Is there a single symbol that represents plus-minus and includes the identity? Is there a single symbol that represents the set of $x\pm 1$ and $x$?
Thus, if $x$ is 8, I'm looking for the most concise way to present the set of $\{7,8,9\}$.
I apologize if this has been asked before; there are many questions on the plus-minus sign, but I couldn't find my answer elsewhere.
 A: If $x\in\Bbb N_{>2}$ then you can use the notation
$$[n]:=\{1,2,\ldots,n\}$$
Hence
$$\{n-1,n,n+1\}=[n+1]\setminus[n-2]$$
In any case it dont seems that it short so much. If you need to denote sets of these kind often then you can previously define a particular notation, by example
$$x_{\{n\}}:=\{x-n+1,x-n,\ldots,x\}$$
A: There's no such symbol in common use (of course, you could define it).
Still, the set of values $$\{(x\pm\tfrac12)\pm\tfrac12\}$$
are the ones in question, where the two ambiguous signs are taken to be independent.
A: From my point of view, the most intuitive and simplest one might be the following:
$$[x-1, x+1]_{\mathbb{Z}}:= [x-1, x+1]\cap \mathbb{Z}$$
':= ' means 'is defined to be'. First, we want three consecutive numbers and this might bring us to the notion of intervals (closed in this case, as we need both ends!). However, we usually talk about intervals in the real number system which is a continuum! That means if we take the notation $[x-1, x+1]$ then it also includes $x-0.5$, $x+0.4$, and $x+\sqrt{2}/2 - \pi/5^2$ etc. All we need to do is to intersect this interval with integers $\mathbb{Z}$. That solves our problem.
