Suppose that $A=\{A_i:i \in I\}$ is a family of sets. Consider the case where we want to define a set $S$ which has a unique element from each set $A_i$ and has no more elements than that.
For example, if $A= \{1,2,3\} , \{ a,b,c \} , \{ !,\#,@ \} \}$ then $S$ may be $\{1,a,!\}$ or $\{1,b,@\}$ but $S$ can't be $\{1,a\}$ as $S$ didn't have an element from the set $\{!,\#,@\}$ and the set $\{1,2,a,@\}$ can't be $S$ as it has two elements from the set $\{1,2,3\}$.
So $S$ has an element from each set $A_i$ and only one element from this $A_i$.
My question is how to define this set formally.
my attempt is , $S \subset \bigcup_{i \in I} A_i$, where for every $i \in I$ , $S \cap A_i =\{a\}$ for some $a \in A_i$.
Does this definition work ?
Are there any better definitions you suggest ( in builder-notation for instance?) ?
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