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

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!

• it means 1. Like $(x,y)\in\Bbb R\times\Bbb R$ – J. W. Tanner Jun 25 '19 at 20:17

$$(a_1,b_1] \times (a_2,b_2] \times \dots \times (a_n,b_n]$$ is a Cartesian product of intervals.
Elements of it are $$n$$-tuples, such as $$(x(t_1), x(t_2), \dots, x(t_n)).$$
• for interpretation 2., there would not be parentheses before $x(t_1)$ and after $x(t_n)$; interpretation 3. would mean an $n$-tuple is in an interval, which doesn't make sense – J. W. Tanner Jun 25 '19 at 20:26
• Okay, number 1 it is, thanks! Also, do you mean 2 would be correct if I instead had $$x(t_1), x(t_2), \dots, x(t_n) \in (a_1,b_1] \times (a_2,b_2] \times \dots \times (a_n,b_n]$$? – JDoeDoe Jun 26 '19 at 14:42
• yes, then each $x(t_i)$ would be an element of the Cartesian product – J. W. Tanner Jun 26 '19 at 14:49