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I needed to find cardinality of irrationals. I have provedthat R\Q is uncountable. Now I need to build a bijection h : R\Q → R . How to do this?

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  • $\begingroup$ "I have provedthat R\Q is uncountable." -- Aren't you done? Or did you want an explicit bijection? $\endgroup$ – user4894 Sep 24 '14 at 5:43
  • $\begingroup$ If you could easily build a bijection from $\mathbb{R}\setminus\mathbb{Q}\to\mathbb{R}$, then you wouldn't need to show that $\mathbb{R}\setminus\mathbb{Q}$ was uncountable. $\endgroup$ – Peter Huxford Sep 24 '14 at 5:43
  • $\begingroup$ But uncountability doesn't assure cardinality same as that of real numbers $\endgroup$ – Silver moon Sep 24 '14 at 5:45
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Can you find a bijection from $\mathbb{Z}\times(\mathbb{Z}\setminus\{0\})$ to $\mathbb{Z}\times\mathbb{Z}$? What about $\mathbb{Q}\times(\mathbb{Z}\setminus\{0\})$ to $\mathbb{Q}\times\mathbb{Z}$? Last hint: $a+b\sqrt{2}$.

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