In set theory, what do the subscripts $A_1,A_2,\dotsc,A_n$ mean?

Question

How are subscripts used in set theory, for example, In set theory, what do the subscripts $A_1,A_2,\dotsc,A_n$ mean?

Answer 1

It just means that $A_1,\ldots,A_n$ are sets. It is often the case that they are distinct, but if it was not mentioned then it might be the case that $A_1=A_2=\ldots=A_n$.

The subscript is just an index. And formally it means that there is a function $A$ whose domain is $\{1,\ldots,n\}$ and we write $A_i$ for the set obtained by $A(i)$. Any additional assumptions need to be mentioned explicitly.

We may assume, of course, that there is some set $X$ such that $A_i\subseteq X$, simply by taking $X=\bigcup_{i=1}^n A_i$. But this is really not different from having $n_1,\ldots,n_k$ some natural numbers or $r_1,\ldots,r_k$ some real numbers.