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 m and n elements, respectively, we know the number of subsets for each set. The number of subsets of a set with k elements is given by 2k. According to the problem:
2m=2n+56
We need to find the values of m and n. To solve this, we can follow these steps:
Start by rewriting the equation:
2m−2n=56
Notice that 2m and 2n 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.