Shorthand for elements in a set and elements in a vector/list?

Question

Really basic notation questions ahead. For both cases below, suppose I have an arbitrary set of integers $S$, e.g., $S = \{2, 4, 6, 8\}$.

1. I want to define a list/vector whose elements are subscripted by the elements in $S$, so that $x = \langle x_2, x_4, x_6, x_8 \rangle$. Is $x = \langle x_i \rangle_{i \in S}$ standard notation?

2. Exact same question, but for a set: I want to define a set whose elements are subscripted by the elements in $S$, so that $Y = \{ y_2, y_4, y_6, y_8\}$. Is $Y = \{ y_i \}_{i \in S}$ standard notation?

3. Does the answer change if $S$ contains all integers from 1 to $s$, i.e., $S = \{1, ..., s\}$?

Answer 1

I disagree with Oscar about part 1 of the question. The elements of a list or vector are ordered, but the elements of a set are not. I don’t think many (any?) mathematicians would write $x = \langle x_i \rangle_{i \in S}$ when $S=\{4,2,8,6\}$. Since $x_i$ is a common notation for the $i$-th element of a sequence or vector, one might be okay if $S=\{1,\dots s\}$, but I would more likely write $x = \langle x_i \rangle_{i=1\dots s}$ (or $\langle x_i \rangle_{i=2,4,6,8}$ ) to make the order clear if there were a reason to use subscript notation.

Answer 2

Yes, I think this it's completely legit, and also commonly used in the literature. Nothing changes if $S$ contains all ingers from $1$ to $s$. In principle $S$ could also be more general than a subset of $\mathbb{N}$. Then the list/vector has not maybe an immediate meaning, but for a continuous set (for example) you can admit a coninuous label and use the same notation.