Notes on Set Theory: Complement of a Set

Universal Set

• The universal set (U) includes all elements involved in the problem.
• Denoted by the uppercase letter U.

Complement of a Set

• The complement of set A includes all elements in the universal set that are not in set A.
• Denoted as A' (A prime).

Example 1

Given:

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

Letter A: Find the Complement of Set A

• Elements in U but not in A (underlined): 1, 3, 5, 7, 9
• Complement of Set A: {1, 3, 5, 7, 9}

Letter B: Venn Diagram Illustration

• Draw a rectangle for U with elements inside: 1, 2, 3, 4, 5, 6, 7, 8, 9
• Circle A inside the rectangle containing: 2, 4, 6, 8
• Shade the complement of A (elements 1, 3, 5, 7, 9) in light blue.

Example 2

Given:

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

Letter A: Find the Complement of A ∪ B

• A ∪ B: {2, 3, 4, 5, 6, 8}
• Elements not in A ∪ B (underlined): 1, 7, 9
• Complement of A ∪ B: {1, 7, 9}

Letter B: Venn Diagram Illustration

• Draw rectangle for U with elements: 1, 2, 3, 4, 5, 6, 7, 8, 9
• Circle A containing: 2, 4, 6, 8
• Circle B containing: 3, 4, 5, 6
• Overlapping area contains common elements 4 and 6.
• Shade complement of A ∪ B in light blue.

Example 3

Given:

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

Letter A: Find the Complement of A ∩ B

• A ∩ B: {4, 6}
• Elements not in A ∩ B (underlined): 1, 2, 3, 5, 7, 8, 9
• Complement of A ∩ B: {1, 2, 3, 5, 7, 8, 9}

Letter B: Venn Diagram Illustration

• Draw rectangle for U with elements: 1, 2, 3, 4, 5, 6, 7, 8, 9
• Circle A containing: 2, 4, 6, 8
• Circle B containing: 3, 4, 5, 6
• Overlapping area contains common elements 4 and 6.
• Shade complement of A ∩ B in light blue.

Example 4

Given:

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

Letter A: Find the Complement of B - A

• B - A: {3, 5}
• Elements not in B - A (underlined): 1, 2, 4, 6, 7, 8, 9
• Complement of B - A: {1, 2, 4, 6, 7, 8, 9}

Letter B: Venn Diagram Illustration

• Draw rectangle for U with elements: 1, 2, 3, 4, 5, 6, 7, 8, 9
• Circle A containing: 2, 4, 6, 8
• Circle B containing: 3, 4, 5, 6
• Overlapping area contains common elements 4 and 6.
• Shade complement of B - A in light blue.