Maddox A Transition to Abstract Mathematics

Learning Mathematical Thinking and Writing
2. Auflage 2008
ISBN: 978-0-08-092271-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 384 Seiten, Web PDF

Constructing concise and correct proofs is one of the most challenging aspects of learning to work with advanced mathematics. Meeting this challenge is a defining moment for those considering a career in mathematics or related fields. Mathematical Thinking and Writing teaches readers to construct proofs and communicate with the precision necessary for working with abstraction. It is based on two premises: composing clear and accurate mathematical arguments is critical in abstract mathematics, and that this skill requires development and support. Abstraction is the destination, not the starting point.

Maddox methodically builds toward a thorough understanding of the proof process, demonstrating and encouraging mathematical thinking along the way. Skillful use of analogy clarifies abstract ideas. Clearly presented methods of mathematical precision provide an understanding of the nature of mathematics and its defining structure.
After mastering the art of the proof process, the reader may pursue two independent paths. The latter parts are purposefully designed to rest on the foundation of the first, and climb quickly into analysis or algebra. Maddox addresses fundamental principles in these two areas, so that readers can apply their mathematical thinking and writing skills to these new concepts. From this exposure, readers experience the beauty of the mathematical landscape and further develop their ability to work with abstract ideas.
* Covers the full range of techniques used in proofs, including contrapositive, induction, and proof by contradiction
* Explains identification of techniques and how they are applied in the specific problem
* Illustrates how to read written proofs with many step by step examples
* Includes 20% more exercises than the first edition that are integrated into the material instead of end of chapter
* The Instructors Guide and Solutions Manual points out which exercises simply must be either assigned or at least discussed because they undergird later results

Autoren/Hrsg.

