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A Transition to Abstract Mathematics Learning Mathematical Thinking and Writing:

A Transition to Abstract Mathematics, Second Edition: Learning Mathematical Thinking and Writing

Author(s): Randall Maddox
Publisher: Academic Press; 2 edition
Date    : 2008
Pages : 384
Format : PDF
OCR    : Yes
Quality  :

Language : English
ISBN-10  : 0123744806
ISBN-13  :

This second edition assists engineering and physical science students on fundamental proof techniques and learning to think and write mathematics
Product Description
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

Contents
Why Read This Book? xiii
Preface xv
Preface to the First Edition xvii
Acknowledgments xxi
0 Notation and Assumptions 1
0.1 Set Terminology and Notation 1
0.2 Assumptions about the Real Numbers 3
0.2.1 Basic Algebraic Properties 3
0.2.2 Ordering Properties 5
0.2.3 Other Assumptions 7
I Foundations of Logic and Proof Writing 9
1 Language and Mathematics 11
1.1 Introduction to Logic 11
1.1.1 Statements 11
1.1.2 Negation of a Statement 13
1.1.3 Combining Statements withAND 13
1.1.4 Combining Statements with OR 14
1.1.5 Logical Equivalence 16
1.1.6 Tautologies and Contradictions 18
vii
viii Contents
1.2 If-Then Statements 18
1.2.1 If-Then Statements Defined 18
1.2.2 Variations on p → q 21
1.2.3 Logical Equivalence and Tautologies 23
1.3 Universal and Existential Quantifiers 27
1.3.1 The Universal Quantifier 28
1.3.2 The Existential Quantifier 29
1.3.3 Unique Existence 32
1.4 Negations of Statements 33
1.4.1 Negations ofAND and OR Statements 33
1.4.2 Negations of If-Then Statements 34
1.4.3 Negations of Statements with the Universal Quantifier 36
1.4.4 Negations of Statements with the Existential Quantifier 37
1.5 HowWe Write Proofs 40
1.5.1 Direct Proof 40
1.5.2 Proof by Contrapositive 41
1.5.3 Proving a Logically Equivalent Statement 41
1.5.4 Proof by Contradiction 42
1.5.5 Disproving a Statement 42
2 Properties of Real Numbers 45
2.1 Basic Algebraic Properties of Real Numbers 45
2.1.1 Properties of Addition 46
2.1.2 Properties of Multiplication 49
2.2 Ordering Properties of the Real Numbers 51
2.3 Absolute Value 53
2.4 The Division Algorithm 56
2.5 Divisibility and Prime Numbers 59
3 Sets and Their Properties 63
3.1 Set Terminology 63
3.2 Proving Basic Set Properties 67
3.3 Families of Sets 71
3.4 The Principle of Mathematical Induction 78
3.5 Variations of the PMI 85
3.6 Equivalence Relations 91
3.7 Equivalence Classes and Partitions 97
3.8 Building the Rational Numbers 102
3.8.1 Defining Rational Equality 103
3.8.2 Rational Addition and Multiplication 104
3.9 Roots of Real Numbers 106
Contents ix
3.10 Irrational Numbers 107
3.11 Relations in General 111
4 Functions 119
4.1 Definition and Examples 119
4.2 One-to-one and Onto Functions 125
4.3 Image and Pre-Image Sets 128
4.4 Composition and Inverse Functions 131
4.4.1 Composition of Functions 132
4.4.2 Inverse Functions 133
4.5 Three Helpful Theorems 135
4.6 Finite Sets 137
4.7 Infinite Sets 139
4.8 Cartesian Products and Cardinality 144
4.8.1 Cartesian Products 144
4.8.2 Functions Between Finite Sets 146
