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Bridge to Abstract Mathematics AMS / MAA TEXTBOOKS VOL 56 Bridge to Abstract Mathematics Ralph W. Oberste-Vorth, Aristides Mouzakitis, and Bonita A. Lawrence 10.1090/text/056 ⃝1 MAA TEXTBOOKS Bridge to Abstract Mathematics, Ralph W. Oberste-Vorth,Aristides Mouzakitis, and Bonita A. Lawrence Calculus Deconstructed: A Second Course in First-Year Calculus, Zbigniew H. Nitecki Combinatorics: A Guided Tour, David R. Mazur Combinatorics: A ProblemOriented Approach, Daniel A. Marcus Complex Numbers and Geometry, Liang-shin Hahn A Course in Mathematical Modeling, Douglas Mooney and Randall Swift CryptologicalMathematics, Robert Edward Lewand Differential Geometryand its Applications, John Oprea Elementary Cryptanalysis, Abraham Sinkov ElementaryMathematical Models, Dan Kalman An Episodic History of Mathematics: Mathematical Culture Through Problem Solving, Steven G. Krantz Essentials of Mathematics, Margie Hale Field Theoryand its Classical Problems, Charles Hadlock Fourier Series, Rajendra Bhatia Game Theoryand Strategy, Philip D. Straffin GeometryRevisited, H. S. M. Coxeter and S. L. Greitzer GraphTheory: A Problem Oriented Approach, Daniel Marcus Knot Theory, Charles Livingston Lie Groups:A Problem-OrientedIntroduction via Matrix Groups, Harriet Pollatsek Mathematical Connections: A Companion for Teachers and Others, Al Cuoco Mathematical Interest Theory, Second Edition, Leslie Jane Federer Vaaler and James W. Daniel Mathematical Modeling in the Environment, Charles Hadlock Mathematics for Business Decisions Part I: Probabilityand Simulation (electronic textbook), Richard B. Thompson and Christopher G. Lamoureux Mathematics for Business Decisions Part 2: Calculus and Optimization (electronictextbook), Richard B. Thompson and Christopher G. Lamoureux Mathematics for SecondarySchool Teachers, Elizabeth G. Bremigan, Ralph J. Bremigan, and John D. Lorch The Mathematics of Choice, Ivan Niven The Mathematics of Games and Gambling, Edward Packel Math Throughthe Ages, William Berlinghoffand FernandoGouvea Noncommutative Rings, I. N. Herstein Non-Euclidean Geometry, H. S. M. Coxeter Number TheoryThrough Inquiry, David C. Marshall, Edward Odell, and Michael Starbird A Primer of Real Functions, Ralph P. Boas A Radical Approachto Lebesgue's Theoryof Integration, David M. Bressoud A Radical Approachto Real Analysis, 2nd edition, David M. Bressoud Real InfiniteSeries, Daniel D. Bonar and Michael Khoury, Jr. Topology Now!, Robert Messer and Philip Straffin Understanding our Quantitative World, Janet Andersen and Todd Swanson Contents Some Notes on Notation xi To the Students xiii To Those Beginning the Journey into Proof Writing . xiii How to Use This Text xiv Do the Exercises! . xiv Acknowledgments xv For the Professors xvii To Those Leading the Development of Proof Writing for Students in a Broad Range of Disciplines . xvii I THE AXIOMATIC METHOD 1 1 Introduction 3 1.1 The History of Numbers 3 1.2 The Algebra of Numbers 4 1.3 The Axiomatic Method . 6 1.4 ParallelMathematical Universes 8 2 Statements in Mathematics 9 2.1 Mathematical Statements 9 2.2 Mathematical Connectives 11 2.3 Symbolic Logic . 16 2.4 Compound Statements in English 20 2.5 Predicates and Quantifiers 21 2.6 Supplemental Exercises . 26 3 Proofs in Mathematics 29 3.1 What is Mathematics? ........... 29 3.2 Direct Proof . ........... 30 3.3 Contraposition and Proof by Contradiction . 33 3.4 Proof by Induction ....... 37 3.5 Proof by Complete Induction . 42 3.6 Examples and Counterexamples 46 3.7 Supplemental Exercises . 48 How to THINK about mathematics: A Summary 52 vii viii Contents How to COMMUNICATE mathematics: A Summary. 52 How to DO mathematics: A Summary. 52 II SET THEORY 53 4 Basic Set Operations 55 4.1 Introduction... 55 4.2 Subsets . 56 4.3 Intersections and Unions 58 4.4 Intersections and Unions of ArbitraryCollections 61 4.5 Differences and Complements 64 4.6 Power Sets . 65 4.7 Russell's Paradox.... 66 4.8 Supplemental Exercises. 68 5 Functions 75 5.1 Functions as Rules .............. 75 5.2 CartesianProducts, Relations, and Functions 76 5.3 Injective, Surjective, and Bijective Functions 82 5.4 Compositions of Functions . 83 5.5 Inverse Functions and Inverse Images of Functions 85 5.6 Another Approach to Compositions 87 5.7 Supplemental Exercises. 89 6 Relations on a Set 93 6.1 Properties of Relations 93 6.2 Order Relations . 94 6.3 Equivalence Relations 98 6.4 Supplemental Exercises. 103 7 Cardinality 107 7.1 Cardinalityof Sets: Introduction . 107 7 .2 Finite Sets. 108 7.3 InfiniteSets . 110 7.4 Countable Sets . 113 7.5 Uncountable Sets . 114 7 .6 Supplemental Exercises. 118 ill NUMBER SYSTEMS 121 8 Algebra of Number Systems 123 8.1 Introduction: A Road Map ....... 123 8.2 PrimaryProperties of Number Systems . 123 8.3 SecondaryProperties . 126 8.4 Isomorphisms and Embeddings. 128 8.5 Archimedean Ordered Fields . 129 Contents ix 8.6 Supplemental Exercises . 