Prove that $\mathcal{P}(\mathbb{N})\setminus\{\emptyset\}$ has a choice function.
(Remark: you are not allowed to use the Axiom of Choice in this part of the question)
Pf: We can define the choice function $f$ for $\mathcal{P}(\mathbb{N})\setminus\{\emptyset\}$ by $f(S)=\text{the least element of $S$}.$
According to Wikipedia,
A choice function (selector, selection) is a mathematical function f that is defined on some collection X of nonempty sets and assigns to each set S in that collection some element f(S) of S. In other words, f is a choice function for X if and only if it belongs to the direct product of X.
Can someone explain the intuition behind this proof? Thank you.
The intuition is that if a set of natural numbers is not empty, then it has a unique least element.
This proves that the relation $\{(S,\min S)\mid S\in\mathcal P(\Bbb N)\setminus\{\varnothing\}\}$ is in fact a function, and by definition a choice function, since $\min S\in S$.