Is there a short mathematical notation for all integers less than $n$ where $n$ itself is some integer? The only thing that comes to mind is $$\mathbb{Z} \cap (-\infty, n),$$ But this is pretty ugly and involves reals. Is there a better way to do this symbolically, or should I just describe such set with words?
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2$\begingroup$ $$\Bbb Z_{<n}$$ $\endgroup$ – user63181 Mar 16 '14 at 6:00
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$ \mathbb{Z}^{< n} $ is sometimes used (I personally like the notation $ \mathbb{Z}_{< n} $ more as I like to reserve the right upper corner for powers, but I don't think I have seen others use it). The $< n$ in the top right can look a little bit weird, because there is no variable on the left side.
Another option is $ k \in \mathbb{Z} $, with $ k < n$. It is also pretty common to just write "for all integers less than $n$" as you did. I think these two options are the most commonly used.
Also, you could write 'for $k \in \{ j \in \mathbb{Z} : j < n \}$', but I don't think many authors would do so. I'm sure there are some other possibilities, but these are the most common ones I can think of.
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$\begingroup$ I didn't know about the subscript notation, that's pretty neat. Thanks. $\endgroup$ – Phonon Mar 16 '14 at 6:13
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$\begingroup$ I don't know how common its use is, though. I just saw it used in the syllabus for one course. I'm hoping some other people can shine some more light on this. $\endgroup$ – Ruben Mar 16 '14 at 6:16
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I would use
$$ \{ x \in \mathbb Z \mid x < n \} . $$
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I have seen the notation $[k<n]$ used several times, first in a combinatorics class when I was an undergrad, and then in another place. By induction, it should hold for all authors.
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1$\begingroup$ I would think this is the Iverson bracket, denoting $1$ is $k<n$ and $0$ otherwise. As a function of $k\in\Bbb Z$ that would be the characteristic function of the set in the question, but noting in the notation you wrote says that $k$ is variable, or that it ranges over $\Bbb Z$. $\endgroup$ – Marc van Leeuwen Apr 3 '14 at 21:04
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Maybe (almost surely) it's not so standard, but sometimes I saw the use of $\bar{n} $ to indicate the set of natural numbers less or equal to $ n $. I suppose there is an analogue notation for integers.
For references: I found this notation in several random pdfs downloaded from Internet, not "ufficial" books (at least book i know). I addee this answer because I found the notation suitable for someone who doesn't want to write "long" formulas :)
