# 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? $\mathbb{N} = \{0, 1, 2,\dotsc \}$, $\mathbb{ℤ}_+ = \{1, 2, \dotsc \}$.

• You can always define your own notation if you state it clearly in your paper/workings. I like to use the notation $I_{m}^{n}$ for the set of integers between $m$ and $n$, so for example $I_{0}^{3} = \{0, 1, 2, 3\}$. The use of $I$ in this way is sufficiently uncommon that it doesn't cause confusion with any 'standard' notation. May 24, 2018 at 7:33
• $x\in \{1,...., 50\}$ is the quickest way and is generally understand. If you really want to worry that it is informal and imprecise, you could say $x \in \mathbb N; 1\le x \le 50$. $x\in [1,50]\cap \mathbb N$ is technically correct but I doubt there's anyone on the planet how wouldn't think that is obtuse and weird. May 24, 2018 at 7:38
• I’d be cautious with $\mathbb{N}_1^{50}$. To me it looks like a Cartesian product of 50 $\mathbb{N}_1$’s. Perhaps the issue is more apparent with $\mathbb{N}^2$, which should be unambiguously $\mathbb{N}\times\mathbb{N}$, but here it isn’t. May 24, 2018 at 8:00
• When I see $x$ I think real, not integer. Conversely, if I see $n$ I automatically think integer and would be surprised to be told real is meant. May 24, 2018 at 9:26
• The goal of writing is not to write as concisely as possible. The goal is to be understood. May 24, 2018 at 15:09

It depends on your own preference on how to write things down, there are countless variations, for example

$x \in \{ n \in \mathbb N : 1 ≤ n ≤ 50\}$

$x \in \{1,2,...,50\}$

$x \in \mathbb N_1^{50}$

• Thanks $x \in \mathbb N_1^{50}$ is very succinct, is that commonly used, as I have not seen that and I like the look of it!
– Tina
May 24, 2018 at 7:47
• As mentioned in the comments, you should define it before using that notation which only makes sense if you use more than a handful of times after defining it. Otherwise I would suggest using the second option. May 24, 2018 at 7:51
• @Tinatim the second option is the best May 24, 2018 at 9:50
• Although I like the second option a lot, I am now wondering what would happen with $x \in \{1, 2, \ldots, 64\}$. May 24, 2018 at 14:03
• I think for cases where your set is defined by a sequence (other than the next element being 1 larger than the previous one) you should provide sufficient elements such as $x \in \{1,2,4,...,64\}$. Again, there are many other efficient/non-efficient ways to write it down. May 24, 2018 at 14:40

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 $\mathbb{R}^+=\{x∈\mathbb{R}|x⩾0\}$.

• That's funny, I also didn't know it was specific to France. It doesn't seem to collide with any other notation. May 24, 2018 at 13:37
• In many places outside France, $\mathbb{R}^+$ would be the open interval of positive numbers, that is the strictly positive numbers. In France, I think the number zero is considered both positive and negative, while in many other places zero is considered neither positive nor negative. May 24, 2018 at 14:28
• Some people will go as far as to write $$\mathbb{R}_{>0}$$ or $$\mathbb{R}_{\ge 0}$$ to try to make sure the reader knows whether zero is included or not. May 24, 2018 at 14:37
• Another convention in French that does not apply to English is to precede an exclamation point by a space ;) May 24, 2018 at 15:51
• @CarstenS It is ! A thin unbreakable space to be precise, and I will fight for it to the death ! May 24, 2018 at 17:58

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).

• This is also pretty much the standard across computer science theory papers. May 24, 2018 at 7:54
• In French (or at least in France) $\mathbb{N}$ always contains $0$, and $\mathbb{N}^*$ is the notation for $\{1,2,3\dots\}$. May 24, 2018 at 15:06

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.

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 $[11,50]$.

• I personally would add a comma after the dots.
– mvw
May 24, 2018 at 8:44
• And I would use $\dotsc$ instead of $\dotsb$.
– mvw
May 24, 2018 at 8:46
• @mvw: a matter of taste. The regexp pattern would be (number\,)*number, the comma is taken in the repetition.
– user65203
May 24, 2018 at 8:48
• I changed $\cdots$ to $\dots$. Feel free to undo it if you're not happy.
May 24, 2018 at 14:58

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.

One possibility is $\{i\}_{i = 1}^{50}$, by analogy with $\sum_{i = 1}^{50}(\cdots)$ and other similar notation.

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.

• Please, for the sake of all that is holy, never do this. Jun 3, 2018 at 8:17