# Bijection

• Nov 17th 2012, 02:47 PM
lovesmath
Bijection
Give an explicit example of a bijection from [0,1] to the set of real numbers.

I'm confused on how to do this one because it is a closed set.
• Nov 17th 2012, 03:08 PM
Re: Bijection
Recall that a finite composition of bijections is also a bijection. [I don't remember the infinite case]

Let f be a bijection between [0,1] and (0,1) and let g be a bijection between (0,1) and R. then g o f is a bijection between [0,1] and R

f(x): [0, 1] -> (0, 1) by

f(0) = 1/2
If n is a positive integer, then f(1/n) = 1/(n + 2)
Otherwise (that is, if x is not zero and cannot be written in the form 1/n for n a positive integer), then f(x) = x.

g(x) = tan($\displaystyle \pi$x - $\displaystyle \pi$/2)
• Nov 17th 2012, 03:20 PM
Plato
Re: Bijection
Quote:

Originally Posted by lovesmath
Give an explicit example of a bijection from [0,1] to the set of real numbers. I'm confused on how to do this one because it is a closed set.

You should not be confused: the real number set is closed as a topological space.

There is a bijection $\displaystyle \left[ {0,1} \right]\mathop \Leftrightarrow \limits^f \left( {0,1} \right)$, this done by shifting a countable set.

$\displaystyle g(x)=\tan\left(x\pi-\frac{\pi}{2}\right)$ is a bijection $\displaystyle \mathbb{R} \Leftrightarrow \left( {0,1} \right)$.

What about a composition of those two?
• Nov 17th 2012, 03:42 PM
emakarov
Re: Bijection