Discrete mathematics Discrete i.e. no continuous

1. Discrete mathematics • Discretei.e. no continuous • Set theory, Combinatorics, Graphs, Modern Algebra(Abstract algebra, Algebraic structures), Logic, classic probability, number theory,Automata and Formal Languages, Computability and decidability etc.

2. Before the 18th century, • Discrete， quantity and space • astronomy, physics • Example: planetary orbital, • Newton's Laws in Three Dimensions • continuous mathematics: • calculus, • Equations of Mathematical Physics, • Functions of Real Variable,Functions of complex Variable • Discrete ? stagnancy

3. in the thirties of the twentieth century, • Turing Machines • Finite • Discrete • Data Structures and Algorithm Design • Database • Compilers • Design and Analysis of Algorithms • Computer Networks • Software • information securityand cryptography • the theory of computation • New generation computers

4. Set theory, • Introductory Combinatorics, • Graphs, • Algebtaic structures, • Logic. • This term: • Set theory, • Introductory Combinatorics , • Graphs, • Algebtaic structures(Group,Ring，Field). • Next term: • Algebtaic structures(Lattices and Boolean Algebras), • Logic

5. 每周一交作业，作业成绩占总成绩的10%； • 平时不定期的进行小测验，占总成绩的20%； • 期中考试成绩占总成绩的20%；期终考试成绩占总成绩的50%

6. 1.离散数学及其应用（英文版·第5版） • 作者：Kenneth H.Rosen 著出版社：机械工业出版社 • 2.组合数学（英文版·第4版）——经典原版书库 • 作者：（美）布鲁迪（Brualdi,R.A.） 著出版社：机械工业出版社 • 3,离散数学暨组合数学（英文影印版） • Discrete Mathematics with Combinatorics  • James A.Anderson,University of South Carolina,Spartanburg • 大学计算机教育国外著名教材系列（影印版）清华大学出版社

7. ⅠIntroduction to Set Theory • The objects of study of Set Theory are sets. As sets are fundamental objects that can be used to define all other concepts in mathematics. • Georg Cantor(1845--1918) is a German mathematician. • Cantor's 1874 paper, "On a Characteristic Property of All Real Algebraic Numbers", marks the birth of set theory. • paradox

8. twentieth century • axiomatic set theory • naive set theory • Concept • Relation,function,cardinal number • paradox

9. Chapter 1 Basic Concepts of Sets 1.1 Sets and Subsets • What are Sets? • A collection of different objects is called a set • S,A • The individual objects in this collection are called the elements of the set • We write “tA” to say that t is an element of A, and We write “tA” to say that t is not an element of A

10. Example:The set of all integers, Z. • Then 3Z, -8Z, 6.5Z • These sets, each denoted using a boldface letter, play an important role in discrete mathematics: • N={0,1,2,…}, the set of natural number • I=Z={…,-2,-1,0,1,2,…}, the set of integers • I+=Z+={1,2,…}, the set of positive integers • I-=Z-={-1,-2,…}, the set of negative integers • Q={p/q|pZ,qZ,q0}, the set of rational numbers • Q+, the set of positive rational numbers • Q-, the set of negative rational numbers

11. 1. Representation of set • (1)Listing elements, One way is to list all the elements of a set when this is possible.. • Example：The set A of odd positive integers less than 10 can be expressed by A={1, 3, 5, 7, 9}。 • B={x1,x2,x3} √

12. (2)Set builder notion: We characterize the property or properties that the elements of the set have in common. • Example:The set A of odd positive integers less than 10 can be expressed by A={x|x is an odd positive integer less than 10} • Example:C={x|x=y3,yZ+} • C describes the set of all cubes of positive integers. • D={x|-1<x<2}

13. (3)Recursive definition • Recursive definitions of sets have three steps: • 1)Basic step: Specify some of the basic elements in the set. • 2)recursive step: Give some rules for how to construct more elements in the set from the elements that we know are already there . • 3) closed step: There are no other elements in the set except those constructed using steps 1 and 2.

14. Example: The set of even nonnegative integers E’={x|x≧0,and x=2y,where yZ} • (1)Basic step:0E+。 • (2)Recursive step: If nE+,then n+2E+. • (3)Closed step:There are no other elements in the set E’ except those constructed using steps (1) and (2). • Example： • (1)Basic step:3S。 • (2)Recursive step: If x and yS, then x+yS。 • (3)Closed step: There are no other elements in the set Sexcept those constructed using steps (1) and (2). • S=? • S={y|y=3x,xZ+}

15. Let aiΣ, sequences of the form a1a2…an are often in computer science. These finite sequences are also called strings. The length of the string S is the number of terms in this string. • The empty string, denoted by , is the string that has no terms. The empty string has length zero. • If x=a1a2…an, and y=b1b2…bm are strings, where ai, bjΣ(1≦i≦n,1≦j≦m), we define the catenation of x and y as the string a1a2…an b1b2…bm . • The catenation of x and y is written as xy, and is another string from Σ, i.e. xy=a1a2…an b1b2…bm. • Note x=x and x=x.

16. Let Σ be an alphabet, we can construct the set Σ+ consisting of all finite nonempty string of elements of Σ: • (1)Basic step: If aΣ, then aΣ+. • (2)Recursive step: If a and xΣ+, then axΣ+. • (3)Closed step: There are no other elements in the set Σ+ except those constructed using steps (1) and (2). • Σ+ element or string: infinite • Length of string: finite, 1,2,3,…

17. Let Σ be an alphabet, we can construct the set Σ* consisting of all finite string of elements of Σ: • (1)Basic step: Σ*. • (2)Recursive step: If xΣ* and aΣ then xaΣ*. • (3)Closed step: There are no other elements in the set Σ* except those constructed using steps (1) and (2).

