How can I define this set?

Question

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?

Answer 1

$\bigcup_nA_n$ is the collection of all of the sets from which you can draw elements, so $\bigcup\bigcup_nA_n$ is the collection of all of the elements that you can use to form members of $B$; in your example

$$\bigcup_nA_n=\big\{\{1\},\{2\},\{3\},\{4\}\big\}\,,$$

and

$$\bigcup\bigcup_nA_n=\{1,2,3,4\}\,.$$

Apparently you want only the non-empty subsets of $B$, so

$$B=\wp\left(\bigcup\bigcup_nA_n\right)\setminus\{\varnothing\}\,.$$