# Notation: how to construct a set with upper bound and lower bound?

For example,

set $A = \left \{ 1, 5, 10, 30 \right \} \in \mathbb{R}^N$

r = $10$

How do I write down a set which takes $A - r$ as lower bound, and $A + r$ as upper bound, containing ranges?

like:

set $C = (A - r)?(A+r) = \left \{ [-9,11], [-5,15], [0,20], [20, 40] \right \} \in \mathbb{R}^?$

Are there any notation that could describe such $?$ operation?

If I want to know if a number is in $C$

Is it okay to use, for example $(A-r)<m<(A+r)$?

• Something like that would be okay, but it's not clear what the $\dots$ means. Is there a pattern we're supposed to recognize? If so, I don't see it. Does it just mean "and some other intervals?" Is it intended that the sets be infinite? There needs to be some context that addresses these questions. Apr 4, 2018 at 19:30
• @saulspatz I just updated the question, and clearfy my point. the ... doesn't matter, I deleted it. The sets themself don't have patterns.
– null
Apr 4, 2018 at 19:52

I would write $C=\{[a-r,a+r]\mid a\in A\}$ When you say, "I want to know if a number is in $C$," I think you must mean "I want to know if some number is in an element of $C$." The elements of $C$ are intervals not numbers, so it really doesn't make sense to ask if a number is in $C$.
There are (at least) two ways to express this. $$\exists c\in C(m\in c)\\ \text{ or }\\ m\in \bigcup_{c\in C} {c}$$ The question marks are not usually used in math, except on the blackboard. You would write, "Is it true that...?"
If I read the question correctly, you start with a point $A$ in $\mathbb{R}^N$. In your example $N=4$. $A$ is a point in that space, not a set.
What you want is the "cube" in $\mathbb{R}^N$ (same $N$) consisting of all the points each coordinate of which differs from the corresponding entry in $A$ by at most $r$.
In your example you could describe that as the product $$[−9,11] \times [−5,15] \times [0,20] \times [20,40].$$