复旦大学：《集合论》课程教学资源（PPT课件）集合论导论 Introduction to Set Theory（张宓）

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

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 of complex variable Discrete stagnancy

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

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

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

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

1离散数学及其应用(英文版) 作者: Kenneth. 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

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 German mathematician. Cantor's 1874 paper, "On a Characteristic Property of All Real Algebraic Numbers", marks the birth of set theory. paradox

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

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 "teA" to say that is an element of A, and We write“tea” to say that is not an element of A

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 3eZ, -8eZ, 6. These sets, each denoted using a boldface letter, play an important role in discrete mathematics: N={0, 1,2,.., the set of natural number ==-2,-1,0,1,2,}, the set of integers It=Z={1,,..}, the set of positive integers I-Z--1,-2,.}, the set of negative integers Q-{p/, qeZ,=0}, the set of rational numbers Qt, the set of positive rational numbers Q, the set of negative rational numbers

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 of odd positive integers less than 10 can be expressed by A={1,3,5,7,9} B={x1,x2,x3 123

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 } √