Solving Problems on Academic Performance Using Set Theory: A Comprehensive Approach

Academic performance is a critical aspect of students' educational journey, often evaluated through various examination results. This article delves into a specific problem related to the academic performance of students in English and Mathematics. Let's explore the step-by-step solution using the principle of set theory, particularly the principle of inclusion-exclusion.

Problem Statement

Consider a scenario where 105 students took an examination for both English and Mathematics. Among these students, 80 passed in English, and 75 passed in Mathematics. Additionally, 10 students failed in both subjects. The task is to determine the number of students who passed in only one subject.

Solution Using Set Theory

Let's define the following sets:

E: Set of students who passed in English. M: Set of students who passed in Mathematics.

From the problem, we have the following information:

Total number of students, U 105 Number of students passing in English, E 80 Number of students passing in Mathematics, M 75 Number of students failing in both subjects, U - E ∪ M 10

Let's begin by finding the number of students passing in at least one subject:

( |E ∪ M| U - (U - E ∪ M) 105 - 10 95 )

Using the principle of inclusion-exclusion, we can find the number of students passing in both subjects, denoted as E ∩ M:

( |E ∪ M| |E| |M| - |E ∩ M| )

Substituting the known values:

95 80 75 - |E ∩ M|

This simplifies to:

|E ∩ M| 155 - 95 60

Now, we can find the number of students who passed in only one subject:

Students passing only in English |E - E ∩ M| 80 - 60 20 Students passing only in Mathematics |M - E ∩ M| 75 - 60 15

Finally, the total number of students passing in only one subject is:

20 15 35

Alternative Approaches to Solving the Problem

There are two alternative methods to approach this problem, which yield the same result.

Approach 1

Total number of students T 105, EnglishPass 80, MathPass 75, and students failing in both B 10.

Visualize the following Venn diagram:

Students who failed in English 105 - 80 25 Students who failed in Mathematics 105 - 75 30

Total number of students failing in both subjects 25 30 - 10 45

Students passing in at least one subject 105 - 45 60

Thus, the number of students passing in only English 80 - 60 20

And the number of students passing in only Mathematics 75 - 60 15

Total number of students passing in only one subject 20 15 35

Approach 2

Total number of students T 105, students failing in both 10, thus T - B 95.

Number of students failing in English 95 - 80 15, and the number of students failing in Mathematics 95 - 75 20.

Total number of students failing in either Mathematics or English 20 15 35

Thus, the number of students passing in either Mathematics or English 95 - 35 60

Number of students passing in only English 80 - 60 20

And the number of students passing in only Mathematics 75 - 60 15

Total number of students passing in either Mathematics or English 20 15 35

Conclusion

The problem of determining the number of students who passed in only one subject using set theory and the principle of inclusion-exclusion is a prime example of the application of mathematical concepts in solving real-world problems. Both the direct method and the alternative approaches highlight the flexibility and utility of set theory in understanding and solving complex scenarios. Whether through direct calculations or visual representations, understanding these principles is invaluable for students and educators alike.