Bijection between $\mathbb R^2$ and $(0,1)$

by Anik Bhowmick   Last Updated September 21, 2018 04:20 AM

[TIFR GS-2013, Part D] Does there exist any bijection between $\mathbb R^2$ and the open interval $(0,1)$ ??

At the first glimpse, I thought about the function $f: \mathbb R^2 \to (0,1)$ defined by $f(x,y) = {0.2}^{x}{0.3}^{y}$. But then I realized that the preimage of any element in $(0,1)$ may not be unique. Here I am stuck with finding any example. Any help would be appreciated.

Tags : functions


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