Let $A_1,..., A_n$ be a family of sets of sets. I want to create a now set as the following:
The set $B$ is made of unions of all possible combinations of elements from any set.
For example: Let $A_1=\{\{1\},\{2\}\}$, $A_2 = \{\{3\}\}$ and $A_3 = \{\{4\}\}$. Then the set $B$ should be:
$$B=\{\{1\},\{2\}, \{3\},\{4\},\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\},\{3,4\},\{1,2,3\},\{1,2,4\},\{2,3,4\},\{1,3,4\},\{1,2,3,4\}\}$$
My question is, how can I formally write this set?
My approach was the following:
First let's put all elements we want to combine in the same set: $\bigcup\limits_n A_n$
Then let's take it's power set: $\mathcal P\left(\bigcup\limits_n A_n\right)$
In this power set we have all the combinations that we want:
Now we can define $B$ as:
$$B = \left\{ \bigcup_{a \in A} a : A \in \mathcal P\left(\bigcup\limits_n A_n\right)\right\}$$
My question is, Am I over complicating? Is there any other way of defining this set?