Discrete mathematics Discrete i.e. no continuousSet theory, Combinatorics, Graphs, Modern

Algebra(Abstract algebra, Algebraic structures), Logic, classic probability, number theory, Automata and Formal Languages, Computability and decidability etc.

Before the 18th century, Discrete ， quantity and spaceastronomy, physics Example: planetary orbital, Newton's Laws in Three Dimensionscontinuous mathematics:calculus, Equations of Mathematical Physics, Functions of Real Variable,Functions of

complex VariableDiscrete ? stagnancy

in the thirties of the twentieth century, Turing MachinesFiniteDiscreteData Structures and Algorithm DesignDatabaseCompilersDesign and Analysis of AlgorithmsComputer NetworksSoftware information security and cryptography the theory of computationNew generation computers

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

每周三交作业，作业成绩占总成绩的 15% ；平时不定期的进行小测验，占总成绩的

15% ；期中考试成绩占总成绩的 20% ；期终考试成绩

占总成绩的 50%zhym@fudan.edu.cn张宓 13212010027@fudan.edu.cnBBS id:abchjsabc 软件楼 1039杨侃 10302010007@fudan.edu.cn

1. 离散数学及其应用（英文版）作者： Kenneth H.Rosen 著出版社：机械工业出版社

2. 组合数学（英文版）——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

Ⅰ Introduction 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

twentieth centuryaxiomatic set theorynaive set theoryConceptRelation,function,cardinal numberparadox

Chapter 1 Basic Concepts of Sets

1.1 Sets and Subsets What are Sets?A collection of different objects is called a setS,AThe individual objects in this collection are

called the elements of the set We write “tA” to say that t is an element of A,

and We write “tA” to say that t is not an element of A

Example:The set of all integers, Z. Then 3Z, -8Z, 6.5Z

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 integersQ={p/q|pZ,qZ,q0}, the set of rational numbersQ+, the set of positive rational numbersQ-, the set of negative rational numbers

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} √

(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,yZ+}C describes the set of all cubes of positive

integers.D={x|-1<x<2}

(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.

Example: The set of even nonnegative integers E’={x|x 0,and x=2y,where y≧ Z}

(1)Basic step:0E+ 。 (2)Recursive step: If nE+,then n+2E+. (3)Closed step:There are no other elements in the set

E’ except those constructed using steps (1) and (2).Example： (1)Basic step:3S 。 (2)Recursive step: If x and yS, then x+yS 。 (3)Closed step: There are no other elements in the set

S except those constructed using steps (1) and (2).S=?S={y|y=3x,xZ+}

Let aiΣ, 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,

bjΣ(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.

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 aΣ, then aΣ+.(2)Recursive step: If a and xΣ+, then

axΣ+.(3)Closed step: There are no other

elements in the set Σ+ except those constructed using steps (1) and (2).

Σ+ element or string: infiniteLength of string: finite, 1,2,3,…

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 xΣ* and aΣ then

xaΣ*.(3)Closed step: There are no other

elements in the set Σ* except those constructed using steps (1) and (2).

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

expressions except those constructed using steps (1) and (2).

A={1, 3, 5, 7, 9},B={x1,x2,x3}, finite elements, 5 3C={x|x=y3,yZ+}, infinite elementsA set S is called finite set if it has n distinct

elements, where nN. 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

Example:A={x|x2+1=0, and x is an real number},

No elementempty 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 setThe universal set is the set of all elements under

consideration in a given discussion. We denote the universal set by U.

(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 setExample: 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.

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

2.SubsetsDefinition 1.1:Let A and B are two sets. If

every element of A is also an element of B, that is, if whenever xA then xB, we say that A is a subset of B or that A is contained in B, and we write AB 。 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.

Venn DiagramsIn Venn diagrams the universal set U is

represented by a rectangle, while sets within U are represented by circles.

AB A⋢B, B ⋢A

Example: A={x|-1<x<2}. 0.5A, but 0.5 is not an integer, so A={x|-1<x<2}⋢Z,

ZQ,(1)For any set A, A. (2)If AB, and BC, then AC

Definition 1.2: Let A and B be sets. We say that A equals B, written A=B, whenever for any x, xA if only if xB. If A and B are not equal, we write AB.

It is easy to see that A=B if only if AB and BA

Definition 1.3: If AB and AB, we write AB and say that A is a proper subset of B.

Example:{a}{a,b} 。Example:S1={a},S2={{a}},S3={a,{a}}

aS3, S1S3

{a}S3,S2S3,

S1S3, S1S2,

Theorem 1.1: For any set A,(1)A ,(2)AA

A={1,2,3},,{1},{2},{3},{1,2},{1,3},{2,3} and {1,2,3} are subsets of A

power set of ADefinition 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|.

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|∪xA or xB}

(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|xA and xB} 。

(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|xA and xB} 。

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

}10,9,8,7,5,3{},10,9,8,7,6{ BA

The complement of A , write , =U-A, is the set of all elements of U that are not elements of A

A A

Definition 1.6: Let A1,A2,…An be sets. If I={1,2,…n}, then

(1)The union of the sets A1,A2,…An, A1 A∪ 2 … A∪ ∪ n={x|there is an iI such that x Ai}.

(2)The intersection of the sets A1,A2,…An, A1∩A2∩…∩An={x|xAi for all iI}.

2.Properties of set operationsTheorem 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

AAUAAUUlawscomplement ,,,:)7(

AAlawsationcomplement :)8(

,,:')9( BABABABAlawssMorganDe

BABA

BABA

,BAxFor ,BAx ,BxorAxHence

,.. BxorAxei

BAxTherefore

BABAHence

BABA

,BAxFor ,BxorAx ,BxorAxHence

BAxei ..

,BAxHence BABATherefore

BABA

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 XP(A)∩P(B)(2)P (A∩B)P (A)∩P (B)For any X P(A∩B)

Example ： (A B)∪ -C=(A-C) (B∪ -C)

CBACBALeftoof )()(:Pr

))()()( lawsveDistributiCBCA

)()( CBCA

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