I'm having problem with understanding power set products of cartesian sets.
I know that power set $P(X)$ holds $2^n$ elements, where n is amount of elements in X.
The problem is following:
Let $A= \{ \{a\}, \{b\} \}$
and $B = \{ \{ a,b\}, \emptyset\}$
Find $P(A) \times P(B)$ and $P(A \times B)$.
I found $P(A) = $ $\{ \emptyset, \{ \{ a\} \}, \{ \{ b\} \},\{ \{ a\}, \{ b\} \} \} $
Not sure if $B = \{ \emptyset, \{ \emptyset\} , \{ \{ a\},\{ b\} \} ,\{ \{ a, b\} \} \} $
or $B = $ $\{ \emptyset, \{ \{ a\}\},\{\{ b\} \} ,\{ \{ a, b\} \} \}$ ?
Edit:
$B = $ $\{ \emptyset, \{ \{ a\}\},\{\{ b\} \} ,\{\emptyset, \{ a, b\} \} \}$,
Edit2:
$B = $ $\{ \emptyset,\{ \emptyset \} , \{\{ a, b\} \},\{\emptyset, \{ a, b\} \} \}$ That is clear.
How to find cartesian products $P(A) \times P(B)$ and $P(A \times B)$ ?
Edit3: Okay, Cartesian product of $P(A) \times P(B)$ contatins 10 elements, because by adding together all products, empty set appears 7 times.
I think powerset B should be: ={●,{●},{{a,b}},{{a,b},●}} As powerset is set of subsets of set as elements. (●- here phi empty set )