Finding the Value of m² + n² from the Number of Subsets

This problem involves two finite sets with a given difference in the number of their subsets. We are given that the first set has \(m\) elements and the second set has \(n\) elements. The total \( \textbf{number of subsets} \) of a set with \(k\) elements is given by \(2^k\).

According to the problem statement, the number of subsets of the first set is greater than the number of subsets of the second set by 56. This can be written as an equation:

\( 2^m - 2^n = 56 \)

Since \(2^m > 2^n\), it implies that \(m > n\). We can factor out the term with the smaller exponent, \(2^n\):

\( 2^n (2^{m-n} - 1) = 56 \)

Now, we need to find the values of \(n\) and \(m-n\) by considering the factors of 56. The term \(2^n\) must be a power of 2, and the term \((2^{m-n} - 1)\) must be an odd integer (since \(2^{m-n}\) is even for \(m-n > 0\)).

Let's look at the factors of 56:

• \(1 \times 56\)
• \(2 \times 28\)
• \(4 \times 14\)
• \(7 \times 8\)

We are looking for a pair of factors where one is a power of 2 and the other is an odd number.

• \(2^n\) must be a power of 2 among the factors: 1, 2, 4, 8.
• \((2^{m-n} - 1)\) must be an odd number among the other factors.

Consider the factor pair \(8 \times 7\):

• If we set \(2^n = 8\), then \(n = 3\) (since \(8 = 2^3\)).
• If we set \(2^{m-n} - 1 = 7\), then \(2^{m-n} = 7 + 1 = 8\).
• Since \(8 = 2^3\), we have \(2^{m-n} = 2^3\), which means \(m-n = 3\).

We have \(n = 3\) and \(m-n = 3\). Substituting the value of \(n\) into the second equation gives:

\( m - 3 = 3 \implies m = 6 \)

So, the values for the number of elements in the two \( \textbf{finite sets} \) are \(m=6\) and \(n=3\).

Let's verify this solution using the original equation:

\( 2^m - 2^n = 2^6 - 2^3 = 64 - 8 = 56 \)

This matches the condition given in the problem about the \( \textbf{number of subsets} \) difference.

Now, we need to find the \( \textbf{value of m\(\text{\textasciicircum}\)2 + n\(\text{\textasciicircum}\)2} \).

\( m^2 + n^2 = 6^2 + 3^2 \)

Calculate the squares:

• \(6^2 = 6 \times 6 = 36\)
• \(3^2 = 3 \times 3 = 9\)

Add the results:

\( m^2 + n^2 = 36 + 9 = 45 \)

The calculated \( \textbf{value of m\(\text{\textasciicircum}\)2 + n\(\text{\textasciicircum}\)2} \) is 45.

Comparing the Result with the Options

We found that \(m^2 + n^2 = 45\). Let's compare this with the given options:

The value 45 is not listed in options 1, 2, or 3. Therefore, the correct option is "None of the above".

Understanding the relationship between the \( \textbf{number of subsets} \) and the \( \textbf{elements} \) in \( \textbf{finite sets} \) is crucial for solving such problems. We successfully used the formula for the number of subsets and solved the exponential equation to find the values of \(m\) and \(n\), and subsequently the desired \( \textbf{value of m\(\text{\textasciicircum}\)2 + n\(\text{\textasciicircum}\)2} \).