Summary of "Discrete Mathematics Lecture 3 | VENN DIAGRAM Concept | Principle of Inclusion & Exclusion By GP Sir"
Summary of the Video:
Discrete Mathematics Lecture 3 | Venn Diagram Concept | Principle of Inclusion & Exclusion By GP Sir
Main Ideas and Concepts:
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Introduction to Venn Diagrams and Sets:
- Explanation of sets and their pictorial representation using Venn diagrams.
- Universal set concept and subsets A and B.
- Basic set operations illustrated via Venn diagrams:
- Union (A ∪ B): All elements in A or B.
- Intersection (A ∩ B): Elements common to both A and B.
- Difference (A - B): Elements in A but not in B.
- Complement (Aᶜ): Elements not in A but in the universal set.
- Use of examples with numbered elements to clarify these operations.
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Principle of Inclusion and Exclusion (PIE):
- Used to find the number of elements in the union of multiple sets without double counting.
- For two sets A₁ and A₂:
- |A₁ ∪ A₂| = |A₁| + |A₂| - |A₁ ∩ A₂|
- For three sets A₁, A₂, A₃:
- |A₁ ∪ A₂ ∪ A₃| = |A₁| + |A₂| + |A₃| - |A₁ ∩ A₂| - |A₂ ∩ A₃| - |A₁ ∩ A₃| + |A₁ ∩ A₂ ∩ A₃|
- Explanation of why intersections are subtracted and then added back to avoid overcounting.
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Example Problems Using PIE and Venn Diagrams:
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Example 1: Students studying Mathematics, Physics, and Biology
- Total students: 100; 30 study none of the subjects.
- Given:
- Asked to find:
- Number studying all three subjects.
- Number studying exactly one subject.
- Steps:
- Use PIE formula to find the intersection of all three sets.
- Construct Venn Diagram to distribute students accordingly.
- Calculate students in exactly one subject by subtracting overlaps.
- Result:
- 5 students study all three.
- 48 students study exactly one subject.
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Example 2: Students studying French, English, and Hindi
- Total students: 120.
- Given:
- French = 20, English = 50, Hindi = 70
- English & French = 5, English & Hindi = 20, Hindi & French = 10
- All three languages = 3
- Asked to find:
- Number studying only Hindi.
- Number studying only French.
- Number studying English but not Hindi.
- Number studying Hindi but not French.
- Steps:
- Use Venn Diagram to allocate students in intersections and exclusive areas.
- Apply PIE for accurate counts.
- Results:
- Hindi only = 43 students.
- French only = 8 students.
- English but not Hindi = 28 students.
- Hindi but not French = 43 students.
-
Example 3: Students studying English, Computer Science, and Music
- Total students: 191.
- Given:
- English & Computer Science = 36, English & Music = 20, Computer Science & Music = 18
- English = 65, Computer Science = 76, Music = 63
- All three subjects = 10
- Asked to find:
- Number studying English and Music but not Computer Science.
- Number studying Computer Science and Music but not English.
- Number not taking any of the three subjects.
- Steps:
- Use PIE and Venn diagrams to find exclusive and overlapping counts.
- Subtract known intersections from totals to find exclusive groups.
- Calculate students outside all three subjects.
- Result:
- 51 students do not study any of these subjects.
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Example 1: Students studying Mathematics, Physics, and Biology
-
Methodology / Instructions for Solving PIE and Venn Diagram Problems:
- Step-by-step approach:
- Write down the total number of elements/students.
- List the counts for each individual set.
- List the counts for each pairwise intersection.
- Note the count for the triple intersection (if any).
- Apply the Principle of Inclusion and Exclusion formula:
- For two sets: sum of individual sets minus intersection.
- For three sets: sum of individuals minus sum of pairwise intersections plus triple intersection.
Category
Educational
