2. #2
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    Re: infinite prove

    Use the bijection f\colon \Bbb N \to 2\Bbb N, f(n):=2n.
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  3. #3
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    Re: infinite prove

    f: N-->2N f(n)=2n

    So if I want to prove its bijective I say:

    n1,n2 ∈N f(n1)=f(n2) so 2n1=2n2 so n1=n2 so f is injective

    n∈N f(n)=2n so n lays in the image of f so f is surjective

    I say f is bijective. Is my prove correct?
    Can I now say that 2N is countably finite ?
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  4. #4
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    Re: infinite prove

    Quote Originally Posted by Burcin View Post
    f: N-->2N f(n)=2n

    So if I want to prove its bijective I say:

    n1,n2 ∈N f(n1)=f(n2) so 2n1=2n2 so n1=n2 so f is injective
    Correct.

    Quote Originally Posted by Burcin View Post
    n∈N f(n)=2n so n lays in the image of f
    Why does n lie in the image of f?
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  5. #5
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    Re: infinite prove

    n∈N and n∈2N so that f(n)=2n so we see that n lie in the image of f?
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  6. #6
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    Re: infinite prove

    Quote Originally Posted by Burcin View Post
    n∈N and n∈2N so that f(n)=2n so we see that n lie in the image of f?
    I don't understand what you are saying. What exactly are you proving (without using the word "surjective")? What is n? Why n∈2N? Not every n is in 2N.
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  7. #7
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    Re: infinite prove

    well I tought for every n∈N there exists a 2n∈2N so that f(n)=2n is this correct proven
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  8. #8
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    Re: infinite prove

    Quote Originally Posted by Burcin View Post
    well I tought for every n∈N there exists a 2n∈2N so that f(n)=2n
    True.

    Quote Originally Posted by Burcin View Post
    is this correct proven
    No. I am still not sure what you are proving.
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  9. #9
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    Re: infinite prove

    I need to prove this question: 2N is the set of all even natural numbers. Prove that 2N is countably infinite. ( N= natural numbers I cant copy the right notation)

    In the forum girdav said that I need to use f;N->2N and f=2n
    I thought I needed to prove that f is bijective to say that 2N is countably infinite
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  10. #10
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    Re: infinite prove

    Quote Originally Posted by Burcin View Post
    I need to prove this question: 2N is the set of all even natural numbers. Prove that 2N is countably infinite. ( N= natural numbers I cant copy the right notation)

    In the forum girdav said that I need to use f;N->2N and f=2n
    I thought I needed to prove that f is bijective to say that 2N is countably infinite
    I understand all this. I agreed that you have proved injectivity of f. We were talking about surjectivity. In post #6 I asked what you are proving without using the word "surjective." It is very important to get the statement right, including the quantifiers ("there exists," "for all"). A proof must match the structure of the claim. For example, there are specific ways to prove claims that start with "For all." Also, a proof must not contain undefined objects like n. All identifiers should be properly introduced.

    So, what exactly are you proving at this point?
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  11. #11
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    Re: infinite prove

    take any n you want 2n∈2N
    if n≥0 then n∈N and f(n)=2n
    so 2n is the image of f so f is surjective
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  12. #12
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    Re: infinite prove

    Quote Originally Posted by Burcin View Post
    take any n you want 2n∈2N
    if n≥0 then n∈N and f(n)=2n
    so 2n is the image of f so f is surjective
    This sort of, somewhat resembles a proof, and if I had evidence that you understand how to write similar proofs, I could possibly accept it. As it is, I doubt this. And I can't help you until you write the exact claim you are proving.
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  13. #13
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    Re: infinite prove

    Quote Originally Posted by Burcin View Post
    n∈N and n∈2N so that f(n)=2n so we see that n lie in the image of f?
    If you are using n as the argument of f, you cannot require that n be in 2N. And by asserting that n is in N, you are going the wrong way.

    Rather, to prove f is "surjective" (one-to-one) you need to prove "If y is in 2N then there exist n in N such that y= f(x)". If y is even then, by definition of "even", there exist n in N such that y= 2n= f(n).
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  14. #14
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    Re: infinite prove

    I understand the theorie behind it all but I don't know how to use it to prove things, because I just started at the university.
    I don't know what you mean do you mean that I cant prove proper that f is surjective?
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  15. #15
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    Re: infinite prove

    So I don't neet to prove if f is surjective and injective to answer the question if 2|N is countably infinite
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