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Math Help - Come up with a bijection?

  1. #1
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    indiana
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    Come up with a bijection?

    Hello,
    I need some help with this problem:
    Let S = {x
    : x2
    )
    I need to come up with a bijection between S and Q.
    Any help appreciated.
    Thanks!

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  2. #2
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    Tejas
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    Re: Come up with a bijection?

    Consider the mapping f:S \to \mathbb{Q} given by:

    f(s) = s^2, s \geq 0

    f(s) = -s^2, s < 0.

    Is this a bijection?
    Thanks from topsquark
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  3. #3
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    indiana
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    Re: Come up with a bijection?

    for s≥0, if I took the inverse of f(s), which equals g(s),
    and showed that f(g(s)) = g(f(s)) and did the same thing for s<0,
    would that show the mapping you provided is a bijection?
    thanks for your help.
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  4. #4
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    Re: Come up with a bijection?

    OK, for one thing the INVERSE of f (if it exists, which you have NOT shown) would be a function g: \mathbb{Q} \to S.

    so you would need to show TWO things:

    1) f \circ g = 1_{\mathbb{Q}}
    2) g \circ f = 1_S

    but you can't simply say "g exists" you have to DEFINE it.

    Perhaps a more straight-forward method is to prove:

    A) f(s_1) = f(s_2) \implies s_1 = s_2 (that is, f is injective)

    B) for EVERY q \in \mathbb{Q}, there exists SOME s \in S with: f(s) = q (that is, f is surjective).
    Thanks from topsquark
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