Is there any standard notation for choosing one element from each group

Question

Let's say I have $n$ sets $A_1, A_2, ..., A_n$.

I want to define an arbitrary set $B$ such that $B$ has exactly one element from each group.

If I write $B = \{a\in A_i|i\in [n]\}$, then $B$ would have all of the elements of $A_1, ..., A_n$.

Is there anything similar such that $B$ contains only one element from each group?

Answer 1

Since there is not necessarily a unique choice of element from each $A_i$ (assuming each is non-trivial), you should just spell it out. Something like "for each $i$, choose an element $a_i \in A_i$..." Then you can let $B$ be the set of these particular chosen elements, $B=\{a_i \mid 1 \leq i \leq n\}$.