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Math Help - Bijection from [1, infinity) to R

  1. #1
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    Bijection from [1, infinity) to R

    The problem says "Find a bijecton f:[1,\infty)\longrightarrow\mathbb{R}."

    The hint goes on to say you won't be able to give a specific formula. My main question is "Is this even possible?" I don't see how it is.
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  2. #2
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    Quote Originally Posted by mathematicalbagpiper View Post
    The problem says "Find a bijecton f:[1,\infty)\longrightarrow\mathbb{R}."

    The hint goes on to say you won't be able to give a specific formula. My main question is "Is this even possible?" I don't see how it is.
    Hmm I would do it this way

    Map 1 to 3.

    Define a bijection between (1,2] and (-\infty, 2] using the usual trick of specifying a point at (2,1) where the number in (1,2] represents an angle in (-pi/2, 0] from the point to the x-axis. (-pi/2 is directly to the left, 0 is straight down).

    For non-integers greater than 2, map them to themselves.

    For integers n greater than two, map them to n+1.
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  3. #3
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    f(x)=(-1)^(⌊x⌋+1)*⌊x/2⌋+(x-⌊x⌋)
    is a bijection

    It bijectively maps these intervals on each other
    [2n-1,2n) -> [n-1,n)
    [2n,2n+1) -> [-n,-n+1)
    for all n>=1
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