Inclusion-Exclusion Principle: A Comprehensive Guide with a Practical Example
Understanding the Inclusion-Exclusion Principle is crucial for solving problems in set theory and probability. This principle helps us to find the number of elements in the union of multiple sets, avoiding the issue of counting elements multiple times. In this article, we will explore the application of the Inclusion-Exclusion Principle through a practical example.
Understanding the Inclusion-Exclusion Principle
The Inclusion-Exclusion Principle states that for any finite unions of sets, the number of elements in the union is equal to the sum of the number of elements in each individual set minus the number of elements in the intersections of every two sets, plus the number of elements in the intersections of every three sets, and so on.
Key Points
Inclusion: Addition of the number of elements in each individual set.
Exclusion: Subtraction of the number of elements in the intersections of every two sets.
Inclusion: Addition of the number of elements in the intersections of every three sets, and so on.
A Practical Example
Let's consider a class consisting of 100 students. We are given that:
20 students know English.
20 students do not know Urdu.
10 students know neither English nor Urdu.
We need to find the number of students who know both Urdu and English.
Step-by-Step Solution
Step 1: Define the Sets
Let:
E: the set of students who know English.
U: the set of students who know Urdu.
We need to find the number of students in the intersection of these two sets, denoted as E ∩ U.
Step 2: Determine the Known Values
From the problem, we know:
Total number of students 100.
Number of students who know English, |E| 20.
Number of students who do not know Urdu 20, which means the number of students who know Urdu, |U| 100 - 20 80.
Number of students who know neither English nor Urdu 10.
Step 3: Calculate the Number of Students Who Know At Least One Language
The number of students who know at least one language (English or Urdu) can be calculated as:
Number of students who know at least one language Total number of students - Number of students who know neither language 100 - 10 90.
Step 4: Apply the Inclusion-Exclusion Principle
Using the Inclusion-Exclusion Principle, we have:
|E ∪ U| |E| |U| - |E ∩ U|
Substituting the known values:
90 20 80 - |E ∩ U|
90 100 - |E ∩ U|
|E ∩ U| 100 - 90 10
Thus, the number of students who know both Urdu and English is 10.
Alternative Approach
Another student suggested that 20 students know both English and Urdu. However, let's break down their logic:
Step 1: Define the Sets Again
From the problem, we have:
20 students who cannot speak both Urdu and English.
20 students who cannot speak Urdu.
Total number of students 100.
Step 2: Calculate the Number of Students Who Know Either or Both Languages
The number of students who either know English or Urdu or both can be calculated as:
100 - 30 70 students.
Since only 20 students know English, it means only those 20 can also speak Urdu.
Hence, according to this logic, the answer is indeed 20. However, we can now see that this approach is not entirely correct as it overlooks the intersection of sets E and U.
Conclusion
The correct number of students who know both Urdu and English, as determined by the Inclusion-Exclusion Principle, is 10. Understanding and applying this principle correctly is crucial for solving similar problems in set theory and probability.
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Further Reading
For more resources on set theory and the Inclusion-Exclusion Principle, refer to the following articles:
Introduction to Set Theory
Inclusion-Exclusion Principle Practice Problems