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 \( 2^m \). 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 equation yields \( 2^n(2^{m-n} - 1) = 56 \). Expressing 56 as a product of powers of 2 gives \( 2^3 \times (2^3 - 1) = 56 \).
Step 3: Comparing Both Sides
By comparing the two forms, we find \( 2^n = 2^3 \) and \( 2^{m-n} - 1 = 7 \). This implies \( n = 3 \) and \( 2^{m-3} = 8 \), which means \( m - 3 = 3 \). Therefore, \( m = 6 \) and \( n = 3 \).
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.