In the question unordered cartesian product an shorthand notation for the unordered cartesian product was discussed but without any standard notation. So my question is what would be the explicit definition of all different sets consisting of exactly one element of each subset of a set S. There can be an arbitrary number of sets in S.
$\{\{a, b\}, \{c\}\}$ should lead to $\{\{a, c\}, \{b, c\}\}$
[Math] Set notation for unordered cartesian product
elementary-set-theorynotation
Best Answer
It's unclear the question but i think i can give an answer:
You can think "the set of all different sets consisting on exactly one element of each subset of a a set $S$" as the cartesian product: \begin{equation} \prod\mathcal{P}(X)\setminus\{\emptyset\}=\{f\mid f:I\to\bigcup_{i\in I}X_i, \text{ }f(i)\in X_i\} \end{equation} where I is an index set for $\mathcal{P}(X)\setminus\{\emptyset\}$, i.e. $\mathcal{P}(X)\setminus\{\emptyset\}=\{X_i\mid i\in I\}$.
(You must sustract the empty set beacause you cannot take the cartesian product of a family which contains an empty set).