Lecture Notes on Set Theory: Intersection and Union of Sets

Intersection of Sets

• Definition: The intersection of two sets is a set that contains all 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: {2, 5, 6, 9}
• Example 2:
  • Set C: {3, 4, 6, 7, 10}
  • Set D: {3, 6, 8, 9}
  • Intersection: {3, 6}
• Example 3:
  • Set F: {a, b, c, d, f, g, j}
  • Set G: {a, c, g, h, k}
  • Intersection: {a, c, g}
• Example 4:
  • Set J: {5, 7, 10, 11}
  • Set K: {2, 4, 8, 13}
  • Intersection: Empty set (no common elements)
• Example 5:
  • Set R: {3, 4, 7, 10}
  • Set S: Empty set
  • Intersection: Empty set (intersection with an empty set is always empty)

Union of Sets

• Definition: The union of two sets is a set containing all elements from both sets, without duplication.
• Example 1:
  • Set A: {1, 2, 3, 4}
  • Set B: {3, 4, 5, 6}
  • Union: {1, 2, 3, 4, 5, 6}
• Example 2:
  • Set C: {3, 5, 9, 11, 13}
  • Set D: {2, 3, 6, 8, 12}
  • Union: {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: {a, b, c, d, e, f, g}
• Example 4:
  • Set X: {2, 5, 8, 12}
  • Set Y: Empty set
  • Union: {2, 5, 8, 12} (union with an empty set results in the original set)

Venn Diagrams

• Understanding Intersections and Unions with Venn Diagrams:
  • Intersection: The overlapping region in a Venn diagram represents the intersection.
  • Union: The entire area covered by circles in a Venn diagram represents the union.
• Example with Venn Diagram:
  • Set A: {3, 4, 5, 7}
  • Set B: {2, 4, 5, 8}
  • Intersection: {4, 5} (middle area of the Venn diagram)
  • Union: {2, 3, 4, 5, 7, 8} (total area covered by both circles)

Conclusion

• Intersections focus on common elements.
• Unions focus on combining all elements from both sets without repeats.
• Venn diagrams provide a visual representation of set relationships.
• Always remember the specific rules for intersections and unions when working with sets.