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Soft question and just starting to learn measure theory. So please be kind, and if you want the question gone, just place a comment.

I got stuck in trying to write on a problem of combinations with replacement (the context not important) the following statement: "Choose $4$ digits from $10$ different options: $(0,9)$".

Straightforward as it is, I started to double-guess myself... Should I have written, $[0,9]$ (closed interval). But, wait, it's not an interval, because it is not on the real line (?): we are choosing just among the integers... What about something like $\{x\in\mathbb N\,|\,0\leq x\leq9\}$. No... So many reasons why not... What about $\{0,9\}$?

So I don't know if the question is worth anyone's time or not, but I don't have an answer.

Antoni Parellada
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    Is your question on how best to write the set of natural numbers from $0$ to $9$? I would go with ${0,1,\cdots,9}$. – Sloan May 11 '16 at 14:38
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    Not ${ 0,9 }$; it is the set with only $0$ and $9$. – Mauro ALLEGRANZA May 11 '16 at 14:38
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    I would argue that ${0,1,\dots,9}$ would make the most sense. Definitely not ${0,9}$ as @MauroALLEGRANZA points out.

    As for an interval, you could write $[0,9] \cap \mathbb{N}$ I suppose, though this kind of notation is more often used for rational numbers within an interval than integers/naturals.

     – sTertooy May 11 '16 at 14:39
  • @Sloan Am I write in assuming that brackets would imply $\mathbb R$? – Antoni Parellada May 11 '16 at 14:40
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    The notation $[n]$ is fairly common for ${1, 2, \ldots , n}$ in combinatorics, but any particular notation for a "shift" of this set does not seem very common. – pjs36 May 11 '16 at 14:41
  • @MauroALLEGRANZA If one were to insist on curly brackets, then, would it be OK to write something completely pedantic like, ${[0,9]}$? – Antoni Parellada May 11 '16 at 14:42
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    ${[0,9]}$, to me, reads as the set with $1$ element, that element being the interval $[0,9]$. – Sloan May 11 '16 at 14:45
  • @Sloan Interval on the real line, right? Nobody would think of integers, correct? – Antoni Parellada May 11 '16 at 14:46
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    I can't speak for everyone, but $[0,9]$ as a closed interval on $\mathbb{R}$ is the most likely interpretation. – Sloan May 11 '16 at 14:47
  • @Sloan And if you were to read out load in a very exacting manner what your proposed notation at the beginning spells out, would you say that ${0,1,\cdots,9}$ is "the *set* of natural number from 0 to 9"? – Antoni Parellada May 11 '16 at 14:51
  • $[0,9]\cap\Bbb Z$ might be fine as well (or $[0,10)\cap \Bbb Z$) – Hagen von Eitzen May 11 '16 at 15:01

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