Question

If U = { 1, 2, 3, 4, 5, 6, 7, 8 } is the universal set, A = { 2, 3, 6, 7 }, B = { 3, 4, 5, 6 }, and C = { 5, 6, 7, 8 }, then n[̅A ∩ (C - B)] + n[̅A ∩ (B ∨ C)] = k, where ̅A is the complement of set A. The value of k is:

• A. 2
• B. 3
• C. 4
• D. 7

Hint:

When working with set operations, remember to first calculate the union and difference, and then evaluate the intersection.

Answer 1

The Correct Option is C

We are given the universal set \( U \), and the sets \( A \), \( B \), and \( C \). First, we need to evaluate the two expressions:
1. \( A \cap (C - B) \): \( C - B = \{ 5, 6, 7, 8 \} - \{ 3, 4, 5, 6 \} = \{ 7, 8 \} \) So, \( A \cap (C - B) = \{ 2, 3, 6, 7 \} \cap \{ 7, 8 \} = \{ 7 \} \), and \( n[A \cap (C - B)] = 1 \).
2. \( A \cap (B \cup C) \): \( B \cup C = \{ 3, 4, 5, 6 \} \cup \{ 5, 6, 7, 8 \} = \{ 3, 4, 5, 6, 7, 8 \} \) So, \( A \cap (B \cup C) = \{ 2, 3, 6, 7 \} \cap \{ 3, 4, 5, 6, 7, 8 \} = \{ 3, 6, 7 \} \), and \( n[A \cap (B \cup C)] = 3 \).
Thus, \( k = 1 + 3 = 4 \).