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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!

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  • 2
    $\begingroup$ it means 1. Like $(x,y)\in\Bbb R\times\Bbb R$ $\endgroup$ – J. W. Tanner Jun 25 at 20:17
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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 $(x(t_1), x(t_2), \dots, x(t_n)).$

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  • $\begingroup$ 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 $\endgroup$ – J. W. Tanner Jun 25 at 20:26
  • $\begingroup$ Okay, number 1 it is, thanks! Also, do you mean 2 would be correct if I instead had $$x(t_1), x(t_2), \dots, x(t_n) \in (a_1,b_1] \times (a_2,b_2] \times \dots \times (a_n,b_n]$$? $\endgroup$ – JDoeDoe Jun 26 at 14:42
  • $\begingroup$ yes, then each $x(t_i)$ would be an element of the Cartesian product $\endgroup$ – J. W. Tanner Jun 26 at 14:49

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