Both intersections (A-B)∩(B-A) and (A∩B) ∩ (A-B) must result in the empty set because the sets in each intersection share no common elements as they are defined by mutually exclusive conditions.
For (a) (A-B)∩(B-A) represents the intersection of the elements that are in A but not in B and the elements that are in B but not in A. Since these are mutually exclusive circumstances, where one set contains only elements not in the other, there can be no overlap, hence the intersection of (A-B) and (B-A) must be the empty set.
For (b), (A∩B) ∩ (A-B) is examining the intersection of A and B, with the set of elements in A that are not in B. Any element that is in the intersection of A and B is by definition not in the set (A-B), since for any element x, if x is in B, it cannot simultaneously be in the set defining elements not in B. Therefore, this intersection is also the empty set.
The final answer in both cases is the empty set, reflecting that there is no common element between sets described in each scenario.
So, both (A-B)∩(B-A) and (A∩B) ∩ (A-B) correctly result in the empty set because the described sets have no elements in common by definition of set subtraction and intersection.