Can anyone give me an example of two sets A and B such that A intersection B is empty but A complement intersection B complement is not?
I will also denote the complement of X by X'.
The OP needs A' ∩ B' ≠ ∅, not A' ∩ B ≠ ∅. Also, what is v and why is (A and B) and (A' and B) = B?
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.