Product and set difference identity: If A = {a, d}, B = {b, c, e} and C = {b, c, f}, evaluate A × (B − C).

Introduction / Context:Cartesian products interact with difference: A × (B − C) = (A × B) − (A × C). We compute B − C and check the identity.

Given Data / Assumptions:

• B − C = {e}
• A = {a, d}

Concept / Approach:Pairs in A × (B − C) must have second coordinate in B but not in C. That equals all pairs from A × B with those from A × C removed.

Step-by-Step Solution:A × (B − C) = {(a,e), (d,e)}(A × B) = {(a,b),(a,c),(a,e),(d,b),(d,c),(d,e)}(A × C) = {(a,b),(a,c),(d,b),(d,c)}Difference = {(a,e),(d,e)}

Verification / Alternative check:Elementwise condition matches exactly: keep y ∈ B and y ∉ C.

Why Other Options Are Wrong:They either include too many pairs or the wrong second components.

Common Pitfalls:Forgetting to remove all pairs whose second coordinate lies in C.

Final Answer:(A × B) − (A × C)