Details

An Introduction to Proof through Real Analysis


An Introduction to Proof through Real Analysis


1. Aufl.

von: Daniel J. Madden, Jason A. Aubrey

CHF 93.00

Verlag: Wiley
Format: EPUB
Veröffentl.: 14.08.2017
ISBN/EAN: 9781119314745
Sprache: englisch
Anzahl Seiten: 448

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Beschreibungen

<p><b> An engaging and accessible introduction to mathematical proof incorporating ideas from real analysis</b></p> <p>A mathematical proof is an inferential argument for a mathematical statement. Since the time of the ancient Greek mathematicians, the proof has been a cornerstone of the science of mathematics. The goal of this book is to help students learn to follow and understand the function and structure of mathematical proof and to produce proofs of their own.</p> <p><i>An Introduction to Proof through Real Analysis </i>is based on course material developed and refined over thirty years by Professor Daniel J. Madden and was designed to function as a complete text for both first proofs and first analysis courses. Written in an engaging and accessible narrative style, this book systematically covers the basic techniques of proof writing, beginning with real numbers and progressing to logic, set theory, topology, and continuity. The book proceeds from natural numbers to rational numbers in a familiar way, and justifies the need for a rigorous definition of real numbers. The mathematical climax of the story it tells is the Intermediate Value Theorem, which justifies the notion that the real numbers are sufficient for solving all geometric problems.</p> <p>• Concentrates solely on designing proofs by placing instruction on proof writing on top of discussions of specific mathematical subjects</p> <p>• Departs from traditional guides to proofs by incorporating elements of both real analysis and algebraic representation</p> <p>• Written in an engaging narrative style to tell the story of proof and its meaning, function, and construction</p> <p>• Uses a particular mathematical idea as the focus of each type of proof presented</p> <p>• Developed from material that has been class-tested and fine-tuned over thirty years in university introductory courses</p> <p><i>An Introduction to Proof through Real Analysis </i>is the ideal introductory text to proofs for second and third-year undergraduate mathematics students, especially those who have completed a calculus sequence, students learning real analysis for the first time, and those learning proofs for the first time.</p> <p><b>Daniel J. Madden, PhD, </b>is an Associate Professor of Mathematics at The University of Arizona, