Is it correct to write "$100 \leq x,y \leq 380$"?

Question

I am solving a problem in which I am looking for integers $x$ and $y$ in between $100$ and $380$ (including $100$ and $380$).

Is it mathematically correct to write $$100 \leq x,y \leq 380,$$ or am I supposed to write them separately: $$100 \leq x \leq 380\quad\text{and}\quad 100 \leq y \leq 380 ?$$

Answer 1

$$100 \leq x \leq 380\quad\text{and}\quad 100 \leq y \leq 380 $$ is always unambiguous, while $$100 \leq \quad x,y \quad\leq 380 $$ is seldom misinterpreted.

To incorporate the information about being integers:

• The integers $x$ and $y$ are between $100$ and $380,$ inclusive.
• $x,y\in\{100,101,\ldots,380\}$
• $(x,y)\in\{100,101,\ldots,380\}^2$
• $(x,y)\in\left\{m\in\mathbb Z\mid 100\le m\le380 \right\}^2$
• $x,y\in\{m\in\mathbb Z\mid 100\le m\le380 \}$

(Strictly speaking, that listing option isn't definitive, since, for example, those set elements conceivably contains precisely the integers, from $100$ to $380,$ that are indivisible by $3.)$

Answer 2

First, don't use $ x$ and $y$ for variables that must be integers. It's only a convention, but it's a strong convention.

With that out of the way, my suggestion is

$$m,n \in \{100, 101,... 380\},$$

assuming you've mentioned the integer restriction somewhere previously.

(I might have omitted the $101$, but RyanG used it, and it can't hurt.)