# What is a good way to compactly write that a number is an integer between a and b?

Specifically, I refer to the following set:

$$\left\{ x\in\mathbb{Z}\mid a\leq x\leq b\right\}$$ where $a\in\mathbb{Z}$ and $b\in\mathbb{Z}$ such that $a<b$.

Alternatively, this can be written as $\mathbb{Z}\cap\left[a,b\right]$, but it still looks a bit ugly.

I am looking for a more compact notation, such as perhaps $\mathbb{Z}_{a}^{b}$. The problem with this is that it is ambiguous, as it can be interpreted as a $b$-dimensional vector space over $\mathbb{Z}_{a}$.

Is there perhaps a good way to compactly write this in a formula?

• The most common way that I've seen is $\{a, a+1, \ldots, b\}$. It will strongly depend on the context - a notation like $\mathbb Z \cap [a,b]$ would to me suggest something more topological. In general your aims should be first clarity, and only then compactness. Commented Oct 29, 2014 at 8:37
• Personally, I'd write $[a,b]\subset \mathbb{Z}$ the first time, and just $[a,b]$ thereafter. Commented Oct 29, 2014 at 8:39
• @FireGarden But $[a,b] \not\subset \mathbb Z$!! Commented Oct 29, 2014 at 8:40
• @mathmo123 It is if you say so.. how is it any different than the set you explicitly wrote? It says the elements from a to b inclusive, as a subset of Z; i.e., integers from a to b inclusive. Commented Oct 29, 2014 at 8:44
• @FireGarden The set $[a,b]$ (which we normally view as a subset of $\mathbb R$) is not a subset of $\mathbb Z$, so to say $[a,b] \subset \mathbb Z$ would be abusing notation slightly. Writing $\mathbb Z \cap [a,b]$ would make sense, but as the OP remarked, is clunky. Commented Oct 29, 2014 at 8:46

The notation $[[a,b]]$ for ${\mathbb Z}\cap [a,b]$ is quite well spread, at least in French litterature. For the special case of $\{1,\dots,n\}$, combinatorists often use $[[n]]$.
• In the UK, I have only ever seen the notation $[[n]]$, never $[[a,b]]$. And even then, only in the world of combinatorics. Commented Oct 29, 2014 at 8:40
• He can probably state something like : "The notation $[[a,b]]$ denotes the set $\{x\in \mathbb Z: a \leq x \leq b\}$." in a notation section. Or anything... Like for example $\mathbb Z_{a\dots b}$ Commented Oct 29, 2014 at 8:50
• Indeed, that's what I will probably do for $\left[\left[a,b\right]\right]$.