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

Step 1: Understanding the Question:
The number of subsets of a set with \(k\) elements is given by \(2^k\). We are given the number of elements in sets A and B, and a relationship between their number of subsets. We need to find the sum of their cardinalities, \(m + n\).


Step 2: Key Formula or Approach:
Number of subsets of \(A = 2^m\)
Number of subsets of \(B = 2^n\)
Given condition: \(2^m - 2^n = 56\)


Step 3: Detailed Explanation:
From the equation \(2^m - 2^n = 56\), we can factor out \(2^n\): \[ 2^n(2^{m-n} - 1) = 56 \] Now, we express 56 as a product of an odd number and a power of 2: \[ 56 = 8 \times 7 = 2^3 \times 7 \] So, we have: \[ 2^n(2^{m-n} - 1) = 2^3 \times 7 \] By comparing the power of 2 and the odd factor on both sides, we get: \[ 2^n = 2^3 \implies n = 3 \] \[ 2^{m-n} - 1 = 7 \] \[ 2^{m-3} = 7 + 1 = 8 \] \[ 2^{m-3} = 2^3 \] Equating the exponents: \[ m - 3 = 3 \implies m = 6 \] We need to find the value of \(m + n\): \[ m + n = 6 + 3 = 9 \]

Step 4: Final Answer:
The value of \(m + n\) is 9.