Let A and B = 0, A' and B = v. This means that (A and B) and (A' and B) = B = 0. So B is the empty set.
This means that for any A', A' and B = 0 since B is the empty set.
So its not possible to satisfy your condition.
I will also denote the complement of X by X'.
A' ∩ B' ≠ ∅ holds for most A and B, more precisely, unless A ∪ B is the whole universal set. So, you just need to find two small disjoint sets.