How Many Students Studied Only One Language? A Practical Application of Inclusion-Exclusion Principle

In a mixed group of students, understanding how many studied only one language can be crucial for educational planning. Consider the following scenario: In one group, 15 students studied Swedish and 12 English. Among this group, 6 students studied both

Introduction to the Problem

The problem at hand is to find out how many students studied only one language. We can use the principle of inclusion-exclusion for a precise calculation.

Understanding the Principles

The principle of inclusion-exclusion is a fundamental concept in set theory that allows us to calculate the number of elements in the union of multiple sets, using the size of individual sets and their intersections.

Defining the Sets

We define the following variables based on the given information:

S the number of students who studied Swedish. E the number of students who studied English. B the number of students who studied both Swedish and English.

From the information given, we have:

S 15 E 12 B 6

Calculating the Number of Students Who Studied Only One Language

To find the number of students who studied only one language, we can use the principle of inclusion-exclusion:

Step 1: Calculate the number of students who studied only Swedish:

Students who studied only Swedish S - B 15 - 6 9

Step 2: Calculate the number of students who studied only English:

Students who studied only English E - B 12 - 6 6

Step 3: Calculate the total number of students who studied only one language:

Total students who studied only one language Students who studied only Swedish Students who studied only English 9 6 15

Therefore, 15 students studied only one language, either Swedish or English.

Alternative Method: Symmetric Difference

Another approach to solving this problem is by using the concept of symmetric difference. The symmetric difference between two sets A and B, denoted as A △ B, is the set of elements which are in either of the sets but not in their intersection.

In this context, the symmetric difference can be expressed as:

(S - B) (E - B)

Calculating using this method:

(15 - 6) (12 - 6) 9 6 15

This confirms that the total number of students who studied only one language is 15.

Conclusion

The application of the principle of inclusion-exclusion and the symmetric difference concept provides a clear and accurate solution to the problem of determining the number of students who studied only one language. Understanding these principles is crucial in various fields, including education and data analysis.

Keywords: inclusion-exclusion principle, students studying languages, symmetric difference