Product distributes over intersection: If A = {a, d}, B = {b, c, e} and C = {b, c, f}, determine A × (B ∩ C).

Introduction / Context:Cartesian product respects intersections: A × (B ∩ C) = (A × B) ∩ (A × C). We confirm this using elementwise reasoning.

Given Data / Assumptions:

• B ∩ C = {b, c}
• A = {a, d}

Concept / Approach:Any pair (x, y) is in A × (B ∩ C) iff x ∈ A and y ∈ B and y ∈ C. This is exactly the definition of being simultaneously in A × B and A × C.

Step-by-Step Solution:A × (B ∩ C) = {(a,b),(a,c),(d,b),(d,c)}(A × B) ∩ (A × C) = same four pairs

Verification / Alternative check:Subset checks both ways show equality straightforwardly.

Why Other Options Are Wrong:Union would include pairs with y in B or C (too many); ϕ is false; “None” contradicts the law above.

Common Pitfalls:Confusing distribution over union versus intersection; note that A × (B ∪ C) = (A × B) ∪ (A × C) (different identity).

Final Answer:(A × B) ∩ (A × C)