Power Sets with Smallest and Largest Possible Cardinalities

Question

Say $|A| = 10$ and $|B| = 6$, what are the largest and smallest possible cardinalities for the following:

1. $\mathcal P(A\cap B)?$ Smallest is when the two sets didn't share anything so the intersection would be an empty set which makes the power set $1$. Largest would be if $B$ shared everything with $A$ so when you had them intersect their intersection would have $6$ elements, thus $\mathcal P(A\cap B) = 2^6.$

2. $\mathcal P(A\cup B)?$ Smallest would be if, again $B$ shared all of the elements with $A$ so their union would make the new set smaller, saying the number of elements in $A \cup B$ would be only $4$, making $\mathcal P(A \cup B) = 2^4$. Largest would be if $A$ and $B$ where two totally different sets, then taking their union would make the new set have $16$ elements, then making $\mathcal P(A \cup B) = 2^{16}.$

3. $\mathcal P(A \times B)?$ Smallest and largest would be the same here because if we take the Cartesian Product of two sets we distribute each element over the over elements in the set. Taking the Cartesian Product of $A$ and $B$ results in 60 new elements. So $\mathcal P(A \times B) = 2^{60}.$

My question is my thinking of power sets and sets correct? My biggest concern is for the Cartesian Product of two sets. If the two sets have the same element, would they still "distributed" through?

Answer 1

For 2, $A\cup B$ must include $A$, so its size is at least $10$.

The other answers are right.