Cartesian product distributes over union: Given A = {a, d}, B = {b, c, e}, and C = {b, c, f}, compute A × (B ∪ C).

Introduction / Context:For sets, the Cartesian product distributes over union: X × (Y ∪ Z) = (X × Y) ∪ (X × Z). This parallels algebraic distributivity.

Given Data / Assumptions:

• A = {a, d}
• B = {b, c, e}
• C = {b, c, f}

Concept / Approach:Use the identity above to avoid listing all ordered pairs explicitly.

Step-by-Step Solution:B ∪ C = {b, c, e, f}A × (B ∪ C) = {(a,x), (d,x) : x ∈ {b,c,e,f}}This equals (A × B) ∪ (A × C) by distributivity

Verification / Alternative check:Explicit enumeration yields 8 ordered pairs, the same as the union of A × B (6 pairs) and A × C (6 pairs) minus their overlap A × {b,c} (4 pairs).

Why Other Options Are Wrong:Intersection gives too few pairs; empty set is impossible.

Common Pitfalls:Confusing set-theoretic union/intersection with arithmetic operations.

Final Answer:(A × B) ∪ (A × C)