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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?

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    $\begingroup$ 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. $\endgroup$
    – Mathmo123
    Commented Oct 29, 2014 at 8:37
  • $\begingroup$ Personally, I'd write $[a,b]\subset \mathbb{Z}$ the first time, and just $[a,b]$ thereafter. $\endgroup$
    – FireGarden
    Commented Oct 29, 2014 at 8:39
  • $\begingroup$ @FireGarden But $[a,b] \not\subset \mathbb Z$!! $\endgroup$
    – Mathmo123
    Commented Oct 29, 2014 at 8:40
  • $\begingroup$ @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. $\endgroup$
    – FireGarden
    Commented Oct 29, 2014 at 8:44
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    $\begingroup$ @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. $\endgroup$
    – Mathmo123
    Commented Oct 29, 2014 at 8:46

1 Answer 1

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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 $[[n]]$.

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    $\begingroup$ In the UK, I have only ever seen the notation $[[n]]$, never $[[a,b]]$. And even then, only in the world of combinatorics. $\endgroup$
    – Mathmo123
    Commented Oct 29, 2014 at 8:40
  • $\begingroup$ I don't think I have ever seen this notation before (maybe once or twice without realizing its exact meaning), but it looks quite reasonable. $\endgroup$
    – Jake
    Commented Oct 29, 2014 at 8:40
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    $\begingroup$ 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 $\mathbb Z_{a\dots b}$ $\endgroup$
    – davcha
    Commented Oct 29, 2014 at 8:50
  • $\begingroup$ Indeed, that's what I will probably do for $\left[\left[a,b\right]\right]$. $\endgroup$
    – Jake
    Commented Oct 29, 2014 at 8:54

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