• Essential concepts of set theory
• 點閱：5
• 作者： Robert S.Y. Wong, Manwai Yuen
• 出版社：Hong Kong Educational Publishing Company
• 出版年：2016
• ISBN：978-988-236-584-1 ; 988-236-584-1
• 格式：PDF
• 附註：Includes index

For thousands of years, it is generally believed that mathematics begins with the natural numbers and counting. But there is something more fundamental than counting. It is the grouping of things. If a child is shown a picture of a farm with sheep and cows here and there and asked to count the number of sheep, the child would first put the sheep in a group mentally and then count the number of sheep in the group. Without grouping, counting cannot happen. Therefore, mathematics begins with the grouping of objects, which is the object of study of set theory.

In this book, we explore the fundamental concepts of sets and related topics: propositional logic, methods of proof, relations and functions. Unlike the technical approach adopted in most books, we use many everyday examples to show that these concepts can be found everywhere in our daily life. The book also has plenty of exercises and solutions to all exercises are provided.

• Chapter 1 Sets(第1頁)
• 1.1 Introduction(第1頁)
• 1.2 Basics of Sets(第1頁)
• 1.3 Set Operations(第7頁)
• 1.4 Universal Sets(第8頁)
• 1.5 Venn Diagrams(第11頁)
• 1.6 Number of Elements in a Set(第13頁)
• 1.7 Power Sets(第16頁)
• 1.8 Indexed Sets(第18頁)
• 1.9 Partitions of Sets(第20頁)
• 1.10 Cartesian Products(第22頁)
• Further Problems(第27頁)
• Chapter 2 Logic(第28頁)
• 2.1 Introduction(第28頁)
• 2.2 Statements(第28頁)
• 2.3 Open Sentences(第29頁)
• 2.4 Conjunction and Disjunction(第32頁)
• 2.5 Negation(第36頁)
• 2.6 Compound Statements(第37頁)
• 2.7 Implication(第38頁)
• 2.8 Use of Implication in Theorems(第40頁)
• 2.9 Biconditional(第43頁)
• 2.11 Logical Equivalence(第47頁)
• 2.12 Application to Set Theory(第50頁)
• 2.13 Quantified Statements(第52頁)
• 2.14 Definition and Characterization(第54頁)
• Further Problems(第56頁)
• Chapter 3 Proof Strategies(第57頁)
• 3.1 Proofs in Mathematics(第57頁)
• 3.2 Trivial and Vacuous Proofs(第59頁)
• 3.3 Proof Strategy and Analysis(第61頁)
• 3.4 Direct Proofs(第62頁)
• 3.5 Proof by Contrapositive(第67頁)
• 3.6 Proof by Cases(第71頁)
• 3.7 Proof Evaluation(第76頁)
• 3.8 More on Proof Involving Sets(第78頁)
• 3.10 Existence Proofs(第86頁)
• 3.11 Principle of Mathematical Induction(第88頁)
• 3.12 Enhanced Mathematical Induction Principle(第94頁)
• 3.13 Strong Principle of Mathematical Induction(第97頁)
• Further Problems(第99頁)
• Chapter 4 Relations(第100頁)
• 4.1 Introduction(第100頁)
• 4.2 Relations between Two Sets(第100頁)
• 4.3 Properties of Relations on a Set(第102頁)
• 4.4 Equivalence Relations(第105頁)
• 4.5 Properties of Equivalence Relations(第108頁)
• 4.6 Arithmetic Congruence(第110頁)
• 4.7 Integers Modulo n(第115頁)
• Further Problems(第119頁)
• Chapter 5 Functions(第120頁)
• 5.1 Introduction(第120頁)
• 5.2 Definition of Functions(第120頁)
• 5.3 Visualizing Relations and Functions(第121頁)
• 5.4 Definition of Functions in Secondary School(第122頁)
• 5.5 Set of All Functions from A to B(第125頁)
• 5.6 Surjective and Injective Functions(第126頁)
• 5.7 Bijective Functions(第129頁)
• 5.8 Operations on Functions(第132頁)
• 5.9 Inverse Functions(第137頁)
• 5.10 Functional Images and Pre-images(第141頁)
• Further Problems(第143頁)
• Revision Exercises(第144頁)