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Math Help - Proving an onto function

  1. #1
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    Proving an onto function

    How do I prove that the function f:\mathbb{Z}^{+}\times \mathbb{Z}^{+}\rightarrow \mathbb{Z}^{+} defined as f\left ( m,n \right )=\binom{m+n-1}{2}+m is an onto function by mathematical induction? The bracket meant "n choose r".

    I started off this way:
    Basis Step: m=1, n=1
    f(1,1) = \binom{1}{2}+1 = 0+1=1
    So f(1,1) is true since f(1,1)=1, the element "1" in the codomain has its preimages m=1 and n=1.

    Then assume true for \exists m,n \in \mathbb{Z}^{+} \; such \; that \; f(m,n) = k.

    Inductive Step:
    Then there exists a m,n \in \mathbb{Z}^{+} such that f(m,n) = k+1
    So \binom{m+n-1}{2}+m = k+1
    \frac{(m+n-1)!}{2!(m+n-1-2)!}+m=k+1
    \frac{(m+n-1)!}{2!(m+n-3)!}+m=k+1

    But how do I continue to show that this will be true for all k+1 and that this is an onto function?

    Are there any "easier tricks/process" when doing induction? I am trying to keep doing induction until I get the hang of it but I am always stuck after the basis step.
    Doesn't feel like a friendly technique.

    Thanks for any help.
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  2. #2
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    Hi

    If f(m,n)=k then what is f(m+1,n-1) ?
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  3. #3
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    Quote Originally Posted by running-gag View Post
    Hi

    If f(m,n)=k then what is f(m+1,n-1) ?
    Thanks but how did you figure out that f(m+1,n-1)=k+1 ? Do I have to keep doing trial and error until I get this?
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  4. #4
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    Quote Originally Posted by xEnOn View Post
    Thanks but how did you figure out that f(m+1,n-1)=k+1 ? Do I have to keep doing trial and error until I get this?
    One way to get k+1 is to have m+1 as last term of the sum but then you change 2 choose m+n-1 into 2 choose m+n. Coming back to m+n-1 can be done by changing n into n-1
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  5. #5
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    I know this is already solved, but if you want to find the ordered pair that maps to some number k from scratch, it helps to realize the connection to triangular numbers here. Any k can be written as n(n-1)/2+j, where 0<=j<n, or in other words k lies between 2 successive triangular numbers. If you find this n and j then you can find your ordered pair.

    For example, if you want to know how to get 43,276, you just need to find that 43,276=293(294)/2+205, and f(205,294-205+1)=43,276. It's not induction, but it might be good to know.

    edit: Also, I was just thinking that if you're still going to use induction you need to be careful with the triangular numbers. If you do the m+1, n-1 thing at a certain point you'll end up with something like f(5,1) and you can't have f(6,0).
    Last edited by LoblawsLawBlog; May 1st 2011 at 09:10 PM.
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