My question is related to set notation.
How can I differentiate with proper mathematical notation between the following 2 subset selection approaches?
From an initial set $S=\left\{q_{i}\right\}_{i=1}^{K}$ with $K$ items, with $q_{i}\in\mathbb{R}$, I want to select one subset $s\subset S$ with $N$ items ($N<K$) out of the total $K$. The selection could be
-
A. Random. For example if $S=\left\{10,2,35,14,58\right\}$ with ($K=5$) and I want to select a subset with $N=3$ items, some possible cases could be $s=\left\{10,14,58\right\}$, $s=\left\{2,35,58\right\}$, $s=\left\{10,2,14\right\}$ etc.
-
B. Sequential. For example, if $S=\left\{10,2,35,14,58\right\}$ with ($K=5$) and I want to select a subset with $N=2$ items, the result would be $s=\left\{10,2\right\}$, for $N=3$ it would be $s=\left\{10,2,35\right\}$, and for $N=4$, $s=\left\{10,2,35,14\right\}$ and so on.
How can I express these 2 cases with proper math notation?
What comes to my mind is simply $s=\left\{q_{i}\right\}_{i=1}^{N}$, which seems to me more like case B. Is that correct? If yes, then how can I express case A and how will this be different from case B?
P.S. There is no specific formula to describe the elements of the initial set $S$. It could be any numbers in $\mathbb{R}$.
Best Answer
I'm not sure I've seen explicit notation for this, but what I would usually see is something like:
I don't think you have to specify a random ordering; this allows for any ordering. On the other hand, if you want to explicitly specify sequentiality, I'd say something like:
Edit: For the second case, you could even say something like...