Can you Use $x \in S$ with Interval Notation? If I have some variable $x$ and I want to show that it takes values from the set of integers $\{1, 2, 3, \dots, n\}$, is it correct to write the following:
$$x \in [1, n]$$
 A: The default notation is that $[1,n]$ is an interval in the set of real numbers:
$$[1,n] = \{x \in \mathbb R \mid 1 \le x \le n\}
$$
and this includes non-integer numbers such as $x = 1.5$ (assuming $n \ge 2$). So no, that's not correct (unless you explained very carefully and very clearly, in what you are writing, what your intention is for the notation $[1,n]$).
What would be correct is to use the intersection operator $\cap$ and to write $x \in [1,n] \cap \mathbb N$, using the standard notation $\mathbb N$ for the set of natural numbers (a.k.a. the positive integers).
However, it is common notation to literally use the ellipsis in this situation,
$$\{1,...,n\} = [1,n] \cap \mathbb N
$$
If you used that in your mathematical writing, you would be almost universally understood.
A: Seth Warner, in his "Modern Algebra" (1965) is often criticised for using unconventional and confusing notation, but it is worth noting that in Section 16 he writes:
"An integer interval is any set $[m, n]$ where $m$ and $n$ are natural numbers satisfying $m \le n$."
Hence yes you can is the answer to your question.
However, If I were you I would be careful in my exposition and (the first time using the notation) say "$x \in [1, n]$ where $[1,n]$ denotes the integer interval $\{1, 2, \ldots, n\}$."
A: Hint: A commonly used notation is
\begin{align*}
[n]:=\{1,2,\ldots,n\}
\end{align*}
This way we can conveniently write
\begin{align*}
x\in[n]
\end{align*}
avoiding interval notation.
