Use cases for interval notation

Question

The notation $x \in [a,b]$ and $a \le x \le b$ are equivalent in how they describe the variable $x$. When writing mathematics, when does the use of one take precedence over the other? In the former case we explicitly point out that $x$ is the element of some set whose shorthand is given by $[a,b]$. In the latter we can infer the set $x$ belongs to, but it is more of an ephemeral reference to what $x$ is "greater" and "less" than in a passing context.

The question: Is there an appropriate use for each notation or is it purely a matter of taste? Can you provide use and non-use examples for each case?

Answer 1

Of course, either can be used in a particular place, but there may (need to) be some change of language around the usage going from one to the other.

• "Let ...". If we write "Let $x \in [a,b]$", it is clear that $x$ is the variable being bound. If we write "Let $a \leq x \leq b$", it is less immediate which of the three symbols is the bound variable. (Surely, $a$ and $b$ are already bound, but the newly bound variable is not the first symbol, so the cognitive load of parsing the sentence increases.) Resolving this is the wordier "Let $x$ be such that $a \leq x \leq b$ ...".
• If the interval containing $x$ depends on prior variables... "Let $x \in [a,b] \subset (a',b')$" is equivalent to "Let $x,a,b \in \Bbb{R}$ satisfy $a' < a \leq x \leq b < b'$". Now the difference is emphasis -- do you intend to emphasize that the closed subset is in the open subset or that the variables are ordered?
• Do you need a set operation? "Let $x \in [a,b] \cap \Bbb{Z}$" versus "Let $x$ be an integer, $a \leq x \leq b$." The more complex the operation, the more likely it is unambiguously expressed by set operations. The order notation leads one to list the various sets in the set expression with ever-more-conflatable meaning. Try $$ x \in [a,b] \cap \left(\Bbb{Z} \cup \left[ \frac{a+b}{2}, \frac{a+3b}{2}\right)\right) $$ starting with "Let $x$ be such that $a \leq x \leq b$ ...".
• There's no hope of "$a \leq x \leq b$" when $x$ is an element of a not linearly ordered set (for instance, $\Bbb{C}$). If you are using many unordered or not linearly ordered sets, use of the order notation stands out loudly. Do you intend that much emphasis? And, you have to specify from which ordered set come $a$, $x$, and $b$.