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).
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AA
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Bϕ
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C{(5, a), (6, a), (7, a), (5, b), (6, b), (7, b)}
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D{(a, 5), (a, 6), (a, 7), (b, 5), (b, 6), (b, 7)}
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E{(a, 2), (a, 3)}
Answer
Correct Answer: {(a, 5), (a, 6), (a, 7), (b, 5), (b, 6), (b, 7)}
Explanation
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)}