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# Section3 1

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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?