4.8.3 Applications 148
4.9 Combinations and Partitions 151
4.9.1 Combinations 151
4.9.2 Partitioning a Set 152
4.9.3 Applications 153
4.10 The Binomial Theorem 157
II Basic Principles of Analysis 163
5 The Real Numbers 165
5.1 The Least Upper Bound Axiom 165
5.1.1 Least Upper Bounds 166
5.1.2 Greatest Lower Bounds 168
5.2 The Archimedean Property 169
5.2.1 Maximum and Minimum of Finite Sets 170
5.3 Open and Closed Sets 172
5.4 Interior, Exterior, Boundary, and Cluster Points 175
5.4.1 Interior, Exterior, and Boundary 175
5.4.2 Cluster Points 176
5.5 Closure of Sets 178
5.6 Compactness 180
6 Sequences of Real Numbers 185
6.1 Sequences Defined 185
6.1.1 Monotone Sequences 186
6.1.2 Bounded Sequences 187
x Contents
6.2 Convergence of Sequences 190
6.2.1 Convergence to a Real Number 190
6.2.2 Convergence to Infinity 196
6.3 The Nested Interval Property 197
6.3.1 From LUB Axiom to NIP 198
6.3.2 The NIP Applied to Subsequences 199
6.3.3 From NIP to LUB Axiom 201
6.4 Cauchy Sequences 202
6.4.1 Convergence of Cauchy Sequences 203
6.4.2 From Completeness to the NIP 205
7 Functions of a Real Variable 207
7.1 Bounded and Monotone Functions 207
7.1.1 Bounded Functions 207
7.1.2 Monotone Functions 208
7.2 Limits and Their Basic Properties 210
7.2.1 Definition of Limit 210
7.2.2 Basic Theorems of Limits 213
7.3 More on Limits 217
7.3.1 One-Sided Limits 217
7.3.2 Sequential Limits 218
7.4 Limits Involving Infinity 219
7.4.1 Limits at Infinity 220
7.4.2 Limits of Infinity 222
7.5 Continuity 224
7.5.1 Continuity at a Point 224
7.5.2 Continuity on a Set 228
7.5.3 One-Sided Continuity 230
7.6 Implications of Continuity 231
7.6.1 The Intermediate Value Theorem 231
7.6.2 Continuity and Open Sets 233
7.7 Uniform Continuity 235
7.7.1 Definition and Examples 236
7.7.2 Uniform Continuity and Compact Sets 239
III Basic Principles of Algebra 241
8 Groups 243
8.1 Introduction to Groups 243
8.1.1 Basic Characteristics of Algebraic Structures 243
8.1.2 Groups Defined 246
Contents xi
8.2 Subgroups 252
8.2.1 Subgroups Defined 252
8.2.2 Generated Subgroups 254
8.2.3 Cyclic Subgroups 255
8.3 Quotient Groups 260
8.3.1 Integers Modulo n 260
8.3.2 Quotient Groups 263
8.3.3 Cosets and Lagrange’s Theorem 267
8.4 Permutation Groups 268
8.4.1 Permutation Groups Defined 268
8.4.2 The Symmetric Group 269
8.4.3 The Alternating Group 271
8.4.4 The Dihedral Group 273
8.5 Normal Subgroups 275
8.6 Group Morphisms 280
9 Rings 287
9.1 Rings and Fields 287
9.1.1 Rings Defined 287
9.1.2 Fields Defined 292
9.2 Subrings 293
9.3 Ring Properties 296
9.4 Ring Extensions 301
9.4.1 Adjoining Roots of Ring Elements 301
9.4.2 Polynomial Rings 304
9.4.3 Degree of a Polynomial 305
9.5 Ideals 306
9.6 Generated Ideals 309
9.7 Prime and Maximal Ideals 312
9.8 Integral Domains 314
9.9 Unique Factorization Domains 319
9.10 Principal Ideal Domains 321
9.11 Euclidean Domains 325
9.12 Polynomials over a Field 328
9.13 Polynomials over the Integers 332
9.14 Ring Morphisms 334
9.14.1 Properties of Ring Morphisms 336
9.15 Quotient Rings 339
Index 345

可惜没有附件可以下载啊，这本书正合我的意

A Transition to Abstract Mathematics Second Edition - Learning Mathematical Thinking and Writing