An introduction to proof through real analysis

• Verfasst von: Madden, Daniel J. [VerfasserIn]
• Verfasst von: Aubrey, Jason A. [VerfasserIn]
• Titel: An introduction to proof through real analysis
• Verf.angabe: Daniel J. Madden and Jason A. Aubrey (The University of Arizona, Tucson, Arizona, USA)
• Verlagsort: Hoboken, NJ
• Verlag: Wiley
• Jahr: 2017
• Umfang: xxiii, 417 Seiten
• Illustrationen: Diagramme
• ISBN: 978-1-119-31472-1
• Schlagwörter: (s)Beweis / (s)Beweistheorie
• Schlagwörter: (s)Beweis / (s)Beweistheorie
• Sprache: eng
• Bibliogr. Hinweis: Erscheint auch als : Online-Ausgabe: Madden, Daniel J., 1948-: An introduction to proof through real analysis. - Hoboken, NJ : Wiley, 2017. - 1 Online-Ressource
• RVK-Notation: SK 130
• K10plus-PPN: 887521738

Abstract

Cover -- Title Page -- Copyright -- Contents -- List of Figures -- Preface -- Introduction -- Part I A First Pass at Defining ℝ -- Chapter 1 Beginnings -- 1.1 A naive approach to the natural numbers -- 1.1.1 Preschool: foundations of the natural numbers -- 1.1.2 Kindergarten: addition and subtraction -- 1.1.3 Grade school: multiplication and division -- 1.1.4 Natural numbers: basic properties and theorems -- 1.2 First steps in proof -- 1.2.1 A direct proof -- 1.2.2 Mathematical induction -- 1.3 Problems -- Chapter 2 The Algebra of the Natural Numbers -- 2.1 A more sophisticated look at the basics -- 2.1.1 An algebraic approach -- 2.2 Mathematical induction -- 2.2.1 The theorem of induction -- 2.3 Division -- 2.3.1 The division algorithm -- 2.3.2 Odds and evens -- 2.4 Problems -- Chapter 3 Integers -- 3.1 The algebraic properties of ℕ -- 3.1.1 The algebraic definition of the integers -- 3.1.2 Simple results about integers -- 3.1.3 The relationship between ℕ and ℤ -- 3.2 Problems -- Chapter 4 Rational Numbers -- 4.1 The algebra -- 4.1.1 Surveying the algebraic properties of ℤ -- 4.1.2 Defining an ordered field -- 4.1.3 Properties of ordered fields -- 4.2 Fractions versus rational numbers -- 4.2.1 In some ways they are different -- 4.2.2 In some ways they are the same -- 4.3 The rational numbers -- 4.3.1 Operations are well defined -- 4.3.2 ℚ is an ordered field -- 4.4 The rational numbers are not enough -- 4.4.1 √2 is irrational -- 4.5 Problems -- Chapter 5 Ordered Fields -- 5.1 Other ordered fields -- 5.2 Properties of ordered fields -- 5.2.1 The average theorem -- 5.2.2 Absolute values -- 5.2.3 Picturing number systems -- 5.3 Problems -- Chapter 6 The Real Numbers -- 6.1 Completeness -- 6.1.1 Greatest lower bounds -- 6.1.2 So what is complete? -- 6.1.3 An alternate version of completeness -- 6.2 Gaps and caps

Abstract

6.2.1 The Archimedean principle -- 6.2.2 The density theorem -- 6.3 Problems -- 6.4 Appendix -- Part II Logic, Sets, and Other Basics -- Chapter 7 Logic -- 7.1 Propositional logic -- 7.1.1 Logical statements -- 7.1.2 Logical connectives -- 7.1.3 Logical equivalence -- 7.2 Implication -- 7.3 Quantifiers -- 7.3.1 Specification -- 7.3.2 Existence -- 7.3.3 Universal -- 7.3.4 Multiple quantifiers -- 7.4 An application to mathematical definitions -- 7.5 Logic versus English -- 7.6 Problems -- 7.7 Epilogue -- Chapter 8 Advice for Constructing Proofs -- 8.1 The structure of a proof -- 8.1.1 Syllogisms -- 8.1.2 The outline of a proof -- 8.2 Methods of proof -- 8.2.1 Direct methods -- 8.2.1.1 Understand how to start -- 8.2.1.2 Parsing logical statements -- 8.2.1.3 Mathematical statements to be proved -- 8.2.1.4 Mathematical statements that are assumed to be true -- 8.2.1.5 What do we know and what do we want? -- 8.2.1.6 Construction of a direct proof -- 8.2.1.7 Compound hypothesis and conclusions -- 8.2.2 Alternate methods of proof -- 8.2.2.1 Contrapositive -- 8.2.2.2 Contradiction -- 8.3 An example of a complicated proof -- 8.4 Problems -- Chapter 9 Sets -- 9.1 Defining sets -- 9.2 Starting definitions -- 9.3 Set operations -- 9.3.1 Families of sets -- 9.4 Special sets -- 9.4.1 The empty set -- 9.4.2 Intervals -- 9.5 Problems -- 9.6 Epilogue -- Chapter 10 Relations -- 10.1 Ordered pairs -- 10.1.1 Relations between and on sets -- 10.2 A total order on a set -- 10.2.1 Definition -- 10.2.2 Definitions that use a total order -- 10.3 Equivalence relations -- 10.3.1 Definitions -- 10.3.2 Equivalence classes -- 10.3.3 Equivalence partitions -- 10.3.3.1 Well defined -- 10.4 Problems -- Chapter 11 Functions -- 11.1 Definitions -- 11.1.1 Preliminary ideas -- 11.1.2 The technical definition -- 11.1.2.1 A word about notation

