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I've received a problem in my course that has confused me a bit and I'm not sure what my instructor means or what I am allowed to assume. The problem reads like this:

"*Find a bijection between the following sets: any two finite intervals of the type $[a,b]$ and $[\alpha,\beta]$ with $a<b$ and $\alpha<\beta$".

If I understand correctly, there can't be a bijection between two finite sets unless they have the same amount of elements (or the same cardinality, but we haven't defined that yet) and if I were to assume this it would be trivially simple because then I could just say that there are $n$ elements in both and map $a$ to $\alpha$ and $b$ to $\beta$ and so forth and I'd be done.

This seems almost too simple and I feel as though I've missed something that would make the exercise more difficult. Have I solved it or is there more to it? How do I see it and how do I go about creating the bijection?

Hagen von Eitzen
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Anton
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    Those are intervals of the real numbers. "Finite" should better be replaced with "bounded". – Hagen von Eitzen Oct 14 '16 at 17:30
  • Make sure you are not confusing the notation $[0,1]$ meaning "the infinite set of real numbers between 0 and 1" with the notation ${0,1}$ meaning "the finite set consisting of just two elements, 0 and 1". Punctuation matters! (as does context) – Erick Wong Oct 14 '16 at 17:36
  • @ErickWong This is unfortunately exactly how it was stated in the assignment. I assume that we do not write [0, 10] to mean the set of integers 0, 1, 2, 3, ... , 10? Is it otherwise not possible that this was what my instructor meant? – Anton Oct 14 '16 at 17:39
  • @Anton I am not saying the assignment is in error, I'm saying you're reading it incorrectly. $[0,10]$ is not generally used to mean the set of integers, but every number between $0$ and $10$, including $2.5$ and $\pi$ (and of course many more). – Erick Wong Oct 14 '16 at 17:52
  • @ErickWong Ok, gotcha thanks for clarifying that! – Anton Oct 14 '16 at 18:01

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Hint: Find the linear map sending $a$ to $\alpha$ and $b$ to $\beta$ and show that it is a bijection.

Sri-Amirthan Theivendran
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