Post on 18-Jan-2016
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
)CD( and )DC(DC
Venn Diagrams
• Venn diagrams show relationships between sets and their elements
Universal Set
Sets A & B
Venn Diagram Example 1
Set Definition Elements
A = {x | x Z+ and x 8} 1 2 3 4 5 6 7 8
B = {x | x Z+; x is even and 10} 2 4 6 8 10
A B
B A
Venn Diagram Example 2
Set Definition Elements
A = {x | x Z+ and x 9} 1 2 3 4 5 6 7 8 9
B = {x | x Z+ ; x is even and 8} 2 4 6 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: (Ac)c = A• Idempotent Laws: A A = A and A A = A • De Morgan’s Laws: (A B)c = Ac Bc and (A B)c = Ac Bc
• Absorption Laws: A (A B) = A and A (A B) = A• Alternate Representation for Difference: A – B = A Bc
• 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 2n elements