Lecture Notes on Sets: Intersection and Union

Introduction to Sets

• Sets are collections of distinct objects or elements.
• Operations on sets include finding intersections and unions.

Intersection of Sets

• Definition: The intersection of two sets is a set of elements that are common to both sets.

• Example 1:

  • Set A: {2, 4, 5, 6, 9}
  • Set B: {2, 3, 5, 6, 7, 9, 10}
  • Intersection (A ∩ B): {2, 5, 6, 9}

• Example 2:

  • Set C: {3, 4, 6, 7, 10}
  • Set D: {3, 6, 8, 9}
  • Intersection (C ∩ D): {3, 6}

• Example 3:

  • Set F: {a, b, c, d, f, g, j}
  • Set G: {a, c, g, h, k}
  • Intersection (F ∩ G): {a, c, g}

• Example 4:

  • Set J: {5, 7, 10, 11}
  • Set K: {2, 4, 8, 13}
  • Intersection (J ∩ K): ∅ (No common elements)

• Example 5:

  • Set R: {3, 4, 7, 10}
  • Set S: ∅ (Empty set)
  • Intersection (R ∩ S): ∅

Union of Sets

• Definition: The union of two sets is a set containing all elements from both sets, without duplicates.

• Example 1:

  • Set A: {1, 2, 3, 4}
  • Set B: {3, 4, 5, 6}
  • Union (A ∪ B): {1, 2, 3, 4, 5, 6}

• Example 2:

  • Set C: {3, 5, 9, 11, 13}
  • Set D: {2, 3, 6, 8, 12}
  • Union (C ∪ D): {2, 3, 5, 6, 8, 9, 11, 12, 13}

• Example 3:

  • Set J: {a, c, d, e}
  • Set K: {a, b, f, e, g}
  • Union (J ∪ K): {a, b, c, d, e, f, g}

• Example 4:

  • Set X: {2, 5, 8, 12}
  • Set Y: ∅ (Empty set)
  • Union (X ∪ Y): {2, 5, 8, 12}
  • Note: The union with an empty set results in the original set.

Intersection and Union in Venn Diagrams

• Venn diagrams visually represent the intersection and union of sets.
• Example:
  • Set A: {3, 4, 5, 7}
  • Set B: {2, 4, 5, 8}
  • Intersection (A ∩ B): {4, 5}
  • Union (A ∪ B): {2, 3, 4, 5, 7, 8}
  • Venn diagrams show overlapping regions for intersections and combined regions for unions.

Conclusion

• Intersection and union are fundamental operations in set theory.
• Understanding how to find these and represent them diagrammatically helps in visualizing relationships between sets.

Thank you for attending the lecture!