Question

Let A and B be two finite sets such that n(A - B), n(A \(\cap\) B), n(B - A) are in an arithmetic progression. Here n(X) denotes the number of elements in a finite set X. If n(A \(\cup\) B) = 18, then n(A) + n(B) is ________

• A. 36
• B. 30
• C. 27
• D. 24

Hint:

Drawing a Venn diagram can be extremely helpful to visualize the relationship between \( n(A-B) \), \( n(B-A) \), and \( n(A \cap B) \). These three quantities represent the three disjoint regions that make up the union \( A \cup B \).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem combines set theory with arithmetic progressions. We need to use the fundamental formulas of set theory and the property of an AP to solve for the required quantity.
Step 2: Key Formula or Approach:
1. If three numbers x, y, z are in AP, then \( 2y = x + z \). 2. The formula for the union of two sets is \( n(A \cup B) = n(A-B) + n(B-A) + n(A \cap B) \). 3. The formula relating union and intersection is \( n(A) + n(B) = n(A \cup B) + n(A \cap B) \).
Step 3: Detailed Explanation:
Let \( x = n(A-B) \), \( y = n(A \cap B) \), and \( z = n(B-A) \). We are given that x, y, z are in an arithmetic progression. This means the middle term is the average of the other two: \[ 2y = x + z \] \[ 2 \cdot n(A \cap B) = n(A-B) + n(B-A) \quad (\text{Equation 1}) \] We are also given that \( n(A \cup B) = 18 \). Using the formula for the union based on disjoint partitions: \[ n(A \cup B) = n(A-B) + n(B-A) + n(A \cap B) \] \[ 18 = (n(A-B) + n(B-A)) + n(A \cap B) \quad (\text{Equation 2}) \] Now, we can substitute the relationship from Equation 1 into Equation 2. \[ 18 = (2 \cdot n(A \cap B)) + n(A \cap B) \] \[ 18 = 3 \cdot n(A \cap B) \] Solving for the size of the intersection: \[ n(A \cap B) = \frac{18}{3} = 6 \] The question asks for \( n(A) + n(B) \). We can use the principle of inclusion-exclusion: \[ n(A) + n(B) = n(A \cup B) + n(A \cap B) \] We are given \( n(A \cup B) = 18 \) and we just found \( n(A \cap B) = 6 \). \[ n(A) + n(B) = 18 + 6 = 24 \] Step 4: Final Answer:
The value of n(A) + n(B) is 24.