Union • Definition ○ Let A and B be sets. ○ The union of the sets A and B, denoted by A∪B, is the set: ○ {x|x∈A∨x∈B} • Example: What is {1,2,3}∪{3, 4, 5}? ○ {1,2,3,4,5} • Venn Diagram Intersection • Definition ○ The intersection of sets A and B, denoted by A ∩ B, is ○ {x|x∈A∧x∈B} • Note ○ If the intersection is empty, then ○ A and B are said to be disjoint. • Example: What is {1,2,3} ∩ {3,4,5} ? ○ {3} • Example: What is {1,2,3} ∩ {4,5,6} ? ○ ∅ • Venn Diagram Complement • Definition ○ If A is a set, then the complement of the A (with respect to U), denoted by Ā is the set U−A ○ Ā={x∈U|x∉A} ○ (The complement of A is sometimes denoted by A^c.) • Example ○ If U is the positive integers less than 100, ○ what is the complement of {x | x 70} ○ {x│x≤70} • Venn Diagram Difference • Definition ○ Let A and B be sets. ○ The difference of A and B, denoted by A–B, is the set containing the elements of A that are not in B. ○ The difference of A and B is also called the complement of B with respect to A. ○ A–B={x|x∈A∧x∉B}=A∩B ̅ • Venn Diagram Set Identities • Identity laws ○ A∪∅=A ○ A∩U=A • Domination laws ○ A∪U=U ○ A∩∅=∅ • Idempotent laws ○ A∪A=A ○ A∩A=A • Complementation law ○ ((A ̅ ) ) ̅=A ̅ • Communtative laws ○ A∪B=B∪A ○ A∩B=B∩A • Associative laws ○ A∪(B∪C)=(A∪B)∪C ○ A∩(B∩C)=(A∩B)∩C • Distributive laws ○ A∩(B∪C)=(A∩B)∪(A∩C) ○ A∪(B∩C)=(A∪B)∩(A∪C) • De Morgan
