Intersection of two Cartesian products: For A = {1,2,3}, B = {2,3,4}, C = {1,3,4}, D = {2,4,5}, compute (A × B) ∩ (C × D).

Introduction / Context:A well-known identity: (A × B) ∩ (C × D) = (A ∩ C) × (B ∩ D). We justify it by unpacking ordered-pair membership conditions.

Given Data / Assumptions:

• A = {1,2,3}, B = {2,3,4}
• C = {1,3,4}, D = {2,4,5}

Concept / Approach:(x,y) lies in both products iff x ∈ A and x ∈ C, and y ∈ B and y ∈ D. Thus x ∈ A ∩ C and y ∈ B ∩ D.

Step-by-Step Solution:A ∩ C = {1,3}B ∩ D = {2,4}(A ∩ C) × (B ∩ D) = {(1,2),(1,4),(3,2),(3,4)}

Verification / Alternative check:Direct enumeration from both sides matches—pairs not satisfying both coordinate conditions drop out.

Why Other Options Are Wrong:(A ∩ D) × (B ∩ C) swaps intersections incorrectly; ϕ is false since intersections are nonempty; “None” contradicts the identity.

Common Pitfalls:Interchanging first and second coordinate intersections; always match first-with-first, second-with-second.

Final Answer:(A ∩ C) × (B ∩ D)