# Notation convention for $\{1,\ldots,n\}$

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.

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

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

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.

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.
• 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.
– user150203
Commented Jan 8, 2019 at 4:40