elementary set theory - Find a bijection between the set of real numbers and the interval $(−1, 1) ≡ {,x ∈ Bbb Rmid − 1 < x < 1,}$.

Find a bijection between the set of real numbers and the interval $(−1, 1) ≡ \{\,x ∈ \Bbb R\mid − 1 < x < 1\,\}$.

Hi am I trying to revise for an an exam and I came across this question which I can not figure out. I tried looking online but I kept just finding graphs.
Thank You

Answer

You have to find bijection map

$g:\mathbb{R} \to \left( { - 1,1} \right)$

Then, first write the any bijection map from $\mathbb{R}$ to any open interval.

And then make its range (co-domain) by some adjustment to required open interval.

Example:

$f\left( x \right) = {\tan ^{ - 1}}x$

then,

$\begin{gathered} Dom\left( f \right) = \mathbb{R} \\ Range\left( f \right) = \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right) \\ \end{gathered}$

then, your job is to arrange your range (co-domain) as
$\left( { - 1,1} \right)$

then, for this multiply your function f(x) by
$\frac{2}{\pi }$,

$\frac{2}{\pi }f\left( x \right) = \frac{2}{\pi }{\tan ^{ - 1}}x$

then take,

$\begin{gathered} g\left( x \right) = \frac{2}{\pi }f\left( x \right) \\ = \frac{2}{\pi }{\tan ^{ - 1}}x \\ \end{gathered}$

And then,

$\begin{gathered} Dom\left( g \right) = \mathbb{R} \\ Range\left( g \right) = \frac{2}{\pi }\left( { - \frac{\pi }{2},\frac{\pi }{2}} \right) \\ = \left( { - 1,1} \right) \\ \end{gathered}$

Hence, the required bijection map $g:\mathbb{R} \to \left( { - 1,1} \right)$ is,

$g\left( x \right) = \frac{2}{\pi }{\tan ^{ - 1}}x$

I hope this will fulfill you required bijection map.