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)
- 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.
- This option uses $[1..w]$ for integers, $[0,w]$ for real numbers, and eventually $[1..w] \times [1..n]$ for 2D integer intervals.
- 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.
- 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?