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Math Help - Bounded Intervals - Cardinality

  1. #1
    pkr
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    Bounded Intervals - Cardinality

    Q: Show that any two intervals (a,b) and (c,d) have the same cardinality

    So I guess I have to show there's a bijective function?

    So f: (a,b) --> (c,d)
    and f(a) = c
    f(b) = d

    In general f(x) = sx + t, just using different letters

    So f(a) = sa + t = c
    and f(b) = sb + t = d

    Subtracting the two (sa + t) - (sb + t) = c - d
    s(a-b) = c-d

    Am I going along the right lines for the proof? How would I write out the final function and prove it's a bijection?
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    Quote Originally Posted by pkr View Post
    Q: Show that any two intervals (a,b) and (c,d) have the same cardinality. How would I write out the final function and prove it's a bijection?
    Write the equaton of the line determined by (a,c) & (b,d).
    f(x) = \left( {\frac{{d - c}}{{b - a}}} \right)\left( {x - a} \right) + c.
    Linear function are bijections.
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  3. #3
    pkr
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    I think i'm being stupid but why is it (x-a)?
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    Quote Originally Posted by pkr View Post
    I think i'm being stupid but why is it (x-a)?
    That is the equation of a line in slope/point form: y=m(x-a)+b.
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