Explanation of $\{A_i\bigcup B_j\}_{i,j\in I\times J}$

Question

It is in my understanding that a family, $\{A_i\}_{i\in I}$ is a collection of sets $A_i$ with $i$ taking on all possible values within $I$. Of course this is the less formal definition; the most formal one being that which defines $\{A_i\}_{i\in I}$ as a graph, but I understand that that definition is seldomly used in practical theory. However, I have none of the idea of what $\{A_i\bigcup B_j\}_{i,j\in I\times J}$ or $\{A_i\bigcap B_j\}_{i,j\in I\times J}$ means. Does it use the same logic? So it's the collection of the sets $\{A_i \bigcup B_j\}$ with $i$ and $j$ taking on all possible values within $I$ and $J$? So there should be $i$ times $j$ terms? Thank you in advance.