Maddox, Randall

Inhaltsverzeichnis

1;Front Cover;1
2;A Transition to Abstract Mathematics;4
3;Copyright Page;5
4;Table of Contents;8
5;Why Read This Book?;14
6;Preface;16
7;Preface to the First Edition;18
8;Acknowledgments;22
9;Chapter 0. Notation and Assumptions;24
9.1;0.1 Set Terminology and Notation;24
9.2;0.2 Assumptions about the Real Numbers;26
9.2.1;0.2.1 Basic Algebraic Properties;26
9.2.2;0.2.2 Ordering Properties;28
9.2.3;0.2.3 Other Assumptions;30
10;Part 1: Foundations of Logic and ProofWriting;32
10.1;Chapter 1. Language and Mathematics;34
10.1.1;1.1 Introduction to Logic;34
10.1.1.1;1.1.1 Statements;34
10.1.1.2;1.1.2 Negation of a Statement;36
10.1.1.3;1.1.3 Combining Statements with AND;36
10.1.1.4;1.1.4 Combining Statements with OR;37
10.1.1.5;1.1.5 Logical Equivalence;39
10.1.1.6;1.1.6 Tautologies and Contradictions;41
10.1.2;1.2 If-Then Statements;41
10.1.2.1;1.2.1 If-Then Statements Defined;41
10.1.2.2;1.2.2 Variations on p . q;44
10.1.2.3;1.2.3 Logical Equivalence and Tautologies;46
10.1.3;1.3 Universal and Existential Quantifiers;50
10.1.3.1;1.3.1 The Universal Quantifier;51
10.1.3.2;1.3.2 The Existential Quantifier;52
10.1.3.3;1.3.3 Unique Existence;55
10.1.4;1.4 Negations of Statements;56
10.1.4.1;1.4.1 Negations of AND and OR Statements;56
10.1.4.2;1.4.2 Negations of If-Then Statements;57
10.1.4.3;1.4.3 Negations of Statements with the Universal Quantifier;59
10.1.4.4;1.4.4 Negations of Statements with the Existential Quantifier;60
10.1.5;1.5 How We Write Proofs;63
10.1.5.1;1.5.1 Direct Proof;63
10.1.5.2;1.5.2 Proof by Contrapositive;64
10.1.5.3;1.5.3 Proving a Logically Equivalent Statement;64
10.1.5.4;1.5.4 Proof by Contradiction;65
10.1.5.5;1.5.5 Disproving a Statement;65
10.2;Chapter 2. Properties of Real Numbers;68
10.2.1;2.1 Basic Algebraic Properties of Real Numbers;68
10.2.1.1;2.1.1 Properties of Addition;69
10.2.1.2;2.1.2 Properties of Multiplication;72
10.2.2;2.2 Ordering Properties of the Real Numbers;74
10.2.3;2.3 Absolute Value;76
10.2.4;2.4 The Division Algorithm;79
10.2.5;2.5 Divisibility and Prime Numbers;82
10.3;Chapter 3. Sets and Their Properties;86
10.3.1;3.1 Set Terminology;86
10.3.2;3.2 Proving Basic Set Properties;90
10.3.3;3.3 Families of Sets;94
10.3.4;3.4 The Principle of Mathematical Induction;101
10.3.5;3.5 Variations of the PMI;108
10.3.6;3.6 Equivalence Relations;114
10.3.7;3.7 Equivalence Classes and Partitions;120
10.3.8;3.8 Building the Rational Numbers;125
10.3.8.1;3.8.1 Defining Rational Equality;126
10.3.8.2;3.8.2 Rational Addition and Multiplication;127
10.3.9;3.9 Roots of Real Numbers;129
10.3.10;3.10 Irrational Numbers;130
10.3.11;3.11 Relations in General;134
10.4;Chapter 4. Functions;142
10.4.1;4.1 Definition and Examples;142
10.4.2;4.2 One-to-one and Onto Functions;148
10.4.3;4.3 Image and Pre-Image Sets;151
10.4.4;4.4 Composition and Inverse Functions;154
10.4.4.1;4.4.1 Composition of Functions;155
10.4.4.2;4.4.2 Inverse Functions;156
10.4.5;4.5 Three Helpful Theorems;158
10.4.6;4.6 Finite Sets;160
10.4.7;4.7 Infinite Sets;162
10.4.8;4.8 Cartesian Products and Cardinality;167
10.4.8.1;4.8.1 Cartesian Products;167
10.4.8.2;4.8.2 Functions Between Finite Sets;169
10.4.8.3;4.8.3 Applications;171
10.4.9;4.9 Combinations and Partitions;174
10.4.9.1;4.9.1 Combinations;174
10.4.9.2;4.9.2 Partitioning a Set;175
10.4.9.3;4.9.3 Applications;176
10.4.10;4.10 The Binomial Theorem;180
11;Part II: Basic Principles of Analysis;186
11.1;Chapter 5: The Real Numbers;188
11.1.1;5.1 The Least Upper Bound Axiom;188
11.1.1.1;5.1.1 Least Upper Bounds;189
