Algebra Textbook: Book 1 - Set Theory/Set Operations

Set Operations Set operations allow us to combine existing sets to create new sets. These operations are fundamental to set theory and have applications throughout mathematics. Union The union of two sets A and B, denoted , is the set containing all elements that belong to either A or B (or both). Example If and , then: Properties of Union For any sets A, B, and C: Commutativity: Associativity: Identity: Idempotence: Intersection The intersection of two sets A and B, denoted , is the set containing all elements that belong to both A and B. Example If and , then: Properties of Intersection For any sets A, B, and C: Commutativity: Associativity: Identity: (where U is the universal set) Idempotence: Null Law: Disjoint Sets Two sets A and B are said to be disjoint if they have no elements in common, i.e., . Difference The difference of sets A and B, denoted (or sometimes ), is the set containing all elements of A that are not in B. Example If and , then: Properties of Difference in general (not commutative) Symmetric Difference The symmetric difference of sets A and B, denoted (or sometimes ), is the set containing elements that belong to exactly one of A or B, but not both. Example If and , then: Properties of Symmetric Difference Commutativity: Associativity: Identity: Self-inverse: Complement The complement of a set A, denoted or or , is the set of all elements in the universal set U that are not in A. = Example If the universal set and , then: Properties of Complement (Double Complement) and De Morgan's Laws: * * Cartesian Product The Cartesian product of two sets A and B, denoted , is the set of all ordered pairs (a, b) where and . Example If and , then: Properties of Cartesian Product in general (not commutative) (distributive over union) (distributive over intersection) (for finite sets) Exercises 1. If , , and , find: a) b) c) d) e) f) 2. Let the universal set be . If and , find: a) b) c) d) e) Verify De Morgan's law: 3. If and , find: a) b) c) d) 4. Draw Venn diagrams to illustrate: a) b) c) d) e) 5. Prove or disprove each of the following: a) b) c) d) 6. If , , and , verify the associative property: a) b)