Question

Let \( A \) and \( B \) be two finite sets with \( m \) and \( n \) elements respectively. The total number of subsets of the set \( A \) is 56 more than the total number of subsets of \( B \). Then the distance of the point \( P(m, n) \) from the point \( Q(-2, -3) \) is:

• A. 10
• B. 6
• C. 4
• D. 8

Answer 1

The Correct Option is A

To resolve the problem, determine the relationship between the cardinalities of sets \( A \) and \( B \), and then compute the distance between points \( P(m, n) \) and \( Q(-2, -3) \).

1. The number of subsets of a set with \( k \) elements is \( 2^k \). For set \( A \) with \( m \) elements, there are \( 2^m \) subsets, and for set \( B \) with \( n \) elements, there are \( 2^n \) subsets.
2. The problem states that \( 2^m = 2^n + 56 \).
3. Find values for \( m \) and \( n \) satisfying this equation:

• Assume \( 2^m = 64 \) (\( 2^6 \)). Then \( 2^n + 56 = 64 \), which implies \( 2^n = 8 \) (\( 2^3 \)). Therefore, \( m = 6 \) and \( n = 3 \).

Calculate the distance between points \( P(6, 3) \) and \( Q(-2, -3) \) using the distance formula:

The distance \( d \) between points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

1. Substitute \( (x_1, y_1) = (6, 3) \) and \( (x_2, y_2) = (-2, -3) \) into the formula:

\(d = \sqrt{((-2) - 6)^2 + ((-3) - 3)^2}\) \(d = \sqrt{(-8)^2 + (-6)^2}\) \(d = \sqrt{64 + 36}\) \(d = \sqrt{100}\) \(d = 10\)

The distance between point \( P(m, n) \) and point \( Q(-2, -3) \) is 10. The correct answer is 10.