Solving Reader Overlap with the Principle of Inclusion-Exclusion: A Practical Example

Introduction

In the world of information consumption, understanding the overlap among different reader demographics is crucial for publishers, marketers, and researchers. This article explores a practical example using the principle of inclusion-exclusion to determine how many individuals read both newspapers A and B in a given population. The problem is presented and solved step-by-step to provide clarity and understanding.

The Scenario: Newspaper Readership in a City

Consider a city with 6,000 literate individuals. Among these:

50 individuals read Newspaper A 45 individuals read Newspaper B 25 individuals read neither Newspaper A nor B

Using the Principle of Inclusion-Exclusion

The principle of inclusion-exclusion is a useful mathematical technique to determine the cardinality of the union of sets. In this case, the sets represent individuals reading different newspapers. Let's break down the steps:

Step 1: Define the Variables

n(Total) Total number of literate individuals 6,000 n(A) Number of individuals who read Newspaper A n(B) Number of individuals who read Newspaper B n(N) Number of individuals who read neither Newspaper A nor B n(A cap B) Number of individuals who read both newspapers A and B

Step 2: Calculate Known Values

We are given:

n(A) 50% of 6,000 3,000 n(B) 45% of 6,000 2,700 n(N) 25% of 6,000 1,500

Step 3: Find the Number of Individuals Reading at Least One Newspaper

To find the number of individuals who read at least one of the newspapers, we subtract the number of individuals who read neither from the total populace:

n(A cup B) n(Total) - n(N) 6,000 - 1,500 4,500

Step 4: Apply the Principle of Inclusion-Exclusion

The principle states:

n(A cup B) n(A) n(B) - n(A cap B)

Substitute the known values:

4,500 3,000 2,700 - n(A cap B)

Solve for n(A cap B):

4,500 5,700 - n(A cap B)

n(A cap B) 5,700 - 4,500 1,200

Verification with Different Approach

Another approach to solve this problem involves using a Venn diagram:

Total individuals who read at least one newspaper 2250 (3000 - 750) Number of individuals reading Newspaper A 1650 Number of individuals reading Newspaper B 1800

From the Venn diagram, we can understand that:

1650 1800 - n(A cap B) 2250

Solving for n(A cap B) gives:

n(A cap B) 1650 1800 - 2250 1200

Conclusion

The number of individuals who read both newspapers A and B is 1,200. This solution emphasizes the practical application of mathematical principles in solving real-world problems related to readership and demographic analysis.