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Is there any convention for a notational shorthand for the set $\{1,\ldots,n\}$ (defined as $\{k\in\mathbb{N} \mid k \le n\}$), where $n\in\mathbb{N}$, that the majority of mathematicians are familiar with?

I find that in some cases in which these sets appear often in the same expression, which can reduce readability, or at least aesthetic cleanness; using some sort of abbreviation would alleviate that.

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    $\begingroup$ In combinatorics it is sometimes written as $[n]$. $\endgroup$
    – Mark
    Commented Jan 7, 2019 at 19:48
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    $\begingroup$ In combinatorial settings $[n]=\{1,2,\ldots ,n\}$ is commonly used. $\endgroup$
    – Anurag A
    Commented Jan 7, 2019 at 19:49
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    $\begingroup$ Sometimes, $\overline{1,n}$ is used. $\endgroup$
    – Litho
    Commented Jan 7, 2019 at 20:04

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I don't know how popular this is but I've seen the convention: $$[n]\equiv\{1,2,3,4,\ldots n\} $$

See for example: http://www.math.cmu.edu/~lohp/docs/math/mop2013/combin-sets-soln.pdf

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In homotopy theory, both $[n]$ and $\mathbf{n}$ are common and, to a lesser extent, $\underline{n}$. None of this matters too much, as long as you define your choice of notation clearly in your writing.

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It depends on the context, but a couple of equivalent formulations I've seen:

  • You could say $\{k\}_{k=1}^n$. I saw this often when considering sets of data points, like below, but I see no reason the notation couldn't extrapolate to any set.

$$\{(x_1,y_1) \; , \; (x_2,y_2) \; , \; ... \; , \; (x_n,y_n)\} = \{(x_i,y_i)\}_{i=1}^n$$

  • In combinatorics, apparently $[n]$ can be used to represent $\{1,...,n\}$ as touched on in the comments and by Archimedesprinciple.
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    $\begingroup$ In general the notation $\left\{ f(k) \right\}_{k = 1}^{n}$ is used to denote a Sequence rather than a Set per sae. There is no absolute way of defining a set but conventional analysis texts tend to use the notation that you have used. $\endgroup$
    – user150203
    Commented Jan 8, 2019 at 4:40

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