Question

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

• A. 9
• B. 13
• C. 8
• D. 10

Hint:

When dealing with differences of powers of 2 (like \(2^m - 2^n\)), always factor out the smaller power to get a product of a power of 2 and an odd number \(2^n(2^{m-n} - 1)\). This makes comparing with prime factorizations straightforward.

Answer 1

The Correct Option is A

To find the value of \( m + n \), we need to explore the number of subsets of two sets \( A \) and \( B \). Given that \( n(A) = m \) and \( n(B) = n \), and that the number of subsets of \( A \) is 56 more than the number of subsets of \( B \), let's analyze the situation:

The number of subsets of a set with \( k \) elements is \( 2^k \). Therefore, the number of subsets of set \( A \) is \( 2^m \), and the number of subsets of set \( B \) is \( 2^n \).
According to the problem:

2^m = 2^n + 56

We are tasked with finding \( m + n \) given the equation above. We can solve this by trial and error or algebraic manipulation.

First, calculate using trial:

1. Assume \( n = 3 \), then \( 2^n = 2^3 = 8 \).
2. Thus, we need \( 2^m = 8 + 56 = 64 \), which implies \( m = 6 \) because \( 2^6 = 64 \).
3. Thus, \( m = 6 \) and \( n = 3 \), therefore \( m+n = 6 + 3 = 9 \).

Finally, the correct answer is that the sum \( m + n \) equals \(9\).