Question

Let \( n(A) = m \) and \( n(B) = n \), if the number of subsets of \( A \) is 56 more than that of subsets of \( B \), then \( m + n \) is equal to:

• A. 9
• B. 13
• C. 8
• D. 10

Hint:

The number of subsets of a set with \( n \) elements is given by \( 2^n \). Factorization techniques are helpful when solving exponential equations.

Answer 1

The Correct Option is A

Step 1: Understanding the Given Condition
The number of subsets of a set \( A \) with \( m \) elements is given by: \[ 2^m \]
Similarly, the number of subsets of a set \( B \) with \( n \) elements is: \[ 2^n \]
It is given that: \[ 2^m - 2^n = 56 \]
Step 2: Expressing the Equation in Factorized Form
Factorizing the given equation: \[ 2^n(2^{m-n} - 1) = 56 \]
Expressing 56 as a product of powers of 2: \[ 2^3 \times (2^3 - 1) = 56 \]
Step 3: Comparing Both Sides
By comparing, we get: \[ 2^n = 2^3 \quad \text{and} \quad 2^{m-n} - 1 = 7 \]
Thus: \[ n = 3, \quad 2^{m-3} = 8 \Rightarrow m - 3 = 3 \]
So: \[ m = 6, \quad n = 3 \]
Step 4: Finding the Sum
The sum of \( m \) and \( n \) is: \[ m + n = 6 + 3 = 9 \]
Step 5: Matching with Options
The correct answer is \( 9 \). Final Answer: (A) 9.