Let's examine each option one by one. Let's take two sets A= {1,2} and B={1,2,3}
Option a:
A'={3} and B'={} we can clearly see B' is a subset of A'. This is true
Option b:
for this let's take A={1,2}
so, P(A)= {{}, {1}, {2}, {1,2}}
now you see there is absolutely no intersection between the two sets, this is because in the powerset "{1}" is an element. You need to understand "{1}" is different than "1".
Thus this is also true.
Option c:
This is false, you see I just proved that there is no intersection between the two sets in other words the intersection is a null set or phi.
Option d:
This is also true, let me clear this with an example.
A={1,2}
then P(A)= {{}, {1}, {2}, {1,2}}
and P(P(A))= {{}, {1}, {2}, {1,2}, {} {1}, {} {2}, {} {1,2}, {1}{2}, {2} {1,2}, {1} {1,2}, {2} {1,2}, {}{1}{2}, {}{1}{1,2}, {1}{2}{1,2}, { {}, {1}, {2}, {1,2} } }
Now you can see the only intersection between these two will be the null set, FYI: after the first P(A) no matter how many times you apply the Powerset operation, you will always get the null set and that will be the only intersection as well.
PS: I think in the P(P(A)) I might have missed out a couple of "{}" here and there but this was just to give you the idea. I hope you'll never struggle with this concept again. :)
