Can interval notation be used to describe a set? If $x$ is an element of the set of integers between 4 and 14 inclusive, some might say
$ x \in Z \hspace{1mm} , \hspace{1mm} 4\le x\le14 $
or 
$ x \in $ { $4,5,...,14$}
I wonder whether this notation seen in an old textbook is considered valid
$ x \in Z [4,14] $
It seems to have disappeared from modern literature.
 A: I think that the use of
$$\; x \in \mathbb{Z} [4,14]\;\text{ to denote the set of integers:}\;\{4, 5, 6, ..., 14\}$$
has "gone by the wayside" (disappeared) in modern literature for a very good reason, as it is rather ambiguous, and (e.g.) looks far too similar to $\mathbb{Z}[p]$, which has an entirely different meaning than what you intend it to denote. (Also, as Sigur comments below, $\mathbb{Z}[a, b]$ can be used to denote the polynomial ring on variables $a, b$.)
The other notations you mention are much less ambiguous, as is $x\in \mathbb{Z} \cap [4, 14]$
A: Given a partial ordered set (or poset) $(P,\leq)$ and $x,y\in P$, the interval $[x,y\textbf{]}$ is defined as $(\mathord{\uparrow}x)\cap(\mathord{\downarrow} y)$, where $\mathord{\uparrow} x=\{p\in P:x\leq p\}$ and $\mathord{\downarrow}y=\{p\in P: p\leq y\}$. Note that $P$ doesn't even need to be totally ordered for intervals to exist.
Specifically about your question, if you make it clear that you're working on the poset $(\mathbb{Z},\leq)$, there shouldn't be any problem with that notation.
