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In a practise exam, the examinator asks: "Give example of a surjection $g: \mathbb R\rightarrow \mathbb R\times \mathbb R$".

The answer key suggests finding a surjection from $\mathbb R\rightarrow [0,\infty)^2$ which can be done by defining the decimal expansion of $x$ as $x=\pm a_0a_1...a_n,b_0b_1...$ then letting $(u,w)$ $$ \begin{cases} u=a_0a_2...a_n,b_0b_2b_4b_6....\\ w=a_1a_3...a_{n-1},b_1b_3b_5... \end{cases} $$ if $n$ is even and \begin{cases} u=a_0a_2...a_{n-1},b_0b_2b_4b_6....\\ w=a_1a_3...a_{n},b_1b_3b_5... \end{cases} if $n$ is odd.

Call this $q(x)$.

Then we simply define $f:[0,\infty)^2\rightarrow \mathbb R \times \mathbb R$ where $f(u,w)=(\ln u, \ln w)$ and defining $ln(0)=1$.

The surjection is then given by $g=q\circ f$

I do not understand why we do have to split into cases if $n$ is odd and even and I do not understand why we can skip every other number in the inputs. Can I get some help about that?

albin
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  • I don't understan d the notation. What does $x=\pm a_0a_1\cdots a_n,b_0b_1$ mean? – lulu Oct 22 '24 at 09:24
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    @lulu I think they are using a decimal comma, so the $a_i$ form the decimal representation of the integer part and $b_i$ the decimal representation of the fractional part. – Jaap Scherphuis Oct 22 '24 at 09:25
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    Nor do I understand the question. Nobody is saying that you have to split into even and odd cases, the hint just points to one particular way of solving the problem. – lulu Oct 22 '24 at 09:26
  • @JaapScherphuis I thought of that, but why does it stop at $b_1$? I guess the OP meant for the $b's$ to continue... – lulu Oct 22 '24 at 09:26
  • @lulu It shouldn't of course. Also, the $f$ should be $f:[0,\infty)\color{red}{^2}\rightarrow \mathbb R \times \mathbb R$ and $g=\color{red}{f\circ q}$. – Jaap Scherphuis Oct 22 '24 at 09:28
  • @JaapScherphuis Yes, this makes sense. – lulu Oct 22 '24 at 09:28
  • Variants of this question have been asked a lot on this site. I've linked to one, but you can find others. – lulu Oct 22 '24 at 09:35
  • Yes sorry there were multiple errors in the OP. I will correct them immediately. – albin Oct 22 '24 at 09:50

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