Is there a quick way to write say positive integers in an interval in mathematical notation? For example I find myself wanting to write $x$ is an element of the integers from $1$ to $50$,
Is this the quickest way? 
$x\in \left[ 1,50\right] \cap \mathbb{N} $
Also is this standard on here? $\mathbb{N} = \{0, 1, 2,\dotsc \}$,
$\mathbb{ℤ}_+ = \{1, 2, \dotsc \}$.
 A: Anyone will understand
$$n\in\{1,2,\dots50\}$$ 
or even
$$n\in\{1,\dots50\}$$ 
without toil.
If it is clear from context that $n$ is an integer,
$$n\in[1,50]$$ is good enough (and is very compact from the standpoint of LaTeX formatting :) ).
Following @EspeciallyLime, $[50]$ is a good option, though you should introduce the notation. This remains compatible with more general intervals like $[11,50]$.
A: As others have said, you should always define non-standard notation, but here is one that you can consider (and is actually valid syntax in some programming languages):

$[a\,..b]$ represents the integers from $a$ to $b$ inclusive.

This is also compatible with the convention for square/round-brackets to denote closed/open interval endpoints:

$[a\,..b)$ represents the half-open interval from $a$ to less than $b$.

Though mixed-bracket interval notation might best be avoided.
A: It depends on your own preference on how to write things down, there are countless variations, for example
$x \in \{ n \in \mathbb N : 1 ≤ n ≤ 50\}$
$x \in \{1,2,...,50\}$
$x \in \mathbb N_1^{50}$  
A: A common convention in French is
$$
x∈⟦1, 50⟧
$$
and I am genuinely surprised to learn that it might not be common elsewhere ! In any case, $\{1, …, 50\}$ or maybe $\{1, 2, …, 50\}$ should be universal and more readable for most people.
For your other question, still from the French perspective,
$$
\mathbb{N} = \{0, 1, …\}\\
\mathbb{N^*} = \{1, 2, …\}\\
$$
though the second one is sometimes frowned upon due to it being an abuse of the $A^*$ notation (where $A$ is a ring) that leads to confusion for the $\mathbb{Z}^*=\{-1, 1\}$ case.
I have never seen $\mathbb{Z}^+$ used, but if I had, I would probably have assumed $\mathbb{Z}^+=\mathbb{N}$, following $\mathbb{R}^+=\{x∈\mathbb{R}|x⩾0\}$.
A: For the specific case that you start at $1$, it is fairly standard in combinatorics to write $[n]$ for $\{1,\ldots,n\}$, so $x\in[50]$ would work. This doesn't really help for other ranges, though - you could write $x\in[50]\setminus[10]$, but you probably shouldn't :)
To answer your other question, I prefer $\mathbb N$ to be $\{0,1,\ldots\}$ and $\mathbb Z_+$ to be $\{1,2,\ldots\}$, but there is no consensus on the first, and it's probably safer to write $\mathbb N_0$, which is unambiguous. Even $\mathbb Z_+$ could be misinterpreted, but I think when writing in English it's standard that this does not include $0$ (when writing in French, I'd expect the standard to be different, but I have no first-hand knowledge of this).
A: I do wonder why so many people believe convoluted notation is better than plainly writing what you mean.
"Let $x \in \mathbb{N}$ with $1 \leq x \leq 50$."
The twin purposes of notation are clarity and precision.  Use of new or rare notation subverts both.  Excessive density subverts clarity.  Use of a single natural language word for exactly its meaning is both clear and precise.
A: One possibility is $\{i\}_{i = 1}^{50}$, by analogy with $\sum_{i = 1}^{50}(\cdots)$ and other similar notation.
A: Another fancy way of writing the set is this one:   

I got this idea when reading Hammerite's answer. However, the formulas are different. Or at least, I hope so. I have never encountered his notation so far, but if it is equivalent to set union, please tell me, in order to delete my answer. However, this is the standard notation for union of sets, the one that I posted.
