Lecture Notes: Operations on Subsets

Universal Set

• Definition: The universal set is the set from which all subsets are derived.
• Example: Natural numbers from 1 to 8.

Venn Diagrams

• A visual representation of sets and subsets.
• Components:
  • Shapes represent sets.
  • Inside the shape: elements in the set.
  • Outside the shape: elements in the universal set that are not in the subset.
  • Overlapping shapes indicate common elements between sets.

Example Sets

• Universal Set (U): {1, 2, 3, 4, 5, 6, 7, 8}
• Subset A: {1, 3, 6, 7}
• Subset B: {3, 6, 8}

Elements Positioning in Venn Diagram

• In A only: 1, 7
• In B only: 8
• In both A and B: 3, 6
• In neither: 2, 4, 5

Set Operations

1. Intersection (A ∩ B)

• Definition: Elements that are in both sets A and B.
• Example: A ∩ B = {3, 6}

2. Union (A ∪ B)

• Definition: Elements that are in set A, set B, or both.
• Example: A ∪ B = {1, 3, 6, 7, 8}

3. Complement (A')

• Definition: Elements in the universal set that are not in set A.
• Example: A' = {2, 4, 5, 8}

4. Relative Complement (A - B)

• Definition: Elements in set A that are not in set B.
• Example: A - B = {1, 7}

5. Symmetric Difference (A △ B)

• Definition: Elements that are in either set A or set B but not in both.
• Calculation: A - B ∪ B - A.
• Example: A △ B = {1, 7, 8}

Alternative Definition for Symmetric Difference

• A △ B = (A ∪ B) - (A ∩ B)

Special Case of Symmetric Difference

• When a set is combined with itself (A △ A), the result is the empty set.
• Example: A △ A = ∅ (no elements are in A but not in A).

Conclusion

• Understanding operations on subsets is essential for set theory.
• Venn diagrams are a helpful tool for visualizing these operations.