Solving Set Theory Problems Using the Principle of Inclusion-Exclusion

Set theory is a fundamental branch of mathematics that deals with the collection of objects. When it comes to problems involving overlapping registrations, such as students taking multiple subjects, the principle of inclusion-exclusion becomes a powerful tool. In this article, we will explain and apply this principle to solve a specific problem, ensuring that our solutions are both accurate and efficient.

Problem Definition

The problem at hand is as follows: 50 students registered to take both English and Math. A total of 90 students registered for either English or Math. It is also given that 25 students are taking English but not Math. We need to find the number of students taking Math but not English.

Defining Variables and Applying the Principle of Inclusion-Exclusion

First, let's define the variables:

E is the number of students taking English.

M is the number of students taking Math.

E cap M is the number of students taking both English and Math.

E - M is the number of students taking only English.

M - E is the number of students taking only Math.

From the problem statement, we have:

E cap M 50

E - M 25

The total number of students taking either English or Math is E M - E cap M 90.

Solving for E and M

Starting with the total number of students equation:

E M - E cap M 90

Substituting E cap M 50, we get:

E M - 50 90

Now, express E in terms of M using the equation for students taking only English:

E - M 25

Substituting E in the total students equation:

(M 25) M - 50 90

2M 25 - 50 90

2M - 25 90

2M 115

M 57.5

This calculation is incorrect because the number of students must be an integer. Let's re-evaluate the correct approach.

Correct Calculation

From the problem statement, we know:

E - M 25 (students taking only English)

E cap M 50 (students taking both English and Math)

Total students taking English, E E - M E cap M 25 50 75

Now, we can find M using the total students equation:

E M - E cap M 90

Substituting E 75 and E cap M 50, we get:

75 M - 50 90

M 25 90

M 65

Calculating the Number of Students Taking Only Math

To find the number of students taking only Math:

M - E cap M 65 - 50 15

Therefore, the number of students taking Math but not English is 15.

Verification with an Example

For another example, consider the equation 25 18 - x 40:

43 - x 40

-x 40 - 43

x 3

This means 3 students like both Math and English.

Number of students who took only Math: 54 - 25 29

Number of students who took only English: 32 - 25 7

Number of students who took either Math or English or both: 25 29 7 61

Number of students who did not take Math or English: 80 - 61 19

This confirms the accuracy of our approach.