Notes on Set Complements and Venn Diagrams

Universal Set

• Definition: The universal set (denoted by U) is the set of all elements involved in the problem.

Complement of a Set

• Definition: The complement of set A is the set of all elements in the universal set that are not in set A.
• Notation: Complement of set A is denoted by A'.

Example 1

• Universal Set: {1, 2, 3, 4, 5, 6, 7, 8, 9}
• Set A: {2, 4, 6, 8}

A. Finding the Complement of Set A

• Elements in U but not in A:
  • 1: Underline (not in A)
  • 2: Circle (in A)
  • 3: Underline (not in A)
  • 4: Circle (in A)
  • 5: Underline (not in A)
  • 6: Circle (in A)
  • 7: Underline (not in A)
  • 8: Circle (in A)
  • 9: Underline (not in A)
• Complement of Set A: {1, 3, 5, 7, 9}

B. Venn Diagram Illustration

• Rectangle represents U.
• Set A (circle) contains: {2, 4, 6, 8}.
• Shade complement of A (light blue).

Example 2

• Universal Set: {1, 2, 3, 4, 5, 6, 7, 8, 9}
• Set A: {2, 4, 6, 8}
• Set B: {3, 4, 5, 6}

A. Finding the Complement of A ∪ B

1. Union of A and B: A ∪ B = {2, 3, 4, 5, 6, 8}
2. Elements in U but not in A ∪ B:
   • 1: Underline
   • 2: Circle
   • 3: Circle
   • 4: Circle
   • 5: Circle
   • 6: Circle
   • 7: Underline
   • 8: Circle
   • 9: Underline

• Complement of A ∪ B: {1, 7, 9}

B. Venn Diagram Illustration

• Draw rectangles and circles for A and B, intersecting.
• Shade the complement of A ∪ B (light blue).

Example 3

• Universal Set: {1, 2, 3, 4, 5, 6, 7, 8, 9}
• Set A: {2, 4, 6, 8}
• Set B: {3, 4, 5, 6}

A. Finding the Complement of A ∩ B

1. Intersection of A and B: A ∩ B = {4, 6}
2. Elements in U but not in A ∩ B:
   • 1: Underline
   • 2: Underline
   • 3: Underline
   • 4: Circle
   • 5: Underline
   • 6: Circle
   • 7: Underline
   • 8: Underline
   • 9: Underline

• Complement of A ∩ B: {1, 2, 3, 5, 7, 8, 9}

B. Venn Diagram Illustration

• Draw rectangles and circles for A and B, indicating intersection (4, 6).
• Shade the complement of A ∩ B (light blue).

Example 4

• Universal Set: {1, 2, 3, 4, 5, 6, 7, 8, 9}
• Set A: {2, 4, 6, 8}
• Set B: {3, 4, 5, 6}

A. Finding the Complement of B - A

1. B - A: Elements in B not in A = {3, 5}
2. Elements in U but not in B - A:
   • 1: Underline
   • 2: Underline
   • 3: Circle
   • 4: Underline
   • 5: Circle
   • 6: Underline
   • 7: Underline
   • 8: Underline
   • 9: Underline

• Complement of B - A: {1, 2, 4, 6, 7, 8, 9}

B. Venn Diagram Illustration

• Draw rectangles and circles for A and B, indicating intersection and shade complement of B - A (light blue).

Conclusion

• Understanding complements and using Venn diagrams are essential for visualizing relationships between sets.