Introduction to Set theory
Ways of Describing Sets
Some Special Sets U
Special Sets Z represents the set of integers –Z + is the set of positive integers and –Z - is the set of negative integers N represents the set of natural numbers ℝ represents the set of real numbers Q represents the set of rational numbers
Subset
Proper Subset
Subsets Symbols a subset exists when a set’s members are also contained in another set notation: means “is a subset of” means “is a proper subset of” means “is not a subset of” Equality of Two Sets
Venn Diagrams Venn diagrams show relationships between sets and their elements Universal Set Sets A & B
Venn Diagram Example 1 Set DefinitionElements A = {x | x Z + and x 8} B = {x | x Z + ; x is even and 10} A B B A
Venn Diagram Example 2 Set DefinitionElements A = {x | x Z + and x 9} B = {x | x Z + ; x is even and 8} A B B A A B
Symmetric Difference: A B = (A – B) (B – A)
Set Identities Commutative Laws: A B = A B and A B = B A Associative Laws: (A B) C = A (B C) and (A B) C = A (B C) Distributive Laws: A (B C) = (A B) (A C) and A (B C) = (A B) (A C) Intersection and Union with universal set: A U = A and A U = U Double Complement Law: (A c ) c = A Idempotent Laws: A A = A and A A = A De Morgan’s Laws: (A B) c = A c B c and (A B) c = A c B c Absorption Laws: A (A B) = A and A (A B) = A Alternate Representation for Difference: A – B = A B c Intersection and Union with a subset: if A B, then A B = A and A B = B
Power Set Power set of A is the set of all subsets of A Theorem: if A B, then P(A) P(B) Theorem: If set X has n elements, then P(X) has 2 n elements