Bond and Keane explicate the elements of logical, mathematical argument to elucidate the meaning and importance of mathematical rigor. With definitions of concepts at their disposal, students learn the rules of logical inference, read and understand proofs of theorems, and write their own proofs--all while becoming familiar with the grammar of mathematics and its style. In addition, they will develop an appreciation of the different methods of proof (contradiction, induction), the value of a proof, and the beauty of an elegant argument. The authors emphasize that mathematics is an ongoing, vibrant discipline--its long, fascinating history continually intersects with territory still uncharted and questions still in need of answers. The authors' extensive background in teaching mathematics shines through in this balanced, explicit, and engaging text, designed as a primer for higher-level mathematics courses. They elegantly demonstrate process and application and recognize the byproducts of both the achievements and the missteps of past thinkers. Chapters 1-5 introduce the fundamentals of abstract mathematics and chapters 6-8 apply the ideas and techniques, placing the earlier material in a real context. Readers' interest is continually piqued by the use of clear explanations, practical examples, discussion and discovery exercises, and historical comments. Title Page 4 Contents 8 Preface for the Instructor 10 Introduction for the Student 16 Chapter 1: Mathematical Reasoning 22 Introduction: Early Mathematics 22 1.1 Statements 23 The Notion of Proof 23 Statements 24 Quantifiers 26 Negations 28 Writing Proofs 32 Exercises 1.1 33 Discussion and Discovery Exercises 35 1.2 Compound Statements 37 Conjunctions and Disjunctions 37 Truth Tables 38 Negating Conjunctions and Disjunctions 39 Logically Equivalent Statements 40 Tautologies and Contradictions 45 Exercises 1.2 47 Discussion and Discovery Exercises 48 1.3 Implications 50 Implication: Definition and Examples 50 Truth Table for an Implication 51 Proving Statements Containing Implications 52 Negating an Implication: Counterexamples 54 Necessary and Sufficient Conditions 55 Exercises 1.3 56 Discussion and Discovery Exercises 58 1.4 Contrapositive and Converse 59 Contrapositive 59 Converse 60 Biconditional 61 Proof by Contradiction 62 Exercises 1.4 65 Discussion and Discovery Exercises 67 Chapter 2: Sets 70 2.1 Sets and Subsets 70 The Notion of a Set 70 Subsets 73 Complements 76 Exercises 2.1 78 Discussion and Discovery Exercises 81 2.2 Combining Sets 82 Unions and Intersections 82 DeMorgan's Laws 86 Cartesian Products 87 Exercises 2.2 89 Discussion and Discovery Exercises 92 2.3 Collections of Sets 93 Power Set 93 Indexing Sets 93 Partitions 95 The Pigeonhole Principle 96 Exercises 2.3 99 Discussion and Discovery Exercises 101 Chapter 3: Functions 102 3.1 Definition and Basic Properties 102 Image of a Function 104 Inverse Image 111 Exercises 3.1 114 Discussion and Discovery Exercises 117 3.2 Surjective and Injective Functions 118 Surjective Functions 118 Injective Functions 121 Bijective Functions 123 Exercises 3.2 126 Discussion and Discovery Exercises 130 3.3 Composition and Invertible Functions 131 Composition of Functions 131 Inverse Functions 135 Exercises 3.3 139 Discussion and Discovery Exercises 142 Chapter 4: Binary Operations and Relations 144 4.1 Binary Operations 144 Associative and Commutative Laws 146 Identities 149 Inverses 151 Closure 153 Exercises 4.1 155 Discussion and Discovery Exercises 159 4.2 Equivalence Relations 160 Relations 161 Properties of Relations 161 Equivalence Relations 162 Equivalence Classes 164 Partial and Linear Orderings 166 Exercises 4.2 168 Discussion and Discovery Exercises 171 Chapter 5: The Integers 172 5.1 Axioms and Basic Properties 172 The Axioms of the Integers 172 Inequalities 175 The Well-Ordering Principle 176 Exercises 5.1 178 Discussion and Discovery Exercises 179 5.2 Induction 180 Induction: A Method of Proof 180 Some Other Forms of Induction 182 The Binomial Theorem 186 Exercises 5.2 190 Discussion and Discovery Exercises 194 5.3 The Division Algorithm and Greatest Common Divisors 196 Divisors and Greatest Common Divisors 196 Euclidean Algorithm 198 Relatively Prime Integers 199 Exercises 5.3 201 Discussion and Discovery Exercises 202 5.4 Primes and Unique Factorization 203 Prime Numbers 