Abstract

11.2 Visualizing functions -- 11.2.1 Graphs in ℝ2 -- 11.2.2 Tables and arrow graphs -- 11.2.3 Generic functions -- 11.3 Composition -- 11.3.1 Definitions and basic results -- 11.4 Inverses -- 11.5 Problems -- Chapter 12 Images and preimages -- 12.1 Functions acting on sets -- 12.1.1 Definition of image -- 12.1.2 Examples -- 12.1.3 Definition of preimage -- 12.1.4 Examples -- 12.2 Theorems about images and preimages -- 12.2.1 Basics -- 12.2.2 Unions and intersections -- 12.3 Problems -- Chapter 13 Final Basic Notions -- 13.1 Binary operations -- 13.2 Finite and infinite sets -- 13.2.1 Objectives of this discussion -- 13.2.2 Why the fuss? -- 13.2.3 Finite sets -- 13.2.4 Intuitive properties of infinite sets -- 13.2.5 Counting finite sets -- 13.2.6 Finite sets in a set with a total order -- 13.3 Summary -- 13.4 Problems -- 13.5 Appendix -- 13.6 Epilogue -- Part III A Second Pass at Defining ℝ -- Chapter 14 ℕ, ℤ, and ℚ -- 14.0.1 Basic properties of the natural numbers -- 14.0.2 Theorems about the natural numbers -- 14.1 The integers -- 14.1.1 An algebraic definition -- 14.1.2 Results about the integers -- 14.1.3 The relationship between natural numbers and integers -- 14.2 The rational numbers -- 14.3 Problems -- Chapter 15 Ordered Fields and the Real Numbers -- 15.1 Ordered fields -- Definitions and easy consequences -- More properties of ordered fields -- 15.2 The real numbers -- Definition -- Properties of ℝ -- Decimals and real numbers -- 15.3 Problems -- 15.4 Epilogue -- 15.4.1 Constructing the real numbers -- Chapter 16 Topology -- 16.1 Introduction -- 16.1.1 Preliminaries -- 16.1.2 Neighborhoods -- 16.1.3 Interior, exterior, and boundary -- 16.1.4 Isolated points and accumulation points -- 16.1.5 The closure -- 16.2 Examples -- 16.3 Open and closed sets -- 16.3.1 Definitions -- 16.3.2 Examples -- 16.4 Problems

Abstract

Chapter 17 Theorems in Topology -- 17.1 Summary of basic topology -- 17.2 New results -- 17.2.1 Unions and intersections -- 17.2.2 Open intervals are open -- 17.2.3 Open subsets are in the interior -- 17.2.4 Topology and completeness -- 17.3 Accumulation points -- 17.3.1 Accumulation points are aptly named -- 17.3.2 For all A ⊆ ℝ, A′ is closed -- 17.4 Problems -- Chapter 18 Compact Sets -- 18.1 Closed and bounded sets -- 18.1.1 Maximums and minimums -- The definition of compact -- Compact sets are closed and bounded sets -- 18.2 Closed intervals are special -- Inseparability of closed intervals -- Sets that are both open and closed -- 18.3 Problems -- Chapter 19 Continuous Functions -- 19.1 First semester calculus -- 19.1.1 An intuitive idea of a continuous function -- 19.1.2 The calculus definition of continuity -- 19.1.3 The official mathematical definition of continuity -- 19.1.4 Examples -- 19.2 Theorems about continuity -- 19.2.1 Three specific functions -- 19.2.2 Multiplying a continuous function by a constant -- 19.2.3 Adding continuous functions -- 19.2.4 Multiplying continuous functions -- 19.2.5 Polynomial functions -- 19.2.6 Composition of continuous functions -- 19.2.7 Dividing continuous functions -- 19.2.8 Gluing functions together -- 19.3 Problems -- Chapter 20 Continuity and Topology -- 20.1 Preliminaries -- 20.1.1 Continuous images mess up topology -- 20.2 The topological definitions of continuity -- 20.3 Compact images -- 20.3.1 The main theorem -- 20.3.2 The extreme value theorem -- 20.3.3 The intermediate value theorem -- 20.4 Problems -- Chapter 21 A Few Final Observations -- 21.1 Inverses of continuous functions -- 21.1.1 A strange example -- 21.1.2 The theorem about inverses of continuous functions -- 21.2 The intermediate value theorem and continuity

Abstract

21.3 Continuity at discrete points -- 21.4 Conclusion -- Index -- EULA