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Transcript

  • 1. Lecture 8 Set June 17 th , 2003
  • 2. Review
    • Survey results.
    • Quiz results.
    • Answer will be posted on the website.
    • Assignment 2 answer will also be posted on the website.
    • First midterm is postponed to next monday.
    • Last day for dropping with w is Next Friday.
  • 3. Sets in real world
    • Some database language is based on set theory, for example, sql.
    • select x
    • from A, B
    • where A.a=B.a
  • 4. Sets
    • Notation: Membership in a set--
    • Not a membership in a set ?
    • Set properties
      • No ordering
      • Each element is listed once.
    • Equal
      • A=B means (  x) (x є A  x є B)
    • Finite set VS infinite set.
  • 5. Set (cont.)
    • How to describe a set
      • List (or partial list ) its element
      • Use recursion to describe how to generate the set elements.
      • Describe a property P that characterizes the set elements.
      • The 3 rd method is the best choice.
    • Examples- Practice in P164.
  • 6. Some conventions
    •  = set of nonnegative integers, including 0
    • Z = set of all integers
    • Q = set of all rational numbers
    • R = set of all real numbers
    • C = set of all complex numbers
    • Ø = empty set (null set)
  • 7. Relationship between sets
    • A is a subset of B
    • A is a proper subset of B
    • A=B 
  • 8. Power Set
    • Sets of Sets S (power sets, P(S) ), its element is all the subset of S.
    • Example, A={1,2,3}, what is the power set of A?
    • S has n elements then how many elements P(S) has?
  • 9. Operation
    • Union U: S U T = {x | x S or x T}
    • Intersection ^: S ^ T = {x | x S and x T}
    • Difference -: S - T = {x | x S and not x T}
    • Examples: S={a,b,c,d,e,f} T={b,c,e,g,h,j} S U T = {a,b,c,d,e,f,g,h,I} , S ^ T = {b,c,e} , S - T = {a,d,f}
  • 10. Operation (cont.)
    • Cartesian Product
    What is the power sets of A in cartesian product ? P(A)=A X A
  • 11. Venn Diagrams
    • Region 1 - Elements in neither S nor T.
    • Region 2 - S - T
    • Region 3 - S ^ T
    • Region 4 - T - S
    • Regions 2 & 3 combined - S
    • Regions 3 & 4 combined - T
    • Regions 2 & 3 & 4 combined - S U T
    • Region 1 is not a set or (Region 1 U S U T) would be the universal set which Russell's paradox won't allow.
  • 12. Set Identities
    • Commutative Properties
      • A  B = B  A
      • A  B = B  A
    • Associative Properties
      • (A  B)  C = A  (B  C)
      • (A  B)  C = A  (B  C)
    • Distributive Properties
      • A  (B  C) = (A  B)  (A  C)
      • A  (B  C) = (A  B)  (A  C)
    • Identity Properties
      • A  = A
      • A  S = A
    • Complement Properties
      • A  A’ = S
      • A  A’ = 
    • De Morgan’s Laws
      • (A  B)’ = A’  B’
      • (A  B)’ = A’  B’
  • 13. Countable and Uncountable sets
    • The number of elements in a finite set is the cardinality of the set.
    • Denumerable Sets
      • Infinite sets in which we’re able to create a list where we can select a first element s 1 , second element s 2 , … until every element of the set will eventually appear in the list.
    • Countable Sets
      • Sets that either finite or denumerable. (we can count, or enumerate all their elements)
  • 14. Countable and Uncountable sets
    • Uncountable Sets
      • Infinite sets that are so big that there is no way to count out the elements and get the whole set in the process.
      • To prove denumerability, we need only exhibit a counting scheme.
  • 15. Cantor’s Set Theory
    • We can determined if two infinite sets are the same “size” (equinumerous) by seeking to find a one-to-one match-up between the elements of each set.
    • A set is infinite if we can remove some of its elements without reducing its size.
  • 16. Examples
    • Set of positive integers is denumerable.
    • Set of all integers Z is denumerable.
    • Set of rational numbers is denumerable.
    • Set Z X Z is denumerable.
    • Practice– Set of positive rational numbers is denumerable.
  • 17. Exercise
    • Exercise 3.1 ---22, 39, 50, 75, 76.
    • You do not have to turn it in. But you might see this question patterns in test or quiz.
  • 18. Introduction of counting
    • A customer is ordering a computer. The choice are 17”, 19” or 21”; 1.5G, 1.7G and 2.0 GHz CPU, 20x, 40x, 48x CD drive.
    • Q: how many different machine configurations are possible?
    • How many different machines can be ordered with 1.7G CPU?

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