Tuple notation for non-negative integers

Question

I have three variables $x_1, x_2, x_3$ that can only be non-negative integers, i.e., $x_1 \in \mathbb{N}_0$, $x_2 \in \mathbb{N}_0$, and $x_3 \in \mathbb{N}_0$. Is it correct to write this in a tuple like $(x_1,x_2,x_3) \in \mathbb{N}_0^3$? If not, what is the correct way?

Answer 1

Unless you're actually considering them as a tuple elsewhere, you can also just write this as

$$x_1, x_2, x_3 \in \mathbb N_0$$

And I would argue that introducing the tuple and the product space solely for the purpose of specifying this constraint makes it more complicated for no reason.

So if you need the tuple, yes, use that notation. But if you don't need the tuple, use this one.

Answer 2

Yes, this is correct. You could also write $$(x_1,x_2,x_3)\in\mathbb{N}_0 \times \mathbb{N}_0 \times \mathbb{N}_0.$$

An alternative that avoids duplication is $$\text{$x_i\in \mathbb{N}_0$ for $i\in \{1,2,3\}$}.$$