I am wondering what notation I should use when writing that some variable is an integer within some range. What is the most common way to do this? Here are some ideas I have but I'm not sure what the usual convention is.

For example, an integer on the interval [0,29], would I denote it as:

$$ x\in [3,29]\cap \mathbb{Z} $$ or something of the sort like $$ x \in \{ k \mid k \in \mathbb{Z} \wedge k \geq3 \wedge k\leq29\} $$ or some combination of both maybe? I know different standards exist but I'm sure there's a "most common one" which I could use, the first that I wrote is the one I've been using so far in texts for personal re-reading. Is it more useful to write it in words when writing something for others to read?

"Let x be an integer less than 30, greater than 2..." or something similar.

Bonus question:

Does this change when talking about strictly natural numbers on a range, how about non-negative integers? i.e x is in a range of [1, n] or [0, z] (where n is some natural number and z some integer), obviously you can change the set in the example but hey, maybe there's a different convention for those. So something like:

"Let x be a natural number less than 30"

This is a little bit of a messy text but hopefully the question is clear

  • 1
    $\begingroup$ Square bracket notation $[a,b]$ is the one I've seen used, with the author making a one-time, up-front declaration that for the given article, the notation $[a,b]$ denotes $\{x \in \mathbb{Z}\mid a \le x \le b\}$. In any case, if you choose some brief notation, and if it's for public consumption, you should declare it in advance. $\endgroup$
    – quasi
    Jan 29 '18 at 18:04

At least in Belgium, high schoolers get taught that $n \in [a, b]$ is the standard notation for a given range of real numbers, so I wouldn't use it straight away. As others have pointed out though, introducing it beforehand is probably the best way to go from a practical standpoint. In my personal opinion, "Let $x \in \mathbb{N} \cap [a, b]$" is the least ambiguous but to people who grew up with $[a, b]$ it'll probably come over as convoluted.


I personally use $[n,m]$ (introducing it once as a notation beforehand), and have seen it used quite often in papers in my field (theoretical computer science). (Moreover, the notation $[n] = \{1,\dots,n\}$ is also pervasive).

However, as an alternative, according to Wikipedia the notations $[n..m]$ or $\{n..m\}$ seem to be at least semi-standard.


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