Question

If \[ A = \{2, 4\}, \quad B = \{3, 4, 5\}, \quad \text{then} \quad (A \cap B) \times (A \cup B) = \]

• A. \( \{(3, 2), (3, 4), (4, 4), (5, 4)\} \)
• B. \( \{(2, 3), (2, 4), (2, 5)\} \)
• C. \( \{(4, 2), (4, 3), (4, 4), (4, 5)\} \)
• D. \( \{(4, 3), (4, 4), (4, 5)\} \)

Hint:

To compute the Cartesian product, first find the intersection and union of the sets, then form the ordered pairs from each element in the intersection with every element in the union.

Answer 1

The Correct Option is C

Step 1: Find the intersection and union of sets.
The intersection \( A \cap B \) is the set of elements common to both sets \( A \) and \( B \): \[ A \cap B = \{4\}. \] The union \( A \cup B \) is the set of all elements in either set \( A \) or \( B \): \[ A \cup B = \{2, 3, 4, 5\}. \]
Step 2: Form the Cartesian product.
The Cartesian product \( (A \cap B) \times (A \cup B) \) is the set of all ordered pairs where the first element is from \( A \cap B \) and the second element is from \( A \cup B \): \[ (A \cap B) \times (A \cup B) = \{(4, 2), (4, 3), (4, 4), (4, 5)\}. \]
Step 3: Conclusion.
Thus, the correct answer is option (C).