Introduction

In this article, we explore a problem involving the performance analysis of 30 students in a combined Biology and Mathematics exam. By utilizing the principle of inclusion-exclusion and visual representations such as Venn diagrams, we derive the number of students who passed both, only one subject, or neither.

Problem Statement

A class of 30 students took exams in Biology and Mathematics. 18 students passed Biology and 17 passed Mathematics. Additionally, 3 students failed both subjects. We need to determine how many students passed both subjects and how many passed only Mathematics.

Step-by-Step Solution

To solve this problem, we will use the principle of inclusion-exclusion, a fundamental concept in set theory.

Using the Principle of Inclusion-Exclusion

The principle states that:

[ |A cup B| |A| |B| - |A cap B| ]

Where:

|A| |B| |A ∩ B|

Here:

|A| 18 (students who passed Biology) |B| 17 (students who passed Mathematics) |A ∩ B| x (students who passed both)

We also know:

[ |A cup B| 30 - 3 27 ] (students who passed at least one subject)

Set Up the Equation

Using the principle of inclusion-exclusion:

[ 27 18 17 - x ]

Solving for x:

[ x 18 17 - 27 ] [ x 25 - 27 ] [ x 8 ] (students who passed both subjects)

Hence, 8 students passed both Biology and Mathematics.

Students Who Passed Only Mathematics

To find the number of students who passed only Mathematics, subtract the number of students who passed both:

[ 17 - 8 9 ] (students who passed only Mathematics)

Students Who Passed Only Biology

Similarly, the number of students who passed only Biology is:

[ 18 - 8 10 ] (students who passed only Biology)

Venn Diagram

A Venn diagram can be drawn with two overlapping circles. The intersection (students who passed both) is 8. This gives us:

8 (passed both) 9 (passed only Mathematics) 10 (passed only Biology) 3 (failed both)

Conclusion

The problem of understanding student performance in combined Biology and Mathematics exams can be effectively analyzed using the principle of inclusion-exclusion and visual tools like Venn diagrams. This method provides a clear and systematic approach to solving such problems in set theory.

By applying these methods, we found:

8 students passed both Biology and Mathematics. 9 students passed only Mathematics. 10 students passed only Biology. 3 students failed both subjects.

The use of Venn diagrams and the principle of inclusion-exclusion not only simplifies the problem but also enhances our understanding of set theory in practical scenarios.