11.1.1.2;5.1.2 Greatest Lower Bounds;191
11.1.1.3;5.2 The Archimedean Property;192
11.1.1.3.1;5.2.1 Maximum and Minimum of Finite Sets;193
11.1.1.4;5.3 Open and Closed Sets;195
11.1.1.5;5.4 Interior, Exterior, Boundary, and Cluster Points;198
11.1.1.5.1;5.4.1 Interior, Exterior, and Boundary;198
11.1.1.5.2;5.4.2 Cluster Points;199
11.1.1.6;5.5 Closure of Sets;201
11.1.1.7;5.6 Compactness;203
11.2;Chapter 6. Sequences of Real Numbers;208
11.2.1;6.1 Sequences Defined;208
11.2.1.1;6.1.1 Monotone Sequences;209
11.2.1.2;6.1.2 Bounded Sequences;210
11.2.2;6.2 Convergence of Sequences;213
11.2.2.1;6.2.1 Convergence to a Real Number;213
11.2.2.2;6.2.2 Convergence to Infinity;219
11.2.3;6.3 The Nested Interval Property;220
11.2.3.1;6.3.1 From LUB Axiom to NIP;221
11.2.3.2;6.3.2 The NIP Applied to Subsequences;222
11.2.3.3;6.3.3 From NIP to LUB Axiom;224
11.2.4;6.4 Cauchy Sequences;225
11.2.4.1;6.4.1 Convergence of Cauchy Sequences;226
11.2.4.2;6.4.2 From Completeness to the NIP;228
11.3;Chapter 7. Functions of a Real Variable;230
11.3.1;7.1 Bounded and Monotone Functions;230
11.3.1.1;7.1.1 Bounded Functions;230
11.3.1.2;7.1.2 Monotone Functions;231
11.3.2;7.2 Limits and Their Basic Properties;233
11.3.2.1;7.2.1 Definition of Limit;233
11.3.2.2;7.2.2 Basic Theorems of Limits;236
11.3.3;7.3 More on Limits;240
11.3.3.1;7.3.1 One-Sided Limits;240
11.3.3.2;7.3.2 Sequential Limits;241
11.3.4;7.4 Limits Involving Infinity;242
11.3.4.1;7.4.1 Limits at Infinity;243
11.3.4.2;7.4.2 Limits of Infinity;245
11.3.5;7.5 Continuity;247
11.3.5.1;7.5.1 Continuity at a Point;247
11.3.5.2;7.5.2 Continuity on a Set;251
11.3.5.3;7.5.3 One-Sided Continuity;253
11.3.6;7.6 Implications of Continuity;254
11.3.6.1;7.6.1 The Intermediate Value Theorem;254
11.3.6.2;7.6.2 Continuity and Open Sets;256
11.3.7;7.7 Uniform Continuity;258
11.3.7.1;7.7.1 Definition and Examples;259
11.3.7.2;7.7.2 Uniform Continuity and Compact Sets;262
12;Part III: Basic Principles of Algebra;264
12.1;Chapter 8. Groups;266
12.1.1;8.1 Introduction to Groups;266
12.1.1.1;8.1.1 Basic Characteristics of Algebraic Structures;266
12.1.1.2;8.1.2 Groups Defined;269
12.1.2;8.1.1 Basic Characteristics of Algebraic Structures;266
12.1.3;8.1.2 Groups Defined;269
12.1.4;8.2 Subgroups;275
12.1.4.1;8.2.1 Subgroups Defined;275
12.1.4.2;8.2.2 Generated Subgroups;277
12.1.4.3;8.2.3 Cyclic Subgroups;278
12.1.5;8.3 Quotient Groups;283
12.1.5.1;8.3.1 Integers Modulo n;283
12.1.5.2;8.3.2 Quotient Groups;286
12.1.5.3;8.3.3 Cosets and Lagrange’s Theorem;290
12.1.6;8.4 Permutation Groups;291
12.1.6.1;8.4.1 Permutation Groups Defined;291
12.1.6.2;8.4.2 The Symmetric Group;292
12.1.6.3;8.4.3 The Alternating Group;294
12.1.6.4;8.4.4 The Dihedral Group;296
12.1.7;8.5 Normal Subgroups;298
12.1.8;8.6 Group Morphisms;303
12.2;Chapter 9. Rings;310
12.2.1;9.1 Rings and Fields;310
12.2.1.1;9.1.1 Rings Defined;310
12.2.1.2;9.1.2 Fields Defined;315
12.2.2;9.2 Subrings;316
12.2.3;9.3 Ring Properties;319
12.2.4;9.4 Ring Extensions;324
12.2.4.1;9.4.1 Adjoining Roots of Ring Elements;324
12.2.4.2;9.4.2 Polynomial Rings;327
12.2.4.3;9.4.3 Degree of a Polynomial;328
12.2.5;9.5 Ideals;329
12.2.6;9.6 Generated Ideals;332
12.2.7;9.7 Prime and Maximal Ideals;335
12.2.8;9.8 Integral Domains;337
12.2.9;9.9 Unique Factorization Domains;342
12.2.10;9.10 Principal Ideal Domains;344
12.2.11;9.11 Euclidean Domains;348
12.2.12;9.12 Polynomials over a Field;351
12.2.13;9.13 Polynomials over the Integers;355
12.2.14;9.14 Ring Morphisms;357
12.2.14.1;9.14.1 Properties of Ring Morphisms;359
12.2.15;9.15 Quotient Rings;362
13;Index;368