hoping someone can help me in understanding the union of the power set of a family of sets. Please tell me if my reasoning is incorrect.
Let $F$ be a family of sets. In particular, let $F$ = {{1,2},{3,4}} = {A,B}.
Then the power set of $F$, denoted $P(F)$, is as follows:
$P(F)$ = { {}, {A}, {B}, {{A, B}} } = { {}, {{1,2}}, {{3,4}}, {{{1,2},{3,4}}} }
I am now wondering what the union of $P(F)$ is, i.e. $U(P(F))$. My understanding is that the union would be all the elements of $P(F)$ but as one big set. So it would look like this:
$U(P(F))$ = { {}, A, B, {A,B} } = { {}, {1,2}, {3,4}, {{1,2},{3,4}} }
I'm not super sure about what I got for $U(P(F))$ so please let me know if I'm wrong. I would really appreciate a second opinion.
Thank you for your time.
Not quite. $U(X)$ is the set of all elements of elements of $X$. In other words $t\in U(X)$ if and only if there exists some $Y$ such that $t\in Y$ and $Y\in X$.
Also, you have a mistake in your $P(F)$ because $\{\{A,B\}\}$ is not a subset of $F$ because its element $\{A,B\}$ is not an element of $F$. So the correct power set is rather $$P(F)=\{\{\},\{A\},\{B\}\{A,B\}\}.$$
With that out of the way, the elements of $\{\}$ are: none; the elements of $\{A\}$ are: only $A$; the elements of $\{B\}$ are: only $B$; the elements of $\{A,B\}$ are: $A$ and $B$. So all in all the elements of $U(P(F))$, i.e., elements we named at least once during this, are: $A$ and $B$. So $$ U(P(F))=\{A,B\}.$$
Your answer wrongly lists $\{\}$ as an element of $U(P(F))$ even though none of the elements of $P(F)$ has $\{\}$ as element. Your other wrong element of $U(P(F))$, namely $\{A,B\}$ seems to be a consequence of the error you had in $P(F)$ mentioned above.