"Finding sets and complements of sets"

Understanding Sets

• Definition of a Set: A set is a collection of distinct objects, called elements.
• Roster Form: Sets can be listed in roster form, which involves writing the elements inside curly braces, e.g., ( A = { 5, 8, 9 } ).

Universal Set

• Universal Set (U): Represents all objects under consideration. In this context, it could be given as ( U = { 5, 6, 7, 8, 9 } ).

Complement of a Set

• Complement Definition: The complement of a set ( A ), denoted as ( A' ), includes all elements in the universal set ( U ) that are not in the set ( A ).
• Mathematical Notation: ( A' = U - A )

Finding Complements

Example to find ( X' ) Given ( U ) and ( X )

1. Let ( U = { 5, 6, 7, 8, 9 } ) and ( X = { 5, 8, 9 } ).
2. Step: Identify elements in ( U ) that are not in ( X ): [ X' = U - X = { 6, 7 } ]

Example to find ( C' ) Given ( U ) and ( C )

1. Let ( C = { 7, 8 } ).
2. Step: Identify elements in ( U ) that are not in ( C ): [ C' = U - C = { 5, 6, 9 } ]

Properties of Complements

• The complement of a complement returns to the original set: ( (A')' = A )
• The union of a set and its complement equals the universal set: ( A \cup A' = U )
• The intersection of a set and its complement is empty: ( A \cap A' = \emptyset )

Summary of Examples

• For the problem given,
  • ( U = { 5, 6, 7, 8, 9 } ) and ( A = { 5, 8, 9 } ), results are: [ A' = { 6, 7 } ]
  • For set ( C ) where ( C = { 7, 8 } ), results are: [ C' = { 5, 6, 9 } ]

Conclusion

• Understanding sets and their complements is crucial for topics in college mathematics where set theory is applicable. This knowledge aids in logical reasoning and problem-solving across various mathematical disciplines.