1. Introduction to Set theory

2. Ways of Describing Sets

3. Some Special Sets U

4. 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

5. Subset

6. Proper Subset

7. 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

8. Venn Diagrams Venn diagrams show relationships between sets and their elements Universal Set Sets A & B

9. Venn Diagram Example 1 Set DefinitionElements 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

10. Venn Diagram Example 2 Set DefinitionElements 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

15. Symmetric Difference: A  B = (A – B)  (B – A)

17. 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

19. 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