Algebra to pick random element from a set

Question

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 ? $$Y = \{?\}$$

Answer 1

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.

Answer 2

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.

Answer 3

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 ?