Lecture Notes on Set Theory: Intersection and Union

Introduction

• The lecture discusses the concepts of intersection and union of sets.
• Examples are provided to illustrate set operations.

Intersection of Sets

• Definition: The intersection of two sets includes elements 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): Empty set (no common elements)

• Example 5:

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

Union of Sets

• Definition: The union of two sets includes all elements from both sets, without duplication.

• 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} (unchanged)

Venn Diagrams

• Example with Venn Diagram:
  • 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}
  • Illustrates intersection and union using overlapping circles.

Conclusion

• Understanding of intersection and union enhances comprehension of set relationships.
• Venn diagrams visually represent these relationships.

• Note: Null sets or empty intersections are denoted by a Greek symbol or by stating the set is empty.
• When combining with an empty set, the union remains the same as the non-empty set.

End of Lecture

Thank you for attending the lecture.