Cartesian product with an intersection — maintain ordered pair order: If A = {a, b}, B = {2, 3, 5, 6, 7}, C = {5, 6, 7, 8, 9}, find A × (B ∩ C).

Introduction / Context:
Cartesian products are ordered pairs (first from the first set, second from the second set). We first intersect B and C, then pair each element of A with each element of that intersection.

Given Data / Assumptions:

• A = {a, b}
• B ∩ C = {5,6,7}

Concept / Approach:
Form all (x, y) with x ∈ A and y ∈ (B ∩ C). Keep the order (x first, y second).

Step-by-Step Solution:
Pairs: (a,5), (a,6), (a,7), (b,5), (b,6), (b,7)

Verification / Alternative check:
Count check: |A| = 2, |B ∩ C| = 3, so |A × (B ∩ C)| = 6; the listed set has 6 pairs.

Why Other Options Are Wrong:
Option (c) reverses order (y, x); others have wrong elements or cardinalities.

Common Pitfalls:
Swapping coordinates or forgetting to intersect before forming pairs.

Final Answer:
{(a, 5), (a, 6), (a, 7), (b, 5), (b, 6), (b, 7)}