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