Question

If \(A=\{x \mid x^2-5x+6=0\}\), \(B=\{2,4\}\), \(C=\{4,5\}\), then \(A \times (B \cap C)=\)

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

Hint:

To evaluate Cartesian products:
First simplify the given sets
Find intersections before forming the product
\(A \times B\) contains ordered pairs \((a,b)\)

Answer 1

The Correct Option is A

Step 1: Find set \(A\) by solving the quadratic equation: \[ x^2-5x+6=0 \] \[ (x-2)(x-3)=0 \] \[ \Rightarrow x=2,3 \] \[ A=\{2,3\} \]
Step 2: Find the intersection of sets \(B\) and \(C\): \[ B=\{2,4\}, \quad C=\{4,5\} \] \[ B \cap C=\{4\} \]
Step 3: Form the Cartesian product: \[ A \times (B \cap C)=\{(2,4),(3,4)\} \]