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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- Jul 5th 2009, 02:23 PMtukilalacreate 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. - Jul 5th 2009, 03:22 PMPlato
- Jul 5th 2009, 05:29 PMTheAbstractionist
A bijection $\displaystyle g:(0,\,1)\to[0,\,1)$ 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 $\displaystyle F\circ g.$