A (simple?) matter of notation

Question

I am currently working on some power series problems, so I deal with the sequence of their (generally complex) coefficients $a_i$, $i\in\mathbb{N}$, denoted in the sequel as $\langle a_n\rangle$ following reference [1]. A sequence is simply a function $$ \mathbb{N}\ni n\mapsto a_n\in\mathbb{C}\tag{1}\label{1} $$ Given a finite/infinite subset $S$ of the range of $\langle a_n\rangle$, I need to analyze the indexes $n$ of each member $a_n\in S$. And now my question arises: does there exists a standard notation for the inverse image of $S$ under the function expressed by \eqref{1}?

Having read many textbooks on real and complex analysis, power series and so on, I did not find anything on such matter: however, perhaps specialists in combinatorics or other similar field are more customary with such concepts so I decided to ask this question in order to not introduce in my work useless notations.

[1] Emanuel Fisher (1983), "Intermediate Real Analysis", Springer Verlag.

Answer 1

Denote the function in (1) by $a$, hence $a: \mathbb N \to \mathbb C$ and $a(n)=a_n$.

Then $\{n \in \mathbb N: a_n \in S\}=a^{-1}(S)$.

Answer 2

If we use the notation $\langle a_n \rangle$ to denote a sequence $a$, where for each $n \in \mathbb{N}$, $a_n := a(n)$, it seems fitting to write the pre-image of $S \subseteq \mathbb{C}$, $\ a^{-1}(S),$ as $\langle a_n \rangle ^{-1}(S).$

Going from what you mentioned under Fred's post, one advantage of this notation is that we are not hiding the fact that $a$ is a sequence. I am not sure whether writing the pre-image this way is standard, however.