I need to construct two bijections: $$f_1 : ((0,1)\times(2,3)) \rightarrow (0,2)\times(5,6)$$ $$f_2 : (\mathbb{Z}\times[0,1))\rightarrow\mathbb{R}$$ I know what bijection means and all conditions that functions have to fulfill in order to be bijective, but I have no idea how should I 'construct' them. I thought of drawing graphs of each set from function $f_1$, but it does not help me to do further steps.
It would be nice if you could show me step-by-step how it should be done.

  • $\begingroup$ Don't get too hung up on the word "construct". If you were just asked to find the bijections, would you know what to do? $\endgroup$ – Henning Makholm Dec 8 '18 at 15:57
  • $\begingroup$ I know that I should somehow find functions which fit these sets, but how can I do it if not by guessing? $\endgroup$ – whiskeyo Dec 8 '18 at 15:59
  • $\begingroup$ Have you tried just "guessing"? This is not a follow-a- method exercise, it's just a check question to make sure you have understood what a bijection is. $\endgroup$ – Henning Makholm Dec 8 '18 at 16:06
  • $\begingroup$ I tried to transform $f_1 : ((0,1)\times(2,3)) \rightarrow (0,2)\times(5,6)$ into $f_1 : A \rightarrow B$, where $A : (0,1)\rightarrow (2,3)$ and $B : (0,2)\rightarrow (5,6)$ and then find functions fitting both sets, but I could not do that. $\endgroup$ – whiskeyo Dec 8 '18 at 16:23
  • $\begingroup$ Hmmm. Can you make just a function that maps (0,1) bijectively to (2,3)? $\endgroup$ – Henning Makholm Dec 8 '18 at 16:33

For $f_2$,The usual euclidean product of sets may be a little confusing here.

Try thinking of "$\mathbb{Z}\times[0,1)$" as the set "To each integer, assign an interval from 0 to 1." Then $f_2(x,t) = x+t$ is a bijection to $\mathbb{R}$, where $x\in\mathbb{Z}$ is the integer part of some real number, and $t\in[0,1)$ is the decimal part of that number.


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