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What is the best way to denote it?

  1. $\forall i\in\{1,\dots,n\}: P(i)$;
  2. $\forall i=1,\dots,n: P(i)$;
  3. $\forall i=\overline{1,n}: P(i)$;
  4. $P(i)$ for $i=1,\dots,n$;
  5. ...
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I find it strange that there is no uniform notation for this common set. Another notation I have seen is $\underline{n}$. – copper.hat Sep 11 '12 at 15:29
I have also seen $[1,n]$ on this site and $[n]$ elsewhere. – Ross Millikan Sep 11 '12 at 15:38
Dem'yanov & Malozemov use $[1:n]$ in "Introduction to minimax". – copper.hat Sep 11 '12 at 15:44
I use $(\forall i\in \mathbb{N}_n)(P(i))$. However in your notations, 1 and 4 are the best. 2,3 are not acceptable. – user59671 Feb 12 '13 at 14:46

Use "$P(i)$ for $i=1,...,n$" to refer to the property being true in those cases. Using existential and universal quantifiers in ordinary mathematical prose (as opposed to formal logic text) is ugly.

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Some mathematicians disagree with excluding quantifies. However there are theorems about logic of quantifiers and some definitions include a long combination of quantifiers (they are frequent in analysis). – user59671 Feb 12 '13 at 14:54

I don't know if I've ever seen the notation $\overline{1, n}$, so I wouldn't use that :) But, maybe others know it, so maybe it's okay. Other than that, I wouldn't use $\forall$ or $:$ ever and most of my experience with such notation is in the class where you first learn it. I would use words but the basic forms of your 1, 2, and 4 seem pretty good.

$P(i)$ is true for all $i$ such that $1 \leq i \leq n$.

$P(i)$ is true for $i = 1, \ldots, n$.

For $i = 1, \ldots, n$, $P(i)$ is true.

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