Power Set Mathematics If given the following sets $A = \{1,2,3\}$ and $B = \{3,4,5\}$, the power sets of each are the following:
$$\mathfrak P(A) = \{\emptyset,(1),(2),(3),(1,2),(1,3),(2,3),(1,2,3)\}\\
\mathfrak P(B) = \{\emptyset, (3),(4),(5),(3,4),(3,5),(4,5),(3,4,5)\}$$

What is
  
  
*
  
*$\mathfrak P(A) \cap \mathfrak P(B)$
  
*$|\mathfrak P(A \cup B)|$
  
*$|\mathfrak P(A) \cup \mathfrak P(B)|$
  

My guess for the intersection one the first one is just $\{3\}$ and ${\emptyset}$ because it is only used elements in both.
I'm not sure about the next two. I do understand how power sets work but im curious on what the difference is between the $\mathfrak P(A \cup B)$ and $\mathfrak P(A) \cup \mathfrak P(B)$. Is there a difference?
How do we approach these types of problems? 
 A: Your assertion is correct since
$$\mathfrak P(A) \cap \mathfrak P(B) = \mathfrak P(A\cap B) = \mathfrak P(\{3\}) = \{\emptyset, \{3\}\}$$
Now for the other two since you already wrote down $\mathfrak P(A)$ and $\mathfrak P(B)$, write down $\mathfrak P(A\cup B) = \mathfrak P(\{1,2,3,4,5\})$ and compare to $\mathfrak P(A) \cup \mathfrak P(B)$. What do you notice?
Using inclusion-exclusion we can see that
$$\begin{align*}
|\mathfrak P(A) \cup \mathfrak P(B)| & = |\mathfrak P(A)| + |\mathfrak P(B)| - |\mathfrak P(A\cap B)| \\
&= 2^{|A|} + 2^{|B|} - 2^{|A\cap B|} \\
& = 2^3 + 2^3 - 2^1 = 8 + 8 - 2 = 14
\end{align*}$$
On the other hand
$$|\mathfrak P(A\cup B)| = 2^{|A\cup B|} = 2^5 = 32$$
A: First of all i would suggest the notation $2^S$ as a notation for "power set of S". Why?
Because the power set of $S$, $2^S$ has $2^{|S|}$ elements.
$2^A \cap 2^B = \{\emptyset,\{3\}\}$ as both elements are included in both $A$ and $B$.
$A \cup B = \{1,2,3,4,5\}$
The powerset of $A \cup B$ must have $2^5 = 32$ elements as there are 5 elements in $A \cup B$:
$2^{A \cup B} = \{\emptyset,\{1\},\{2\},\{3\},\{4\},\{5\},\{1,2\},\{1,3\},\{1,4\},\{1,5\},\{2,3\},\{2,4\},\{2,5\},\{3,4\},\{3,5\},\{4,5\},\{1,2,3\},\{1,2,4\},\{1,2,5\},\{1,3,4\},\{1,3,5\},\{1,4,5\},\{2,3,4\},\{2,3,5\},\{2,4,5\},\{3,4,5\},\{1,2,3,4\},\{1,2,3,5\},\{1,2,4,5\},\{1,3,4,5\},\{2,3,4,5\},\{1,2,3,4,5\}\}$
The power set of $A$ has $2^3 = 8$ elements (same for power set of $B$). The union therefore has a maximum of 16 elements (two less since $\emptyset$ and $\{3\}$ are included in both):
$2^A \cup 2^B = \{\emptyset,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\},\{4\},\{5\},\{3,4\},\{3,5\},\{4,5\},\{3,4,5\}\}$
When to include a set  to $2^A$?
Whenever all elements of the set are elements of $A$ i.e a subset of $A$.
