Venn Diagrams 2: Three-Set Venn Diagrams

Introduction

• Topic: Venn Diagrams 2 (Three Sets)
• Continuation of Previous Lecture: Venn Diagrams 1 (Two Sets)
• Objective: How to solve questions with three-set Venn diagrams

Important Points

Three-Set Venn Diagrams

• There are three sets: A, B, C
• Universal Set: A+B+C+N (where N is neither in A, B, nor C)
• Disjoint Sets: If there is no interaction of any set, all interaction values are 0.

Non-Disjoint Sets

• When there are interaction values, they must be included.
• Interaction:
  • A ∩ B
  • B ∩ C
  • A ∩ C
  • A ∩ B ∩ C (interaction of all three)
• Formula: A + B + C - (A ∩ B) - (B ∩ C) - (A ∩ C) + A ∩ B ∩ C

Questions and Solutions

Question 1: How many residents had only one mode?

• Method: Add only A, only B, only C
• Answer: 134

Question 2: How many residents had at least two modes?

• Method:
  • Add only M∩B, M∩C, B∩C
  • Add A∩B∩C
• Answer: 266

Question 3: How many residents had no mode?

• Answer: 0

Question 4: If every resident had a maximum of two vehicles?

• Method:
  • Add N, only one mode, and only two modes
• Answer: 1300 members

Question 5: How many residents had a motorcycle and a car but not a bicycle?

• Answer: 50

Conclusion

• Venn Diagrams make it easier to analyze various situations.
• Attention is needed to how 'and' or 'or' is used in the question.
• The key to solving various questions through Venn Diagrams is to clearly diagram them.