Math Problem Statement

two finite sets having m and n elements. the total number of subsets of the first set is 56 more than the the total number of subsets of the second set. find the values of m and n.

Solution

Given two finite sets with mm and nn elements, respectively, we know the number of subsets for each set. The number of subsets of a set with kk elements is given by 2k2^k. According to the problem:

2m=2n+562^m = 2^n + 56

We need to find the values of mm and nn. To solve this, we can follow these steps:

  1. Start by rewriting the equation:

2m2n=562^m - 2^n = 56

  1. Notice that 2m2^m and 2n2^n are powers of 2, and so we can express 56 as a difference of two powers of 2. Let's list the differences between some powers of 2 to find a pair that matches 56:
2^6 - 2^5 &= 64 - 32 = 32 \\ 2^7 - 2^6 &= 128 - 64 = 64 \\ 2^8 - 2^7 &= 256 - 128 = 128 \\ 2^5 - 2^4 &= 32 - 16 = 16 \\ 2^6 - 2^4 &= 64 - 16 = 48 \\ 2^6 - 2^3 &= 64 - 8 = 56 \\ \end{align*}$$ From the above calculations, we see that: $$2^6 - 2^3 = 64 - 8 = 56$$ Thus, $$m = 6$$ and $$n = 3$$ is a solution. To verify, substitute $$m = 6$$ and $$n = 3$$ back into the equation: $$2^6 = 64, \quad 2^3 = 8 \quad \text{and} \quad 64 - 8 = 56$$ This confirms that $$m = 6$$ and $$n = 3$$ satisfy the given condition. Hence, the values of $$m$$ and $$n$$ are: $$\boxed{m = 6, \ n = 3}$$ Would you like more details or have any questions? Here are 8 related questions that expand on this topic: 1. How do you calculate the number of subsets of a given set? 2. What is the general formula for the number of subsets of a set with $$k$$ elements? 3. Can you explain why the difference between two powers of 2 can sometimes match specific numbers? 4. What are some properties of powers of 2 that make them useful in solving such problems? 5. How would the problem change if the difference were a different number, like 48 instead of 56? 6. What are some real-life applications of understanding the number of subsets of a set? 7. Could there be another pair of values for $$m$$ and $$n$$ that satisfies the equation $$2^m - 2^n = 56$$? 8. How can you generalize this approach to solve similar problems with different differences between the number of subsets? **Tip:** Always verify your solution by substituting the values back into the original equation to ensure they satisfy the given conditions.

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Math Problem Analysis

Mathematical Concepts

Exponents
Set Theory

Formulas

Number of subsets of a set with k elements: 2^k

Theorems

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Suitable Grade Level

High School