132 9 The Natural Numbers 137 9 .1 Introduction . 137 9.2 Zero, the Natural Numbers, and Addition . 138 9.3 Multiplication ............... 141 9 .4 Supplemental Exercises . 143 Summary of the Properties of the Nonnegative Integers 144 10 The Integers 145 10.1 Introduction: Integers as Equivalence Classes . 145 10.2 A Total Ordering of the Integers . 146 10.3 Addition of Integers . 147 10.4 Multiplication of Integers . 149 10.5 Embedding the Natural Numbers in the Integers . 151 10.6 Supplemental Exercises . 152 Summary of the Properties of the Integers 153 11 The Rational Numbers 155 11.1 Introduction: Rationals as Equivalence Classes . 155 11.2 A Total Ordering of the Rationals . 156 11.3 Addition of Rationals . 157 11.4 Multiplication of Rationals . 158 11.5 An Ordered Field Containing the Integers . 159 11.6 Supplemental Exercises . 161 Summary of the Properties of the Rationals 164 12 The Real Numbers 165 12.1 Dedekind Cuts . 165 12.2 Order and Addition of Real Numbers . 166 12.3 Multiplication of Real Numbers . 168 12.4 Embedding the Rationals in the Reals . 169 12.5 Uniqueness of the Set of Real Numbers . 169 12.6 Supplemental Exercises . 172 13 Cantor's Reals 173 13.1 Convergence of Sequences of Rational Numbers . 173 13.2 Cauchy Sequences of Rational Numbers . 175 13.3 Cantor's Set of Real Numbers . 177 13.4 The Isomorphism from Cantor's to Dedekind's Reals . 178 13.5 Supplemental Exercises ................ 180 14 The Complex Numbers 181 14.1 Introduction .... 181 14.2 Algebra of Complex Numbers . 181 14.3 Order on the Complex Field . 183 14.4 Embedding the Reals in the Complex Numbers . 184 X Contents 14.5 Supplemental Exercises ............................185 IV TThfE SCALES 187 15 Time Scales 189 15.1 Introduction ... 189 15.2 PreliminaryResults .......... 189 15.3 The Time Scale and its Jump Operators . 191 15.4 Limits and Continuity .. 193 15.5 Supplemental Exercises . 197 16 The Delta Derivative 199 16.1 Delta Differentiation ....... 199 16.2 Higher Order Delta Differentiation . 203 16.3 Properties of the Delta Derivative . 204 16.4 Supplemental Exercises ...... 208 V HINTS 209 17 Hints for (and Comments on) the Exercises 211 Hints for Chapter 2 . 211 Hints for Chapter 3 . 211 Hints for Chapter 4 . 212 Hints for Chapter 5 . 213 Hints for Chapter 6 . 214 Hints for Chapter 7 . 214 Hints for Chapter 8 . 215 Hints for Chapter 9 . 216 Hints for Chapter10 . 216 Hints for Chapter 11 . 217 Hints for Chapter12 . 218 Hints for Chapter 13 . 219 Hints for Chapter 14 . 220 Hints for Chapter 15 . 220 Hints for Chapter 16 . 220 Bibliography 223 Index 225 About the Authors 231 Some Notes on Notation We use this notation forcommonly used sets of numbers: N is the set of natural numbers { 1, 2, 3, 4, 5, ...} Zn is the set of the firstn integers { 1, 2, 3, ..., n} Z is the set of integers Z+ is the set of nonnegative integers { 0, 1, 2, 3, 4, ...} Z * is the set of nonzero integers Q is the set of rational numbers Q + is the set of nonnegative rational numbers Q � is the set of positive rational numbers JR is the set of real numbers JR+ is the set of nonnegative real numbers JR� is the set of positive real numbers C is the set of complex numbers (a,b) is the open interval{x E JR I a< x < b} [a, b] is the closed interval{x E JR I a ::::: x ::::: b} (a,b] is the interval{x E JR la< x::::: b} [a,b) is the interval{x E JR la::::: x < b} Most authors use Zn = { 0, 1, 2, ..., n - 1 }. The choice here is more natural in our discussion of cardinalityin Chapter 7. We use this notation forsubsets and proper subsets: AcB A is a subset of B (A may equal B) A�B A is a proper subset of B (A -:/:- B) We never use A � B. Many authors use A � B forsubsets and A C B forproper subsets. When you read "A C B" somewhere, be careful to determine its meaning. We use this notation forfunctions: / : X ➔ Y f is a function fromdomain X to codomain Y / I s the restriction of f to a subset S of its domain xi xii Some Notes on Notation The symbol □ will denote the end of a proof; it may be read as "Q.E.D." abbreviating the Latin quod eratdemonstrandum, which translates the Greek of Euclid of Alexandria 07tEp EOEL OEL�O(L. To the Students 'Mathematicians,' [Uncle Petros] continued, 'find the same enjoyment in their studies that chess players findin chess. In fact,the psychological make-up of the truemathematician is closer to that of the poet or the musical composer, in other words of someone concernedwith the creation of Beauty and the search forHarmony and Perfection. Heis the polar opposite of the practical man, the engineer, the politician or the ... ' - he paused for a moment seeking something even more abhorred in his scale of values - ' ... indeed, the businessman.' - Apostolos Doxiadis [3] To Those Beginning the Journey into Proof Writing As undergraduates in the 1970s, we, the authors of this text, learnedhow to construct proofs essentially by osmosis.
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