Question

A and B are two sets having 3 and 6 elements respectively. Consider the following statements: - Statement (I): Minimum number of elements in \( A \cup B \) is 3 - Statement (II): Maximum number of elements in \( A \cap B \) is 3 Which of the following is correct?

• A. Statement (I) is false, statement (II) is true.
• B. Both statements (I) and (II) are true.
• C. Both statements (I) and (II) are false.
• D. Statement (I) is true, statement (II) is false.

Hint:

To find the minimum or maximum number of elements in union or intersection of sets, consider the possible overlaps or lack of overlap.

Answer 1

The Correct Option is A

Given: - \( n(A) = 3 \) - \( n(B) = 6 \) From the set theory, we know: - The minimum number of elements in \( A \cup B \) is \( n(A \cup B) \geq \max(n(A), n(B)) - n(A \cap B) \). - The maximum number of elements in \( A \cap B \) is \( \min(n(A), n(B)) \). The minimum number of elements in \( A \cup B \) occurs when \( A \) and \( B \) have no common elements, so: \[ n(A \cup B) = 3 + 6 = 9. \] Thus, statement (I) is false because the minimum number of elements in \( A \cup B \) cannot be 3. The maximum number of elements in \( A \cap B \) is 3, which is the smaller of the two set sizes. Therefore, statement (II) is true. Thus, the correct answer is option (1).