Question

A set A has 3 elements and another set B has 6 elements. Then:

• A. 3 ≤ n(A ∪ B) ≤ 6
• B. 3 ≤ n(A ∪ B) ≤ 9
• C. 6 ≤ n(A ∪ B) ≤ 9
• D. 0 ≤ n(A ∪ B) ≤ 9

Hint:

When calculating the number of elements in the union of two sets, consider both the total number of elements and the overlap between the sets.

Answer 1

The Correct Option is C

Let the number of elements in set A be \( |A| = 3 \) and the number of elements in set B be \( |B| = 6 \). The number of elements in the union of two sets \( A \cup B \) is given by the formula: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B), \] where \( n(A \cap B) \) is the number of elements common to both sets A and B. The maximum value of \( n(A \cup B) \) occurs when the sets A and B have no common elements, i.e., \( n(A \cap B) = 0 \). In this case: \[ n(A \cup B) = 3 + 6 = 9. \] The minimum value of \( n(A \cup B) \) occurs when sets A and B are identical, i.e., \( n(A \cap B) = 3 \) (since set A has 3 elements). In this case: \[ n(A \cup B) = 3 + 6 - 3 = 6. \] Therefore, the number of elements in \( A \cup B \) is between 6 and 9, inclusive. Hence, the correct range is: \[ 6 \leq n(A \cup B) \leq 9. \]