Lecture Notes: Set Operations and Venn Diagrams

Introduction to Sets

• Defining Sets:
  • Members listing
  • Set-builder notation
• Types of Relations: Between sets

Set Operations

• Intersection:

  • Visualized in Venn diagrams as overlapping circles
  • Disjoint Sets: No overlap, no common elements
  • Intersection Notation: Inverted U (∩)
  • Example:
    • Set A = {1, 2, 3}
    • Set B = {2, 3, 4}
    • Intersection = {2, 3}
  • Set-builder Notation: Intersection of A and B is {x | x ∈ A and x ∈ B}
  • Properties:
    • Binary operation
    • Result is a new set

• Union:

  • Union Notation: Denoted by U
  • Example:
    • Set A = {1, 2, 3}
    • Set B = {2, 3, 4}
    • Union = {1, 2, 3, 4}
  • Set-builder Notation: Union of A and B is {x | x ∈ A or x ∈ B}
  • Common elements included once

Venn Diagrams

• Purpose: Visual representation of set operations and relations
• Components: Sets as circles or enclosed areas
• Subsets and Supersets:
  • Subsets: Smaller region within a larger set
  • Supersets: Larger region encompassing subsets
  • Example:
    • Natural numbers ⊆ Whole numbers ⊆ Rational numbers
    • Rational numbers ∪ Irrational numbers = Real numbers

Conclusion

• Venn diagrams help visualize set operations like intersections and unions.
• Next lecture: More complex operations using Venn diagrams.