# 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.

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