The question is how to construct a bijection between [2, 4] and (0, 1), but is there a general formula to do so?
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2All that’s required is a minor adaptation of the first answer to this question. – Brian M. Scott May 14 '20 at 17:22
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Yeah I understand the solution for both [0, 1] to (0, 1] and from [0, 1] to (0, 1) but somehow I don't seem to find that minor adaptation – Kat W. May 14 '20 at 18:16
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2Just compose with a bijection between $[a,b]$ and $[0,1]$: the linear function that takes $0$ to $a$ and $1$ to $b$ is the simplest. – Brian M. Scott May 14 '20 at 18:18
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I recommend first solving these two simpler problems:
- Find a bijection between $[0,1]$ and $(0,1]$.
- Find a bijection between $[0,1]$ and $(0,1)$.
Problem (1) already contains the essential new difficulty of this set of problems. Problem (2) is a chance to double-down on the new idea from the solution to problem (1), and should lead pretty nicely to a solution to your original problem.
One way to go from problem (2) to your exact problem is to find a bijection between $[2,4]$ and $[0,1]$ (which should be easy, with a continuous function) and then compose the two bijections together.
Greg Martin
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Yeah I understand the solution for both [0, 1] to (0, 1] and from [0, 1] to (0, 1) but somehow I don't seem to find that minor adaptation that's needed here. Thanks for your answer. – Kat W. May 14 '20 at 18:17
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I added another sentence to address this. By the way, the remarks in your comment are really great things to include in the question statement to begin with; providing that context is important for people's feedback to be aimed at the part of the problem that you actually want help with. – Greg Martin May 14 '20 at 20:00