IfA (x:x Is A Prime Number, X < 12 ) And B={x:x Is A Prime Factor Of 30; Then Show That N(AB)+n(AB) N (A)+n (B) 6x4=24
In this article, we will delve into the world of prime numbers and sets to solve a problem involving the intersection and union of two sets, A and B. We will start by defining the sets A and B, and then proceed to find the number of elements in the intersection and union of these sets.
Defining Set A
Set A is defined as the set of all prime numbers less than 12. This means that the elements of set A are the prime numbers 2, 3, 5, 7, and 11.
Defining Set B
Set B is defined as the set of all prime factors of 30. The prime factors of 30 are 2, 3, and 5. Therefore, the elements of set B are 2, 3, and 5.
Finding the Intersection of Sets A and B
The intersection of two sets is the set of elements that are common to both sets. In this case, the intersection of sets A and B is the set of elements that are prime numbers less than 12 and are also prime factors of 30. The only element that satisfies this condition is 2, 3, and 5.
Finding the Union of Sets A and B
The union of two sets is the set of all elements that are in either set. In this case, the union of sets A and B is the set of all prime numbers less than 12 and all prime factors of 30. The elements of the union of sets A and B are 2, 3, 5, 7, and 11.
Finding the Number of Elements in the Intersection and Union of Sets A and B
Now that we have found the intersection and union of sets A and B, we can find the number of elements in each. The intersection of sets A and B has 3 elements, and the union of sets A and B has 5 elements.
Solving the Problem
The problem asks us to show that n(AB)+n(AB) n (A)+n (B) 6x4=24. To solve this problem, we need to find the number of elements in the intersection and union of sets A and B, and then use these values to find the value of the expression.
Step 1: Finding the Number of Elements in the Intersection of Sets A and B
The intersection of sets A and B has 3 elements.
Step 2: Finding the Number of Elements in the Union of Sets A and B
The union of sets A and B has 5 elements.
Step 3: Finding the Number of Elements in the Intersection of Sets A and B and the Union of Sets A and B
The intersection of sets A and B and the union of sets A and B is the set of elements that are in both the intersection and union of sets A and B. In this case, the intersection of sets A and B and the union of sets A and B is the set of elements that are prime numbers less than 12 and are also prime factors of 30. The only element that satisfies this condition is 2, 3, and 5.
Step 4: Finding the Value of the Expression
Now that we have found the number of elements in the intersection and union of sets A and B, and the intersection of sets A and B and the union of sets A and B, we can find the value of the expression.
n(AB) = 3 n(AB) n (A) = 3 x 5 = 15 n (B) = 3 n(AB) + n(AB) n (A) + n (B) = 3 + 15 + 3 = 21
However, the problem states that n(AB)+n(AB) n (A)+n (B) 6x4=24. To find the value of the expression, we need to find the value of 6x4.
Step 5: Finding the Value of 6x4
To find the value of 6x4, we need to multiply 6 by 4.
6 x 4 = 24
Conclusion
In the previous article, we delved into the world of prime numbers and sets to solve a problem involving the intersection and union of two sets, A and B. However, we may still have some questions about the problem and its solution. In this article, we will address some of the most frequently asked questions about the problem.
Q: What is the definition of a prime number?
A: A prime number is a positive integer that is divisible only by itself and 1. For example, the prime numbers less than 12 are 2, 3, 5, 7, and 11.
Q: What is the definition of a set?
A: A set is a collection of unique objects, known as elements or members. For example, the set A = {2, 3, 5, 7, 11} is a set of prime numbers less than 12.
Q: What is the intersection of two sets?
A: The intersection of two sets is the set of elements that are common to both sets. For example, the intersection of sets A and B is the set of elements that are prime numbers less than 12 and are also prime factors of 30.
Q: What is the union of two sets?
A: The union of two sets is the set of all elements that are in either set. For example, the union of sets A and B is the set of all prime numbers less than 12 and all prime factors of 30.
Q: How do we find the number of elements in the intersection and union of sets A and B?
A: To find the number of elements in the intersection and union of sets A and B, we need to identify the elements that are common to both sets and the elements that are in either set.
Q: How do we find the value of the expression n(AB)+n(AB) n (A)+n (B)?
A: To find the value of the expression n(AB)+n(AB) n (A)+n (B), we need to find the number of elements in the intersection and union of sets A and B, and then use these values to find the value of the expression.
Q: Why is the value of the expression not equal to 24?
A: The value of the expression is not equal to 24 because the number of elements in the intersection and union of sets A and B is not equal to 24. The correct value of the expression is 21.
Q: What is the significance of the problem?
A: The problem is significant because it involves the intersection and union of two sets, which is a fundamental concept in mathematics. Understanding this concept is essential for solving problems in mathematics and other fields.
Q: How can I apply the concept of intersection and union of sets to real-life problems?
A: The concept of intersection and union of sets can be applied to real-life problems in various fields, such as computer science, data analysis, and engineering. For example, in computer science, the intersection and union of sets can be used to find the common elements between two datasets, and the union of sets can be used to find the total number of elements in both datasets.
Conclusion
In this article, we have addressed some of the most frequently asked questions about the problem involving the intersection and union of two sets, A and B. We hope that this article has provided a better understanding of the problem and its solution. If you have any further questions, please feel free to ask.