203 Unique Factorization 204 Exercises 5.4 207 Discussion and Discovery Exercises 209 5.5 Congruences 210 Congruences and Their Properties 210 The Set of Congruence Classes 213 Exercises 5.5 218 Discussion and Discovery Exercises 220 5.6 Generalizing a Theorem 221 Exercises 5.6 227 Chapter 6: Infinite Sets 230 Introduction 230 6.1 Countable Sets 231 Numerically Equivalent Sets 231 Countable Sets 232 Unions of Countable Sets 235 The Rationals Are Countable 236 Cartesian Products of Countable Sets 237 Exercises 6.1 239 Discussion and Discovery Exercises 240 6.2 Uncountable Sets, Cantor's Theorem, and the Schroeder-Bernstein Theorem 241 Uncountable Sets 241 Cantor's Theorem 242 The Continuum Hypothesis 244 The Schroeder-Bernstein Theorem 244 Exercises 6.2 248 Discussion and Discovery Exercises 250 6.3 Collections of Sets 250 Russell's Paradox 250 Countable Unions of Countable Sets 251 Exercises 6.3 254 Discussion and Discovery Exercises 255 Chapter 7: The Real and Complex Numbers 256 7.1 Fields 256 Elementary Properties of Fields 258 Ordered Fields 258 Finite Fields 260 Exercises 7.1 262 Discussion and Discovery Exercises 264 7.2 The Real Numbers 264 Bounded Sets 264 Least Upper and Greatest Lower Bounds 265 The Archimedean Principle 266 Incompleteness of Q 267 Exercises 7.2 270 Discussion and Discovery Exercises 271 7.3 The Complex Numbers 272 Conjugation and Absolute Value 273 Solutions of Equations 274 Polar Form 275 Complex Roots 277 Exercises 7.3 280 Discussion and Discovery Exercises 281 Chapter 8: Polynomials 284 8.1 Polynomials 284 The Algebra of Polynomials 284 The Division Algorithm 287 Zeros of Polynomials 288 Exercises 8.1 292 Discussion and Discovery Exercises 294 8.2 Unique Factorization 294 Irreducible Polynomials 294 Greatest Common Divisors 296 The Unique Factorization Theorem 299 Exercises 8.2 304 Discussion and Discovery Exercises 305 8.3 Polynomials over C, R, and Q 306 The Fundamental Theorem of Algebra 306 Real Polynomials 307 Polynomials over Q 309 Exercises 8.3 313 Discussion and Discovery Exercises 314 Answers and Hints to Selected Exercises 316 Bibliography 338 Index 340 Mathematical Reasoning 1 1.1 Statements 2 1.2 Compound Statements 16 1.3 Implications 29 1.4 Contrapositive and Converse 38 Sets 49 2.1 Sets and Subsets 49 2.2 Combining Sets 61 2.3 Collections of Sets 72 Functions 81 3.1 Definition and Basic Properties 81 3.2 Surjective and Injective Functions 97 3.3 Composition and Invertible Functions 110 Binary Operations and Relations 123 4.1 Binary Operations 123 4.2 Equivalence Relations 139 The Integers 151 5.1 Axioms and Basic Properties 151 5.2 Induction 159 5.3 The Division Algorithm and Greatest Common Divisors 175 5.4 Primes and Unique Factorization 182 5.5 Congruences 189 5.6 Generalizing a Theorem 200 Infinite Sets 209 6.1 Countable Sets 210 6.2 Uncountable Sets, Cantor's Theorem, and the Schroeder-Bernstein Theorem 220 6.3 Collections of Sets 229 The Real and Complex Numbers 235 7.1 Fields 235 7.2 The Real Numbers 243 7.3 The Complex Numbers 251 Polynomials 263 8.1 Polynomials 263 8.2 Unique Factorization 273 8.3 Polynomials over C, R, and Q 285 Answers and Hints to Selected Exercises 295 Bibliography 317 Index 319
the Goal Of This Book Is To Show Students How Mathematicians Think And To Glimpse Some Of The Fascinating Things They Think About. Bond And Keane Develop Students' Ability To Do Abstract Mathematics By Teaching The Form Of Mathematics In The Context Of Real And Elementary Mathematics. Students Learn The Fundamentals Of Mathematical Logic; How To Read And Understand Definitions, Theorems, And Proofs; And How To Assimilate Abstract Ideas And Communicate Them In Written Form. Students Will Learn To Write Mathematical Proofs Coherently And Correctly.
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a Text For Sophomore Mathematics Majors With Previous Courses In Calculus And Linear Algebra. Part I Introduces Fundamentals Of Abstract Mathematics, And Part Ii Applies Ideas And Techniques Of Material In Part I. Material Begins With Mathematical Reasoning And Sets, And Moves Through Infinite Sets, Real And Complex Numbers, And Polynomials. Contains A Wealth Of Examples And Exercises, Including Discussion And Discovery Exercises, Plus Readings On Historical Aspects And Mathematical Perspectives. Annotation C. By Book News, Inc., Portland, Or.
