Question

Let \( A = \{ 1, 2, 5 \} \), \( B = \{ 1, 2, 3, 4 \} \), and \( C = \{ 2, 5, 6 \} \) be the three sets. If \( D = [A \times (B \cap C)] \cap [(A - B) \times C] \), then which of the following is true?

• A. \( (1, 2) \in D \)
• B. \( (5, 6) \in D \)
• C. \( (5, 2) \in D \)
• D. \( (5, 1) \in D \)

Hint:

For set products and differences, carefully follow the set operations to generate pairs as per the question's instructions.

Answer 1

The Correct Option is C

First, calculate the intersection \( B \cap C \): \[ B \cap C = \{ 2 \} \] Now, calculate the sets \( A \times (B \cap C) \) and \( (A - B) \times C \): \[ A \times (B \cap C) = \{ (1, 2), (2, 2), (5, 2) \} \] \[ A - B = \{ 5 \} \] \[ (A - B) \times C = \{ (5, 2), (5, 5) \} \] The intersection of these two sets gives: \[ D = \{ (5, 2) \} \] Thus, the correct answer is \( (5, 2) \).