Set A And Set B Have { M $}$ And { N $}$ Elements, Respectively. If The Total Number Of Subsets Of Set A Is 480 More Than The Total Number Of Subsets Of Set B, Find The Values Of { M $}$ And { N $}$.[1.5 Points]
Introduction
In the realm of mathematics, subsets play a crucial role in understanding the properties of sets. Given two sets, A and B, with m and n elements respectively, we are tasked with finding the values of m and n when the total number of subsets of set A is 480 more than the total number of subsets of set B. In this article, we will delve into the world of subsets, exploring the concept of power sets and the relationship between the number of elements in a set and the number of its subsets.
The Power Set
A power set is a set that contains all possible subsets of a given set. For example, if we have a set A = {a, b, c}, then the power set of A, denoted as P(A), is {{}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}. The power set of a set with n elements has 2^n subsets.
The Relationship Between the Number of Elements and the Number of Subsets
The number of subsets of a set with n elements is given by 2^n. This is because each element in the set can either be included or excluded from a subset, resulting in 2 possibilities for each element. Therefore, the total number of subsets is 2^n.
The Problem
We are given that set A has m elements and set B has n elements. The total number of subsets of set A is 480 more than the total number of subsets of set B. Mathematically, this can be represented as:
2^m = 2^n + 480
Simplifying the Equation
To simplify the equation, we can subtract 2^n from both sides, resulting in:
2^m - 2^n = 480
Using Properties of Exponents
We can use the property of exponents that states a^m - a^n = a^(m-n) when a is not equal to 1. Applying this property to our equation, we get:
2^(m-n) = 480
Finding the Value of m-n
To find the value of m-n, we can take the logarithm base 2 of both sides of the equation. This gives us:
m-n = log2(480)
Evaluating the Logarithm
Using a calculator or a logarithm table, we can evaluate the logarithm base 2 of 480. This gives us:
m-n ≈ 8.97
Rounding to the Nearest Integer
Since m and n are integers, we can round the value of m-n to the nearest integer. This gives us:
m-n ≈ 9
Finding the Values of m and n
Now that we have the value of m-n, we can find the values of m and n. We know that m-n = 9, so we can write:
m = n + 9
Substituting the Value of m
Substituting the value of m into the original equation, we get:
2^(n+9) = 2^n + 480
Simplifying the Equation
Using the property of exponents that states a^(m+n) = a^m * a^n, we can simplify the equation to:
2^n * 2^9 = 2^n + 480
Cancelling Out 2^n
Cancelling out 2^n from both sides of the equation, we get:
2^9 = 480
Evaluating the Exponent
Using a calculator or a logarithm table, we can evaluate the exponent 2^9. This gives us:
2^9 = 512
Finding the Value of n
Since 2^9 is not equal to 480, we made an error in our previous steps. Let's go back and re-evaluate our equation.
2^m - 2^n = 480
Using Trial and Error
We can use trial and error to find the values of m and n. Let's start by trying different values of m and n.
Trying m = 10 and n = 8
If m = 10 and n = 8, then:
2^10 - 2^8 = 1024 - 256 = 768
Trying m = 11 and n = 8
If m = 11 and n = 8, then:
2^11 - 2^8 = 2048 - 256 = 1792
Trying m = 10 and n = 7
If m = 10 and n = 7, then:
2^10 - 2^7 = 1024 - 128 = 896
Trying m = 10 and n = 6
If m = 10 and n = 6, then:
2^10 - 2^6 = 1024 - 64 = 960
Trying m = 10 and n = 5
If m = 10 and n = 5, then:
2^10 - 2^5 = 1024 - 32 = 992
Trying m = 10 and n = 4
If m = 10 and n = 4, then:
2^10 - 2^4 = 1024 - 16 = 1008
Trying m = 10 and n = 3
If m = 10 and n = 3, then:
2^10 - 2^3 = 1024 - 8 = 1016
Trying m = 10 and n = 2
If m = 10 and n = 2, then:
2^10 - 2^2 = 1024 - 4 = 1020
Trying m = 10 and n = 1
If m = 10 and n = 1, then:
2^10 - 2^1 = 1024 - 2 = 1022
Trying m = 10 and n = 0
If m = 10 and n = 0, then:
2^10 - 2^0 = 1024 - 1 = 1023
Trying m = 9 and n = 7
If m = 9 and n = 7, then:
2^9 - 2^7 = 512 - 128 = 384
Trying m = 9 and n = 6
If m = 9 and n = 6, then:
2^9 - 2^6 = 512 - 64 = 448
Trying m = 9 and n = 5
If m = 9 and n = 5, then:
2^9 - 2^5 = 512 - 32 = 480
Conclusion
We have found that if m = 9 and n = 5, then the total number of subsets of set A is 480 more than the total number of subsets of set B. Therefore, the values of m and n are 9 and 5, respectively.
References
- [1] "Set Theory" by John L. Kelley
- [2] "Discrete Mathematics" by Kenneth H. Rosen
Frequently Asked Questions: Set A and Set B =============================================
Q: What is the relationship between the number of elements in a set and the number of its subsets?
A: The number of subsets of a set with n elements is given by 2^n. This is because each element in the set can either be included or excluded from a subset, resulting in 2 possibilities for each element.
Q: How do we find the total number of subsets of a set?
A: To find the total number of subsets of a set, we use the formula 2^n, where n is the number of elements in the set.
Q: What is the difference between the total number of subsets of set A and set B?
A: The total number of subsets of set A is 480 more than the total number of subsets of set B.
Q: How do we find the values of m and n?
A: We can use trial and error to find the values of m and n. We can start by trying different values of m and n, and then check if the difference between the total number of subsets of set A and set B is 480.
Q: What are the values of m and n?
A: The values of m and n are 9 and 5, respectively.
Q: Why did we try different values of m and n?
A: We tried different values of m and n because we wanted to find the values that satisfy the condition that the total number of subsets of set A is 480 more than the total number of subsets of set B.
Q: What is the significance of the values of m and n?
A: The values of m and n are significant because they represent the number of elements in set A and set B, respectively. The values of m and n are used to calculate the total number of subsets of each set.
Q: How do we use the values of m and n to calculate the total number of subsets of each set?
A: We use the formula 2^n to calculate the total number of subsets of each set, where n is the number of elements in the set.
Q: What is the relationship between the values of m and n and the total number of subsets of each set?
A: The values of m and n are related to the total number of subsets of each set through the formula 2^n. The values of m and n are used to calculate the total number of subsets of each set.
Q: Why is it important to understand the relationship between the values of m and n and the total number of subsets of each set?
A: It is important to understand the relationship between the values of m and n and the total number of subsets of each set because it helps us to calculate the total number of subsets of each set accurately.
Q: How do we apply the concept of subsets to real-world problems?
A: We can apply the concept of subsets to real-world problems by using it to calculate the total number of possible outcomes in a situation. For example, if we have a set of possible outcomes for a coin toss, we can use the concept of subsets to calculate the total number of possible outcomes.
Q: What are some examples of real-world problems that involve subsets?
A: Some examples of real-world problems that involve subsets include:
- Calculating the total number of possible outcomes in a coin toss
- Calculating the total number of possible outcomes in a roll of a die
- Calculating the total number of possible outcomes in a deck of cards
- Calculating the total number of possible outcomes in a situation where we have multiple choices
Q: How do we use the concept of subsets to solve real-world problems?
A: We use the concept of subsets to solve real-world problems by applying the formula 2^n to calculate the total number of possible outcomes in a situation. We can also use the concept of subsets to identify the possible outcomes in a situation and to calculate the probability of each outcome.