Let we have the following set
$$
\{y_1, y_2, ..., y_L\}
$$
My question:
For short should I use $\{y_k\}$, $\{y_k\}_k$ or $\{y_k\}_{k=1,..,L}$?
My idea:
$\{y_k\}$ is a one member set, so it is incorrect.
$\{y_k\}_{k=1,..,L}$ is correct.
$\{y_k\}_k$ is OK from the context.
Thanks.
The notation $\{y_k\}$ is indeed a set of arbitrary many indexed elements. We may use ${\{y_k\}}_{k\in\Bbb N^+}$ or $\{y_k: k\in\Bbb N^+\}$ to indicate the set has countable infinite elements, and likewise ${\{y_k\}}_{k\in\{1..n\}}$ , ${\{y_k\}}_{k=1}^n$ , or $\{y_k:k\in\{1,..,n\}\}$ to indicate finite set size, or a particular subset of interest (here, the first $n$ terms).
Note that when the elements are themselves sets, this notation can be employed for arbitrary unions and arbitrary intersections.
$\displaystyle\bigcup_{k\in\Bbb N^+}\{y_k\} = \{x:\exists k\in\Bbb N^+: x\in y_k\}$
$\displaystyle\bigcap_{k=12}^{26}\{y_k\} = \{x:\forall k\in \Bbb N: 12\leq k\leq 26, x\in y_k\}$
So, yes, ${\{y_k\}}, {\{y_k\}}_{k\in\Bbb N^+}, {\{y_k\}}_{k\in\{1..n\}}, {\{y_k\}}_{k=1}^n$ are all reasonable.
$\{y_1,y_2,\ldots,y_L\}$ is also perfectly fine.
The purpose of notation is precision and clarity. The notation $\{y_1, y_2, ..., y_L\}$ has both.
