Understanding and Solving Sets Problems: An Example with English and Maths
When dealing with problems related to sets, especially those related to subjects like English and Maths, it's crucial to understand the principles of set theory. This includes the concepts of union, intersection, and complementary sets. Let's delve into a specific example where 200 students took either English or Maths, or both, and explore how to determine the number of students who took exactly one of the subjects.
Solving the Problem
The problem states that 200 students took either English or Maths, or both. We are given that 3/5 of the students took English and 4/5 took Maths. Let's break down the steps to solve it.
Step 1: Calculate the Number of Students Who Took English and Maths
The number of students who took English is ( frac{3}{5} times 200 120 ). The number of students who took Maths is ( frac{4}{5} times 200 160 ).Step 2: Find the Total Number of Students Taking at Least One Subject
Given that the total number of students taking either subject is 200, we can calculate the number of students who took both subjects using the following steps:
The sum of the students taking English and Maths is 120 160 280. However, this sum counts the students who took both subjects twice. The number of students who took both subjects is 280 - 200 80.Step 3: Find the Number of Students Who Took Only One Subject
The number of students who took only English is 120 - 80 40. The number of students who took only Maths is 160 - 80 80.The total number of students who took exactly one subject is 40 80 120.
Alternative Method: Using Basic Set Theory
We can also solve this problem using the basic principles of set theory. The formula to find the number of elements in the union of two sets is:
A ∪ B A B - A ∩ B
Where: A is the set of students who took English. B is the set of students who took Maths. A ∪ B is the total number of students who took either subject. A ∩ B is the number of students who took both subjects.
Using the given values:
A 120 B 160 A ∪ B 200Solving for A ∩ B:
A ∪ B A B - A ∩ B
200 120 160 - A ∩ B
A ∩ B 120 160 - 200 80
Thus, the number of students who took both subjects is 80.
Conclusion
By breaking down the problem and applying the principles of set theory, we can accurately determine the number of students who took exactly one of the subjects. This involves calculations for union, intersection, and careful consideration of each subset of students.
Practical Application
Understanding these concepts can be applied to various real-world scenarios, such as survey data, statistical analysis, and educational assessment. Proficiency in set theory can provide valuable insights and help in making informed decisions based on data.