I'm currently in the following framework: $n \in \mathbb{N}$, $I \subseteq \{1, \dots, n\}$ and $X \subseteq \mathbb{R}$. Now, I need to define a set which contains all vectors $v \in \mathbb{R}^n$ such that the components of $v$ with indices in $I$ are elements of $X$ and the other components are zero. My naive approach to define this set looks like this
$$\{(x_1, \dots, x_n) \in \mathbb{R}^n: x_i \in X, i \in I, x_j = 0, j \in \{1, \dots, n\} \setminus I\}.$$
I don't like this definition as it contains two indices $i, j$ for the components. Also it is not immediately clear that the cases are well defined (each index is covered by a case and no index appears in both terms). In a normal cases environment one could write "else" for the second case. However a cases environment does not look nice within the braces of the surrounding set.
Is there some notation I am missing? How can I write this set more clearly? Of course I could use a description of the set as I did in my introduction of this question.
Best Answer
You can describe it very simply if you do a little preliminary setting up. Let $$Y_i=\begin{cases}X,&\text{if }i\in I\\\{0\},&\text{otherwise;}\end{cases}$$ the desired set is then $\prod\limits_{i=1}^nY_i$.
If you want everything in the curly braces, it’s going to be a bit ugly. One fairly readable version is
$$\{\langle x_1,\ldots,x_n\rangle\in\Bbb R^n:x_i\in X\text{ for all }i\in I,\text{ and }x_i=0\text{ otherwise}\}\,.$$