create function

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July 5th 2009, 02:23 PM
tukilala

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.
July 5th 2009, 03:22 PM
Plato

Quote:

Originally Posted by tukilala

2 bijection(injective and surjective) functions:
1. from (0, 1) to (0,infinite )
2. from (0,1) to [0,infinite)

1. $f(x)=\tan \left( \frac{\pi x}{2} \right)$
July 5th 2009, 05:29 PM
TheAbstractionist

Quote:

Originally Posted by tukilala

2. from (0,1) to [0,infinite)

A bijection $g:(0,\,1)\to[0,\,1)$ is defined by

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

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

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