Question

A set \(A\) has 3 elements and another set \(B\) has 6 elements. Then

• A. \(3 \le n(A\cup B) \le 6\)
• B. \(3 \le n(A\cup B) \le 9\)
• C. \(6 \le n(A\cup B) \le 9\)
• D. \(0 \le n(A\cup B) \le 9\)

Hint:

For two finite sets: \[ \max(n(A),n(B)) \le n(A\cup B) \le n(A)+n(B) \] Always consider extreme cases of intersection.

Answer 1

The Correct Option is C

Step 1: Use the formula for cardinality of union of two sets: \[ n(A\cup B)=n(A)+n(B)-n(A\cap B) \]
Step 2: Given: \[ n(A)=3,\quad n(B)=6 \]
Step 3: The intersection \(A\cap B\) can have: \[ 0 \le n(A\cap B) \le 3 \] (since the smaller set has 3 elements)
Step 4: Find minimum value of \(n(A\cup B)\): \[ n(A\cup B)_{\min}=3+6-3=6 \]
Step 5: Find maximum value of \(n(A\cup B)\): \[ n(A\cup B)_{\max}=3+6-0=9 \]
Step 6: Hence, \[ 6 \le n(A\cup B) \le 9 \]