Constrained variables in set-builder notation

Question

I have a set of variables $x_i \in \mathbb{R}$ that are subject to upper bounds $X_i$ such that $x_i \leq X_i $ for each $ i \in \{1, 2, ..., n\}$. I want to write this in set-builder notation. I tried $\{x_i\ |\ x_i \leq X_i, i \in \{1, 2, ..., n\}\}$ but it does not seem right to include the $\{1, 2, ..., n\}$ set in the definition of the $x_i$'s. What is the correct way to do it?

Answer 1

Here are two ways: $$\{x\in\mathbb{R}^n \mid x_i \le X_i \quad \forall i\in\{1,\dots,n\}\}$$ $$\{(x_1,\dots,x_n)\in\mathbb{R}^n \mid x_i \le X_i \quad \forall i\in\{1,\dots,n\}\}$$

Two more ways, using interval notation: $$(-\infty,X_1] \times \dots \times (-\infty,X_n]$$ $$\times_{i=1}^n (-\infty,X_i]$$

Answer 2

Using set-builder notation: $$\{x_i\mid 1\le i\le n\,\text{ and }\, x_i\le X_i\}.$$

Combining set-builder notation and the listing method (so, a bit unorthodox): $$\{x_1,\ldots,x_i,\ldots,x_n\mid x_i\le X_i\}.$$