How to interpret $(x(t_1), x(t_2), \dots, x(t_n)) \in (a_1,b_1] \times (a_2,b_2] \times \dots \times (a_n,b_n]$?

Question

I need help with the following, how should I interpret this notation?

$$ (x(t_1), x(t_2), \dots, x(t_n)) \in (a_1,b_1] \times (a_2,b_2] \times \dots \times (a_n,b_n] $$

1. Does it mean "$x(t_1)$ is a element of the set $(a_1,b_1]$" and "$x(t_2)$ is a element of the set $(a_2,b_2]$", and so forth? So we have \begin{align} x(t_1)&\in (a_1,b_1] \\ x(t_2)&\in (a_2,b_2] \\ &\vdots \\ x(t_n)&\in (a_n,b_n] \end{align}

2. Or does the notation mean something like this \begin{align} x(t_1)&\in (a_1,b_1] \times (a_2,b_2] \times \dots \times (a_n,b_n] \\ x(t_2)&\in (a_1,b_1] \times (a_2,b_2] \times \dots \times (a_n,b_n] \\ &\vdots \\ x(t_n)&\in (a_1,b_1] \times (a_2,b_2] \times \dots \times (a_n,b_n] \end{align} I don't know how to phrase this notation. Does it have a meaning in this context?

3. Or maybe something like this \begin{align} (x(t_1), x(t_2), \dots, x(t_n)) &\in (a_1,b_1] \\ (x(t_1), x(t_2), \dots, x(t_n)) &\in (a_2,b_2] \\ &\vdots \\ (x(t_1), x(t_2), \dots, x(t_n)) &\in (a_n,b_n] \end{align} Same here, I don't know how to phrase this notation. Does it have a meaning in this context?

Thanks!

Answer 1

You should use interpretation 1.

$ (a_1,b_1] \times (a_2,b_2] \times \dots \times (a_n,b_n] $ is a Cartesian product of intervals.

Elements of it are $n$-tuples, such as $(x(t_1), x(t_2), \dots, x(t_n)).$