Question

If \( A = \{1, 2, 5, 6\} \) and \( B = \{1, 2, 3\} \), then \( A \times B \cap B \times A \) is equal to:

• A. \( \{(1, 1), (2, 1), (6, 1), (3, 2)\}
• B. \( \{(1, 1), (2, 1), (3, 2), (2, 2)\}
• C. \( \{(1, 1), (2, 1), (6, 1), (2, 2), (3, 2)\}
• D. \( \{(1, 1), (2, 1), (6, 1), (2, 2), (3, 2), (2, 5)\} \)

Hint:

To find the intersection of two Cartesian products, list all pairs in both products and identify common pairs.

Answer 1

The Correct Option is B

We are asked to find the intersection of \( A \times B \) and \( B \times A \). Step 1: Find \( A \times B \) The Cartesian product \( A \times B \) consists of all ordered pairs where the first element is from \( A \) and the second element is from \( B \): \[ A \times B = \{(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (5, 1), (5, 2), (5, 3), (6, 1), (6, 2), (6, 3)\} \] Step 2: Find \( B \times A \) The Cartesian product \( B \times A \) consists of all ordered pairs where the first element is from \( B \) and the second element is from \( A \): \[ B \times A = \{(1, 1), (1, 2), (1, 5), (1, 6), (2, 1), (2, 2), (2, 5), (2, 6), (3, 1), (3, 2), (3, 5), (3, 6)\} \] Step 3: Find the intersection The intersection \( A \times B \cap B \times A \) is the set of pairs that are in both products: \[ A \times B \cap B \times A = \{(1, 1), (2, 1), (3, 2), (2, 2)\} \] Thus, the correct answer is \( \{(1, 1), (2, 1), (3, 2), (2, 2)\} \).