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 “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 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 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 integers Q={p/q|pZ,qZ,q0}, 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 } √