Notation for an interval when you don't know which bound is greater

Is there a notation in English written mathematics for $$\textit{the interval of all points lying between two real numbers $a$ and $b$}$$ when you don't know which of $a$ and $b$ is greater?

Which one is greater is completely irrelevant for what I am writing, and I would like to avoid making the text heavier as much as possible.

Suggestions that have been made so far that rely on external notions: $$[\min\{a,b\}, \max\{a,b\}]\qquad \operatorname{Conv}(a,b)$$

Suggestions for a brand new notation: $$(a,b]^*\qquad (\{a,b\}]\qquad (a\nearrow b]\qquad /a,b/\qquad \left(\begin{matrix}a\\b\end{matrix}\right]^\star$$

$^\star$ intervals open at the lower bound and closed at the higher bound, whichever of $a$ and $b$ they are.

Some other options:

• Assume wlog that $a<b$
• Make explicit that the notation $[a,b]$ doesn't imply $a<b$.

Solution 1:

One possibility is $\operatorname{Conv}(a,b)$: the convex hull of $a$ and $b$. Maybe this should really be $\operatorname{Conv}(\{a,b\})$, but I think it is forgivable to omit the curly braces - or even to write $\operatorname{Conv}\{a,b\}$, which keeps it clear that order does not matter.

When $a,b \in \mathbb R$, this just gives us the closed interval $[a,b]$ or $[b,a]$; for points $a,b \in \mathbb R^n$, this gives us the line segment from $a$ to $b$.

It generalizes to $\operatorname{Conv}\{a,b,c\}$ which is the smallest closed interval containing all three of $a,b,c \in \mathbb R$, and so on.

Solution 2:

Assuming you're meaning the closed interval for the notation I'm going to write, something that will always work is

$$[\min\{a,b\}, \max\{a,b\}]$$

Another possibility is

$$[a,b] \cup [b,a]$$

But I think that there is no standard notation, so you could create yours explaining it.

Solution 3:

Without loss of generality, let's assume $a<b$. Consider the interval $[a,b]$...

If that's not working, define some intuitive variable names like $m:=\min(a,b) , M:=\max(a,b)$, where $m$ stands for min, and $M$ stands for max.
Or use $l$ and $u$ for lower and upper, or $l$ and $h$ for low and high. As long as you couple it with a sentence, people will see the variables as acronyms for their intuitive meaning.

Solution 4:

When no convenient standard notation exists for something you need to use repeatedly, you are entitled to make up a new notation for it, for example $(a\nearrow b)$ or $[a\nearrow b]$. Another suggestion is $(\{a,b\})$ or $[\{a,b\}]$. The idea behind the first notation is that $a$ and $b$ are placed in a "rising" sequence, while in the second the braces indicate a neglect of the existing order of $a$ and $b$. Be warned, though, that people are critical of new notation; so choose it carefully!

Solution 5:

Probably, the simplest notation in this case is $I$ (together with some words):

Let $I$ be the interval of all points lying between $a$ and $b$. Then...

... Then, the interval $I$ of all points lying between $a$ and $b$ satisfies...

... Then ... where $I$ is the interval of all points lying between $a$ and $b$.

None of these sentences seems heavy. Instead, they seem are very simple and clear (in my opinion).