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Questions and Answers
Given two finite sets, A and B, what does the Inclusion-Exclusion Principle state about the cardinality of their union?
Given two finite sets, A and B, what does the Inclusion-Exclusion Principle state about the cardinality of their union?
Using the Inclusion-Exclusion Principle, if set A has 10 elements, set B has 15 elements, and their intersection has 5 elements, how many elements are in their union?
Using the Inclusion-Exclusion Principle, if set A has 10 elements, set B has 15 elements, and their intersection has 5 elements, how many elements are in their union?
What does |A ∩ B| represent in the context of the Inclusion-Exclusion Principle for two sets A and B?
What does |A ∩ B| represent in the context of the Inclusion-Exclusion Principle for two sets A and B?
If |A ∪ B| = 30, |A| = 18, and |B| = 22, what is the cardinality of the intersection of sets A and B, |A ∩ B|?
If |A ∪ B| = 30, |A| = 18, and |B| = 22, what is the cardinality of the intersection of sets A and B, |A ∩ B|?
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In a group of students, 25 are taking math, 20 are taking science, and 10 are taking both. How many students are taking either math or science, or both, according to the Inclusion-Exclusion Principle?
In a group of students, 25 are taking math, 20 are taking science, and 10 are taking both. How many students are taking either math or science, or both, according to the Inclusion-Exclusion Principle?
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Flashcards
Inclusion-Exclusion Principle
Inclusion-Exclusion Principle
A formula used to count the number of elements in the union of multiple sets by including and excluding overlapping elements.
Union of Sets
Union of Sets
The union of sets includes all elements from the sets without duplicates.
Overlapping Elements
Overlapping Elements
Elements that are common to two or more sets.
Counting Method
Counting Method
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Example Application
Example Application
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Study Notes
Inclusion-Exclusion Principle
- The principle describes how to find the size of a union of sets.
- For two finite sets A and B, the size of their union (A ∪ B) is equal to the sum of the sizes of A and B, minus the size of their intersection (A ∩ B).
- This is expressed as: n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
- For three finite sets A, B, and C, the size of their union (A ∪ B ∪ C) is given by:
- n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ C) - n(B ∩ C) + n(A ∩ B ∩ C)
Example 1
- Demonstrates the principle for three sets.
- Shows how to calculate the size of the union of finite sets, considering shared elements.
Example 2
- Contains a word problem:
- In a town of 10,000 families, percentages of families purchasing specific newspapers are given.
- 40% buy newspaper A, 20% buy newspaper B, 10% buy newspaper C.
- Specific percentages overlap between the newspapers (A and B, etc).
- Finally, 2% purchase all three newspapers.
- The inclusion-exclusion principle is used to calculate the number of families purchasing:
- Specific combinations of newspapers (exactly one newspaper, two newspapers, or all three newspapers).
Further Calculations
- Calculation of the number of families purchasing each combination or sub-combination, and the total number of families purchasing at least two newspapers.
- The results are expressed as percentages of the total number of families in the town (10000).
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Description
Test your understanding of the Inclusion-Exclusion Principle with this quiz. You'll tackle problems regarding the size of unions of sets and practical examples involving percentages in real-life scenarios. Challenge yourself with questions that blend theory and application.
