Algebra to pick random element from a set Lets define set $G$:
$$G = \{ 1,2, \dots,n \space | \space n \in \mathbb{N} \} \text{ and }\mathbb{N} \rightarrow \mathbb{R}_+$$
What is the algebraic notation to build set $Y$ by picking randomly 20 elements from $G$, without replacement ?
$$Y = \{?\}$$
 A: You can write this as $$Y \subseteq G \space \text{ with }\space \#Y=20 \space \text{($\#$ stands for cardinality).}$$ 
If you want to use this for a proof, it satisfies to write for example
$$Y \subseteq G \space \text{ with }\space \#Y=20$$ and $\forall n \in Y$ chosen arbitrarily. This notes that the values are chosen randomly. I hoped this helped you out.
A: Would this suffice?
$$Y = \{ y \text{ }| \text{ } y = f(x), f:[n] \mapsto N, |f| = n\}$$
This reads:
$Y$ is the set of elements from some bijective function $f$ which maps the first $n$ naturals to some natural number.
The fact $f$ is bijective implies there are no duplicates.
A: So to summary, we define set G:
$$G = \left \{ 1,2,...,n \: |\: n\in \mathbb{N}  \right \}\:\text{where }n\geq 20$$
Set Y is built with randomly sampled unique elements from G so that:
$$Y_{|G|} = \left \{ y \: |\:y=f(x), f:\left [ |G| \right ] \mapsto \mathbb{N},|f|=|G| \right \}$$
With probability P(e) for elements e from G to be in Y:
$$P(e)=\frac{(N-n)!}{N!}\: with \: e \in G, \: N=|G|, \: n=|Y|$$
Right ?
