If A and B be any two sets such that A - B = A, then A ∩ B is

For any two sets A and B, the set difference A − B = {x : x ∈ A and x ∉ B} keeps exactly the elements of A that are absent from B. If that result still equals the whole of A, no element of A has been removed — which is possible only when no element of A is present in B at all. Two sets with no elements in common are called disjoint sets.

1. Write the given condition: A − B = A.

2. Apply the definition of set difference, A − B = {x ∈ A : x ∉ B}. Setting this equal to A means every element x ∈ A must satisfy x ∉ B — so no element of A lies in B.

3. Since no element of A is in B, sets A and B have no elements in common; A and B are disjoint.

4. By the definition of intersection, two disjoint sets share no common elements, so A ∩ B = ∅.

• A ∩ B = A would require A ⊆ B, i.e. every element of A also lying in B — the exact opposite of what A − B = A gives. The two can agree only in the degenerate case A = ∅, so A is not the value of the intersection in general.

• A ∩ B = B would require B ⊆ A, i.e. every element of B also lying in A. The given condition says nothing that places B inside A, and the two agree only when B = ∅, so B is not the value in general.

• A ∩ B = A ∪ B forces A = B, because the intersection is contained in the union and equals it only when the two sets coincide. Together with A − B = A this would need A = B = ∅, so A ∪ B is not the value in general.

• The stem fixes A and B as any two sets satisfying A − B = A, so the answer must be the value that holds for every such pair — and only ∅ does. The other three coincide with the intersection at most in an isolated empty-set case, never as the general result.

Cross-check with a concrete example: let A = {1, 2, 3} and B = {4, 5}. Then A − B = {1, 2, 3} = A, which matches the given condition, and indeed A ∩ B = { } = ∅ — consistent with the general result above.