Download Z - ={-1,-2,…}

 Discrete mathematics
 Discrete i.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.









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
complex Variable
 Discrete ? stagnancy
of













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 security and cryptography
the theory of computation
New 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%
 [email protected]
 张宓 [email protected]
 BBS id:abchjsabc 软件楼1039
 杨侃 [email protected]
 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
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






twentieth century
axiomatic set theory
naive set theory
Concept
Relation,function,cardinal number
paradox
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
 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
 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,yZ+}
 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: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
S except those constructed using steps (1) and (2).
 S=?
 S={y|y=3x,xZ+}
 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.
 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,…
 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).
 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
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
 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.
 (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.
 Example: Let S={,{}}。 Elements of
S are  and {}
 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.
 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
 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
 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.
 Example:{a}{a,b}。
 Example:S1={a},S2={{a}},S3={a,{a}}
 aS3, S1S3
 {a}S3,S2S3,
 S1S3, S1S2,
 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|.
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}
 (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A and 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A and xB}。
The complement of A , write A , A =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
A  {6,7,8,9,10},
B  {3,5,7,8,9,10}
 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∪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}.
 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
(7)complement laws :   U , U  ,
A A U,
A A  
(8)complementation laws : A  A
(9) De Morgan ' s laws : A  B  A  B ,
A B  A  B,
A B  A  B
A B  A  B
For x  A  B,
x  A  B, Hence x  A or x  B,
i.e.x  A or x  B ,
Therefore x  A  B
Hence A  B  A  B
A  B  A B
For x  A  B ,
x  A or x  B , Hence x  A or x  B,
i.e.x  A  B
Hence x  A  B,
A B  A  B
Therefore A  B  A  B
 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 any X P(A∩B)
Example：(A∪B)-C=(A-C)∪(B-C)
Pr oof : Left  ( A  B)  C  ( A  B)  C
 ( A  C )  ( B  C ) )( Distributi ve laws)
 ( A  C)  ( B  C)
 Exercise:P11 2,4,8,12,34, 39,40