Sums and Products over sets of sets

Question

Say I have a set of sets eg. $A:=\{\{1,2\},\{2,3\}\}$. Does this formula $$\sum_\limits{A_i \in A}\prod_{j\in A_i}x_j$$ yield $x_1x_2+x_2x_3$? Or does it even make sense to define a sum over a set of sets?

Answer 1

I assume the product in the sum was meant to be $\prod_{j\in A_i}x_j$. As long as some indexing set $I$ (possibly a set of sets) and the corresponding value for each index is something compatible with multiplication or addition then $\prod_{i\in I}$ or $\sum_{i\in I}$ respectively makes sense (a formal polynomial in this case). So even though the sum of $A_i$s themselves may not make sense, we still have $\prod_{j\in A_i}x_j$ which makes sense as $A_i$ varies over elements of $A$ which $\sum_{A_i\in A}$ sums over.