# Short notation for intervals of real and natural numbers

Are there short, common, and intuitive notations for intervals of real and natural numbers that explicitly use $\mathbb R$ and $\mathbb N$ as basis of their notation?

The main purpose of the notation is denote and distinguish between discrete and continuous intervals.

Currently, based on the following notation examples

$$\mathbb{R}^{>a}=\{x\in\mathbb{R} \mid x>a\} \\ \mathbb{R}^{\neq a}=\{x\in\mathbb{R} \mid x\neq a\} \\ \mathbb{N}^{+}=\{x\in\mathbb{N} \mid x>0\}$$

I would like to define intervals of real numbers and natural positive numbers -- and explicitly distinguish between real and natural numbers -- as follows.

$$\mathbb{R}^{[0,1]}=\{x\in\mathbb{R} \mid 0\leq x \leq 1\} \\ \mathbb{N}^{[1,w]}=\{x\in\mathbb{N} \mid 1\leq x \leq w\}$$

I prefer this notation over $[0,1]$ and $\{1,2,...,w\}$, since it is short, emphasizes the distinction between $\mathbb R$ and $\mathbb N$, and allows me to use the same interval notation for both types of numbers. I also regard it as more explicit than, e.g., $[0,1]$, which may be misinterpreted as $\{0,1\}$ by the occasional non-mathaffine reader, especially since $0$ and $1$ represent the possible values of a bit.

Currently, I would define this notation at the beginning of my thesis and do not use it for other purposes, such as "the set of all functions from the set $[0,1]$ into the set $R$" [1].

I also use the notation for defining two-dimensional ranges as follows.

$$\mathbb{N}^{[1,w]}\times \mathbb{N}^{[1,h]}$$

Are there better, i.e., short, well-known, and intuitive interval notations for the described purposes?

Edit (considering other alternatives):

Goal. I want to denote, i.e., name the following sets to repeatedly refer to them in my thesis (see new naming ideas in preceding braces).

• The set natural numbers $\{1,2,...,w\}$ -- ($\mathbb N _w$)
• The set natural numbers $\{1,2,...,h\}$ -- ($\mathbb N _h$)
• The range of real numbers $[0,w]$ -- ($\mathbb R _w$)
• The range of real numbers $[0,h]$ -- ($\mathbb R _h$)
• The range of real numbers $[0,1]$ -- (just use $[0,1]$, as $\mathbb R _{1}$ would be too confusing)
• The set of two-tuples $(x,y)$ with $x\in [0,w], y\in[0,h]$ -- ($\mathbb R _w \times \mathbb R _h$)
• The set of two-tuples $(x,y)$ with $x\in \mathbb N \cap [1,w], y\in \mathbb N \cap [1,h]$ -- ($\mathbb N _w \times \mathbb N _h$)

Moreover, I also just found $[1.. w]$ to denote integer intervals, which seems much more convenient than $\mathbb N \cap [1,w]$, i.e., is easier to read, as it does not require you to process the set operation.

Options. As a result, my notation options are the following (presented as example text, to allow for evaluation of readability)

1. This option uses $\mathbb N \cap [1,w]$ for integers, $[0,w]$ for real numbers, and eventually $\mathbb N \cap [1,w] \times \mathbb N \cap [1,n]$ for 2D integer intervals.
2. This option uses $[1..w]$ for integers, $[0,w]$ for real numbers, and eventually $[1..w] \times [1..n]$ for 2D integer intervals.
3. This option uses $\mathbb N _w$ for integers, $\mathbb R _w$ for real numbers, and eventually $\mathbb N _w \times \mathbb N _h$ for 2D integer intervals.
4. This option uses $N _w$ for integers, $R _w$ for real numbers, and eventually $N _w \times N _h$ for 2D integer intervals.

Evaluation

• Option 1 is hardly readable (does not easily convey the message).
• Options 2 to 4 are OK.
• Options 3 and 4 are a little more readable (but need to introduced once).
• Option 4 moreover avoids redefining existing interpretations of $\mathbb N _?$ and $\mathbb R _?$.

Question: Did I overlook something, i.e., what are the potential problems of Options 2, 3, and 4?

• But they are typographically inconvenient... – Benicio Sep 11 '15 at 8:58
• Try $\mathbb{N} \cap [1, w]$. – Benicio Sep 11 '15 at 8:58
• $\Bbb R^{[0,1]}$ already has a meaning; it's the set of functions from $[0,1]$ to $\Bbb R$. (I think that $[a,b]_A$ should be a notation for $[a,b]\cap A$, but I doubt that would ever be widely accepted.) – Akiva Weinberger Sep 11 '15 at 11:33

My currently preferred notations are listed below. However I have not understood your general problem yet, which might deserve a new notation.

$$\mathbb{R}^{>a}=\{x\in\mathbb{R} \mid x>a\}$$

$(a, \infty)$

$$\mathbb{R}^{\neq a}=\{x\in\mathbb{R} \mid x\neq a\}$$

$\mathbb{R} \setminus \{ a \}$

$$\mathbb{N}^{+}=\{x\in\mathbb{N} \mid x>0\}$$

If $\mathbb{N}$ starts at $1$: $\mathbb{N}$ in contrast to $\mathbb{N}_0 = \mathbb{N} \cup \{ 0 \}$

If $\mathbb{N}$ starts at $0$: $\mathbb{N} \setminus \{ 0 \}$

$$\mathbb{R}^{[0,1]}=\{x\in\mathbb{R} \mid 0\leq x \leq 1\}$$

$[0,1]$ or maybe $\mathbb{R} \cap [0,1] = [0,1] \cap \mathbb{R}$. If I see nothing mentioned I usually assume $\mathbb{R}$ as base set, like with the first case.

$$\mathbb{N}^{[1,w]}=\{x\in\mathbb{N} \mid 1\leq x \leq w\}$$

$\mathbb{N} \cap [1,w] = [1,w] \cap \mathbb{N}$

I see sometimes $[\![1,n]\!]$ instead of $\{1,\dots,n\}.$

• OK, but like $[1..n]$ this may not be known to the untrained reader and may be easily misread as a real numbered interval $[1,n]$. – Juve Jan 31 '16 at 12:03
• In France it is often used in probability books : I agree that it could be an "official" notation only in this particular context. – Balloon Jan 31 '16 at 12:29

In my document, I now use the following notations and conventions.

1. I use $[1,w]$ for real numbered intervals and $\{1,2,...,w\}$ for integer intervals.

2. If I reuse a specific interval several times, I introduce and name it once, e.g., as $\mathbb{N}_w=\{1,2,...,w\}$, and then refer to it using its name, e.g., $\mathbb{N}_w$.

3. Thereby, throughout the document, $\mathbb{N}_n$ always denotes an integer interval from $1$ to $n$, while $\mathbb{R}_n$ always denotes a real numbered interval between and including $0$ and $n$.

4. However, I do not use this notation unless having introduced each specific $\mathbb{N}_?$ and $\mathbb{R}_?$ explicitly.

Using short names for my common intervals, I can now easily and consistently denote multidimensional ranges, e.g., $\mathbb{R}_w \times \mathbb{R}_h$, or express membership, e.g., $x \in \mathbb{N}_w, y \in \mathbb{N}_h$.

I believe that using a subscript index for $\mathbb{R}$ and $\mathbb{N}$ is safe enough and creates the least confusion with other common notations. I now prefer the longer notation $\{1,2,...,n\}$ over $[1..n]$ as it is more common, and over $\mathbb{N}\cap[1,n]$ as it does not require the reader to process the expression, i.e., as it is more explicit.