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

Especially Lime
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Tina
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    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. – Bilbottom May 24 '18 at 07:33
  • When one need to deal with first $n$ natural numbers or first $n$ positive integers a lot, some authors prefer to use $[n]$ as a short hand of ${ 0, 1, \ldots, n-1 }$ or ${ 1, 2, \ldots, n }$. As long as you define it ahead of its use, everything is fine. – achille hui May 24 '18 at 07:36
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    $x\in {1,...., 50}$ 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. – fleablood May 24 '18 at 07:38
  • @mvw is there any reason why you do that? – Tina May 24 '18 at 07:48
  • @fleablood, ok thanks, so what I have done is not standard, I quite like $x \in \mathbb N_1^{50}$, but I forgot I could do x∈{1,....,50} – Tina May 24 '18 at 07:49
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    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. – adfriedman May 24 '18 at 08:00
  • I read integers in your first line. But now I see you wanted only positive ones in your title. – mvw May 24 '18 at 08:42
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    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. – AakashM May 24 '18 at 09:26
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    Possible duplicate of Notation for integer between two values – Bence Mélykúti May 24 '18 at 10:00
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    @fleablood: I always write x ∈ [1, 50] ∩ ℤ. To me this notation feels very natural (and easy to read and understand). – Andreas Rejbrand May 24 '18 at 13:13
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    The goal of writing is not to write as concisely as possible. The goal is to be understood. – Najib Idrissi May 24 '18 at 15:09
  • You could say $x \in { n }_{n=1}^{50}$. – Steven Alexis Gregory May 25 '18 at 06:21
  • @AakashM Are you a former Fortran programmer? – badjohn May 25 '18 at 14:57
  • @badjohn no, just an amateur mathematician. But that said, i j and k are still my first choices for loop variables... – AakashM May 29 '18 at 07:32
  • @AakashM I am not sure whether Fortran started the habit or just followed an earlier habit. Its variables did not need to be declared and were assumed to be INTEGER if they started with I through N and REAL (float) if they did not. i, j, k are very popular loop variables and my first choices as well though I know some programmers who use X, Y, and Z. – badjohn May 29 '18 at 08:07
  • $x=1,2,\dots,50$, not so rigorous but commonly used – Sil Jun 03 '18 at 13:26

8 Answers8

18

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

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

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

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

Tesla
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    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 '18 at 07:47
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    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. – Tesla May 24 '18 at 07:51
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    @Tinatim the second option is the best – minseong May 24 '18 at 09:50
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    The third notation could be understood as a cartesian product. Couldn't it? – Eric Duminil May 24 '18 at 12:12
  • Yes it could, that is why it is worth defining it when using it quite often in the same paper or similar. Any variation is possible if defined appropriately I guess as long as it does not intersect "too much" with conventional notation. – Tesla May 24 '18 at 13:44
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    Although I like the second option a lot, I am now wondering what would happen with $x \in {1, 2, \ldots, 64}$. – Andrea Lazzarotto May 24 '18 at 14:03
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    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. – Tesla May 24 '18 at 14:40
  • I think ${1,....,50}$ being acceptable is the exception, not the rule. And if anyone were to say ${1,....,50}$ should not be acceptable I wouldn't argument. $x\in {1,2,4,...,64}$ is not acceptable as the reader shouldn't have to second guess and it is the responsible of the writer to be specific. I inconsistently make an exception for ${1,..,50}$ because any technical definition seems to heave. Bascially I think we can define ... to mean all unit incremental, but we can't have it mean "and you know what I mean". – fleablood May 24 '18 at 15:18
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    @fleablood: Human mathematicians aren’t computer parsers. We can and do quite comfortably use notations that mean “you know what I mean”, and leave it to good judgement to judge when that’s in danger of causing ambiguity. – Peter LeFanu Lumsdaine May 24 '18 at 18:21
  • $x \in \mathbb N_1^{50}$ --> WHY ONLY NOW DO I KNOW THIS?!?!?!! – BCLC May 24 '18 at 18:31
  • @PeterLeFanuLumsdaine I agree, but it's subjective. $x \in {1,2,4,...,64}$ is probably ${2^i|i \in \mathbb N}$ but it's not unequivicably unambiguously certain. I'd definitely put it in danger of causeing ambiguity range. – fleablood May 24 '18 at 18:47
  • The first one is overly convoluted. You can just write $x \in \mathbb N, 1 ≤ n ≤ 50$ – Acccumulation May 24 '18 at 19:12
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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 $\mathbb{R}^+=\{x∈\mathbb{R}|x⩾0\}$.

Evpok
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    That's funny, I also didn't know it was specific to France. It doesn't seem to collide with any other notation. – Eric Duminil May 24 '18 at 13:37
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    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. – Jeppe Stig Nielsen May 24 '18 at 14:28
  • @JeppeStigNielsen Yes, at that point, I stopped reasserting “this is the French convention” :) – Evpok May 24 '18 at 14:30
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    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. – Jeppe Stig Nielsen May 24 '18 at 14:37
  • I can attest to this notation being standard in France and misunderstood (in my experience) in anglophone Canada. Doesn't stop me from using it ;-) – Alexandre Cassagne May 24 '18 at 14:59
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    Another convention in French that does not apply to English is to precede an exclamation point by a space ;) – Carsten S May 24 '18 at 15:51
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    @CarstenS It is ! A thin unbreakable space to be precise, and I will fight for it to the death ! – Evpok May 24 '18 at 17:58
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    @JeppeStigNielsen: Why is that going far? I use that all the time, and it's only one extra character than the plus version! =) – user21820 May 25 '18 at 06:07
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    @CarstenS: Not just exclamation points, but any punctuation symbol which has two parts : ! ? ; :. It makes it easy to remember : 2 spaces around ! ? ; :, 1 space after , .. – Eric Duminil May 25 '18 at 08:37
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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).

Especially Lime
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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.

Eric Towers
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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 $[11,50]$.

wlad
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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.

user21820
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One possibility is $\{i\}_{i = 1}^{50}$, by analogy with $\sum_{i = 1}^{50}(\cdots)$ and other similar notation.

Hammerite
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Another fancy way of writing the set is this one:

enter image description here

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.

Newton fan 01
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