notation for sequences and families of sets

Question

Maybe this is more about language and convention than actual math.

Literally, the notation $\{x_i:i\in I\}$ or $\{x_i\}_{i\in I}$ means the range of the function $x$ with domain $I$. But then many people seem to use this symbol to stand for the actual function $x$ itself. For example, somebody might write "let $\{x_n\}$ be a sequence". Am I correct in understanding that there are two different meanings for the same symbol?

Answer 1

I think the reality is people use notation they can get away with until someone (an editor or referee) complains. Literal reading of $\{x_n\}$ would be a set with one element in it. But if context had been to talk about sequences then I suppose many of in math would be forgiving of seeing n as an implied parameter that varies over the appropriate context. Advice? When you write, be more clear until such time as it is clear what you subset of the dialect of math will permit.

Answer 2

In my opinion, the notation $\{x_i\}_{i\in I}$ is preferable to $\{x_i:i\in I\}$.

An infinite sequence in which the indexing variable is countable is also denoted as an infinite ordered-tuple as $(x_1,x_2,\dots,x_k,\dots)$ in the literature.