18. Arithmetic expressions • (B) A numeral is an arithmetic expression. • (R) If e1 and e2 are arithmetic expressions, then • all of the following are arithmetic expressions: • e1+e2, e1−e2, e1*e2, e1/e2, (e1) • (C)There are no other arithmetic expressionsexcept those constructed using steps (1) and (2).

19. A={1, 3, 5, 7, 9},B={x1,x2,x3}, finite elements, • 5 3 • C={x|x=y3,yZ+}, infinite elements • A set S is called finite set if it has n distinct elements, where nN. In this case, n is called the cardinality of S and is denoted by |S|. A set that is not finite is called infinite set. • Σ*,Σ+,C,D,S are infinite sets, A,B are finite sets. • P={x|x is an prime number less than 6}, 2,3,5, |P|=3

20. Example:A={x|x2+1=0, and x is an real number}, • No element • empty set,|A|=0. • The set that has no elements in it is denoted by {} or the symbol  and is called the empty set. • Note: {} is not an empty set. It is a set with one element which the element is the empty set. • {}, but . • universal set • The universal set is the set of all elements under consideration in a given discussion. We denote the universal set by U.

21. (1)The order in which the elements of a set are listed is not important. • {a,b,c},{a,c,b},{b,a,c},{b,c,a},{c,a,b},and {c, b, a} are all representations of the same set. • (2) In the listing of the elements of a set, repeated elements aren't allowed. • (3)A set can be an element of another set • Example: S={{a,b},{a,b,c},{d,e}} • Note:{a,b,c} is also a set consisting of elements a,b,c。a,b, and c aren’t elements of S.

22. Example：Let S={,{}}。 Elements of S are  and {}

23. 2.Subsets • Definition 1.1:Let A and B are two sets. If every element of A is also an element of B, that is, if whenever xA then xB, we say that A is a subset of B or that A is contained in B, and we write AB。If there is an element of A that is not in B, then A is not a subset of B, and we write A⋢B.

24. Venn Diagrams • In Venn diagrams the universal set U is represented by a rectangle, while sets within U are represented by circles. AB A⋢B, B ⋢A

25. Example: A={x|-1<x<2}. 0.5A, but 0.5 is not an integer, so A={x|-1<x<2}⋢Z, • ZQ, • (1)For any set A, A. • (2)If AB, and BC, then AC

26. Definition 1.2: Let A and B be sets. We say that A equals B, written A=B, whenever for any x, xA if only if xB. If A and B are not equal, we write AB. • It is easy to see that A=B if only if AB and BA • Definition 1.3: If AB and AB, we write AB and say that A is a proper subset of B.

27. Example:{a}{a,b}。 • Example:S1={a},S2={{a}},S3={a,{a}} • aS3, S1S3 • {a}S3,S2S3, • S1S3, S1S2,

28. Theorem 1.1: For any set A, • (1)A ,(2)AA • A={1,2,3},,{1},{2},{3},{1,2},{1,3},{2,3} and {1,2,3} are subsets of A • power set of A • Definition 1.4: Given a set A, the power set of A is the set of all subsets of the set A. The power set of A is denoted by P(A). • |A|=k，|P (A)|=? • Theorem 1.2: If A is a finite set, then |P (A)|=2|A|.

29. 1.2 Operations on Sets • 1.Definition of operations on sets • Definition 1.5：Let A and B be two subsets of universal set U, • (1)The union of A and B, write A∪B, is the set of all elements that are in A or B. i.e. A∪B={x|xA or xB}

30. (2)The intersection of A and B, write A∩B, is the set of all elements that are in both A and B . i.e.A∩B= {x|xAand xB}。 (3) The difference of A and B, write A-B, is the set of all elements that are in A but are not in B. i.e.A-B={x|xAand xB}。

31. The complement of A , write , =U-A, is the set of all elements of U that are not elements of A • Example:A={1,2,3,4,5},B={1,2,4,6},C={7,8}, U={1,2,3,4,5,6, 7,8,9,10}。 A∪B={1,2,3,4,5,6}, A∩B={1,2,4},A∩C=, A-B={3,5},A-C=A

32. Definition 1.6: Let A1,A2,…An and An be sets. If I={1,2,…n}, then • (1)The union of the sets A1,A2,…An, A1∪A2∪…∪An={x|there is an iI such that x Ai}. • (2)The intersection of the sets A1,A2,…An, A1∩A2∩…∩An={x|xAi for all iI}.

33. 2.Properties of set operations • Theorem 1.3: The operations defined on sets satisfy the following properties: • (1)commutative laws :A∪B=B∪A; A∩B=B∩A • (2)associative laws: A∪(B∪C)=(A∪B)∪C; • A∩(B∩C)=(A∩B)∩C • (3)distributive laws: • A∪(B∩C)=(A∪B)∩(A∪C) • A∩(B∪C)=(A∩B)∪(A∩C) • (4)idempotent laws: A∪A=A； A∩A=A • (5)domination laws A∪U=U; A∩= • (6)identical laws: A∪=A; A∩U=A

36. Example：Let A and B be two sets. Then P(A)∩P(B)=P(A∩B) • Proof:(1)P(A)∩P(B)P(A∩B) For any XP(A)∩P(B) (2)P(A∩B)P(A)∩P(B) For anyX P(A∩B)

37. Example：(A∪B)-C=(A-C)∪(B-C)

38. Exercise:P11 2,4,8,12,34, 39,40