A Transition to Abstract Mathematics

by Maddox

• ISBN13: 9780123744807
• ISBN10: 0123744806
• Edition: 2nd
• Format: Hardcover
• Copyright: 2008-09-04
• Publisher: Elsevier Science

Summary

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

Table of Contents

Why Read This Book?	p. xiii
Preface	p. xv
Preface to the First Edition	p. xvii
Acknowledgments	p. xxi
Notation and Assumptions	p. 1
Set Terminology and Notation	p. 1
Assumptions about the Real Numbers	p. 3
Basic Algebraic Properties	p. 3
Ordering Properties	p. 5
Other Assumptions	p. 7
Foundations of Logic and Proof Writing	p. 9
Language and Mathematics	p. 11
Introduction to Logic	p. 11
Statements	p. 11
Negation of a Statement	p. 13
Combining Statements with AND	p. 13
Combining Statements with OR	p. 14
Logical Equivalence	p. 16
Tautologies and Contradictions	p. 18
If-Then Statements	p. 18
If-Then Statements Defined	p. 18
Variations on p to q	p. 21
Logical Equivalence and Tautologies	p. 23
Universal and Existential Quantifiers	p. 27
The Universal Quantifier	p. 28
The Existential Quantifier	p. 29
Unique Existence	p. 32
Negations of Statements	p. 33
Negations of AND and OR Statements	p. 33
Negations of If-Then Statements	p. 34
Negations of Statements with the Universal Quantifier	p. 36
Negations of Statements with the Existential Quantifer	p. 37
How We Write Proofs	p. 40
Direct Proof	p. 40
Proof by Contrapositive	p. 41
Proving a Logically Equivalent Statement	p. 41
Proof by Contradiction	p. 42
Disproving a Statement	p. 42
Properties of Real Numbers	p. 45
Basic Algebraic Properties of Real Numbers	p. 45
Properties of Addition	p. 46
Properties of Multiplication	p. 49
Ordering Properties of the Real Numbers	p. 51
Absolute Value	p. 53
The Division Algorithm	p. 56
Divisibility and Prime Numbers	p. 59
Sets and Their Properties	p. 63
Set Terminology	p. 63
Proving Basic Set Properties	p. 67
Families of Sets	p. 71
The Principle of Mathematical Induction	p. 78
Variations of the PMI	p. 85
Equivalence Relations	p. 91
Equivalence Relations	p. 91
Equivalence Classes and Partitions	p. 97
Building the Rational Numbers	p. 102
Defining Rational Equality	p. 103
Rational Addition and Multiplication	p. 104
Roots of Real Numbers	p. 106
Irrational Numbers	p. 107
Relations in General	p. 111
Functions	p. 119
Definition and Examples	p. 119
One-to-one and Onto Functions	p. 125
Image and Pre-Image Sets	p. 128
Composition and Inverse Functions	p. 131
Composition of Functions	p. 132
Inverse Functions	p. 133
Three Helpful Theorems	p. 135
Finite Sets	p. 137
Infinite Sets	p. 139
Cartesian Products and Cardinality	p. 144
Cartesian Products	p. 144
Functions Between Finite Sets	p. 146
Applications	p. 148
Combinations and Partitions	p. 151
Combinations	p. 151
Partitioning a Set	p. 152
Applications	p. 153
The Binomial Theorem	p. 157
Basic Principles of Analysis	p. 163
The Real Numbers	p. 165
The Least Upper Bound Axiom	p. 165
Least Upper Bounds	p. 166
Greatest Lower Bounds	p. 168
The Archimedean Property	p. 169
Maximum and Minimum of Finite Sets	p. 170
Open and Closed Sets	p. 172
Interior, Exterior, Boundary, and Cluster Points	p. 175
Interior, Exterior, and Boundary	p. 175
Cluster Points	p. 176
Closure of Sets	p. 178
Compactness	p. 180
Sequences of Real Numbers	p. 185
Sequences Defined	p. 185
Monotone Sequences	p. 186
Bounded Sequences	p. 187
Convergence of Sequences	p. 190
Convergence to a Real Number	p. 190
Convergence to Infinity	p. 196
The Nested Interval Property	p. 197
From LUB Axiom to NIP	p. 198
The NIP Applied to Subsequences	p. 199
From NIP to LUB Axiom	p. 201
Cauchy Sequences	p. 202
Convergence of Cauchy Sequences	p. 203
From Completeness to the NIP	p. 205
Functions of a Real Variable	p. 207
Bounded and Monotone Functions	p. 207
Bounded Functions	p. 207
Monotone Functions	p. 208
Limits and Their Basic Properties	p. 210
Definition of Limit	p. 210
Basic Theorems of Limits	p. 213
More on Limits	p. 217
One-Sided Limits	p. 217
Sequential Limits	p. 218
Limits Involving Infinity	p. 219
Limits at Infinity	p. 220
Limits of Infinity	p. 222
Continuity	p. 224
Continuity at a Point	p. 224
Continuity on a Set	p. 228
One-Sided Continuity	p. 230
Implications of Continuity	p. 231
The Intermediate Value Theorem	p. 231
Continuity and Open Sets	p. 233
Uniform Continuity	p. 235
Definition and Examples	p. 236
Uniform Continuity and Compact Sets	p. 239
Basic Principles of Algebra	p. 241
Groups	p. 243
Introduction to Groups	p. 243
Basic Characteristics of Algebraic Structures	p. 243
Groups Defined	p. 246
Subgroups	p. 252
Subgroups Defined	p. 252
Generated Subgroups	p. 254
Cyclic Subgroups	p. 255
Quotient Groups	p. 260
Integers Modulo n	p. 260
Quotient Groups	p. 263
Cosets and Lagrange's Theorem	p. 267
Permutation Groups	p. 268
Permutation Groups Defined	p. 268
The Symmetric Group	p. 269
The Alternating Group	p. 271
The Dihedral Group	p. 273
Normal Subgroups	p. 275
Group Morphisms	p. 280
Rings	p. 287
Rings and Fields	p. 287
Rings Defined	p. 287
Fields Defined	p. 292
Subrings	p. 293
Ring Properties	p. 296
Ring Extensions	p. 301
Adjoining Roots of Ring Elements	p. 301
Polynomial Rings	p. 304
Degree of a Polynomial	p. 305
Ideals	p. 306
Generated Ideals	p. 309
Prime and Maximal Ideals	p. 312
Integral Domains	p. 314
Unique Factorization Domains	p. 319
Principal Ideal Domains	p. 321
Euclidean Domains	p. 325
Polynomials over a Field	p. 328
Polynomials over the Integers	p. 332
Ring Morphisms	p. 334
Properties of Ring Morphisms	p. 336
Quotient Rings	p. 339
Index	p. 345
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