# Pi product + set builder notation - how to read

From the low-discrepancy wiki I have:

$$\displaystyle \prod_{i=1}^s[a_i, b_i) = \{𝑥 \in \mathbb{R}^s : a_i \le x_i \le b_i\}$$ where $$0 \le a_i < b_i \le 1$$

How can I read it? Should I first make do a cartesian product of $$[a_i,b_i)$$? For example for $$s=3$$ I would have: $$[a_1, b_1) \times [a_2, b_2) \times [a_3, b_3) = \{ \{a_1, a_2, a_3\}, \{a_1, a_2, b_3\}, \{a_1, b_2, a_3\}, \{a_1, b_2, b_3\}, \{b_1, a_2, a_3\}, \{b_1, a_2, b_3\}, \{b_1, b_2, a_3\}, \{b_1, b_2, b_3\}\}$$

then I would put it into set-builder notation.. but for first $$\{a_1, a_2, a_3\}$$ I don't have $$b_i$$ value.

• What is your question? Apr 23, 2019 at 9:50
• @lisyarus I am sorry. Stackexchenge posted my question when I was trying to add a link. I have edited it. Apr 23, 2019 at 10:06

You should read, $$\displaystyle \prod_{i=1}^s[a_i,b_i)$$ as, "the product of all intervals closed $$a_i$$, open $$b_i$$ from $$i=1$$ to $$s$$". And $$\displaystyle \{\tilde x\in \Bbb R^s:\tilde x=(x_1,x_2,\dots x_s) \text{ where }x_i\in[a_i,b_i),\forall i=1,2\dots s\}$$ as, "the collection of all $$\tilde x$$ ($$x$$ tilde) in $$\Bbb R^s$$ such that $$x$$ tilde is of the form of $$s$$-tuple $$(x_1,x_2,\dots x_s)$$ where $$x_i$$ belongs to $$[a_i,b_i)$$ for all $$i=1,2,... s$$".
Further, $$\displaystyle \prod_{i=1}^s[a_i,b_i]=[a_1,b_1]\times [a_2,b_2]\times ... \times [a_s,b_s]$$ where $$\times$$ is the cartesian product defined here.
• After I compute $\displaystyle \prod_{i=1}^s[a_i, b_i) = \{𝑥 \in \mathbb{R}^s : a_i \le x_i \le b_i\}$ I will get collection of s-tuples: $\{ s-tuple, s-tuple...\}$. So, for $s = 1$ I will get $\{a_1, a_1 + v, a_1 + 2*v, ...a_1 + k*v\}$ where $v$ is some "small value" and $a_1 + k*v$ is smaller than $b_1$. So, for $v=0.4$, $a_1=0$ and $b_1=1$ I will get $\{0,0.4,0.8\}$. For $s=2$ I need to compute the Cartesian product: $\{0,0.4,0.8\}\times\{0,0.4,0.8| = \{\{0,0\},\{0,0.4\},\{0,0.8\},\{0.4,0\},\{0.4,0.4\},\{0.4,0.8\},\{0.8,0\},\{0.8,0.4\},\{0.8,0.8\}\}$. Is this correct? How do I find $v$? Apr 29, 2019 at 11:40
• I think you have confused with math.stackexchange.com/questions/2115752/… this one. Further for $s=1$ you simply get an interval which cannot be written in your form $\{a1,a1+v...\}$ etc. Apr 30, 2019 at 2:33
• As for link: instead of $\{\{0,0\},...\}$ I should write $\{(0,0)\}$ and $\{0, 0.4, 0.8\}$ as $\{(0), (0.4), (0.8)\}$. That is what you mean? As for $s=1$ do you mean that I have forgotten parenthesis ($()$)? May 9, 2019 at 16:06