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Thread: create function

  1. #1
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    create function

    can someone help me please build 2 bijection(injective and surjective) functions:
    1. from (0, 1) to (0,infinite )
    2. from (0,1) to [0,infinite)

    thanks.
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  2. #2
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    Quote Originally Posted by tukilala View Post
    2 bijection(injective and surjective) functions:
    1. from (0, 1) to (0,infinite )
    2. from (0,1) to [0,infinite)
    1. $\displaystyle f(x)=\tan \left( \frac{\pi x}{2} \right)$
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  3. #3
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by tukilala View Post
    2. from (0,1) to [0,infinite)
    A bijection $\displaystyle g0,\,1)\to[0,\,1)$ is defined by

    $\displaystyle g(x)\ =\ \left\{\begin{array}{cl}0 & x=\dfrac12\\[7mm]
    \dfrac1{n-1} & x=\dfrac1n,\ n\in\mathbb Z,\ n\ge3\\[7mm]
    x & \mbox{otherwise}
    \end{array}\right.$

    Define $\displaystyle F:[0,\,1)\to[0,\,\infty)$ by $\displaystyle F(0)=0$ and $\displaystyle F(x)=f(x)$ for $\displaystyle 0<x<1$ where $\displaystyle f$ is your bijection in (1). Then the required bijection for (2) is $\displaystyle F\circ g.$
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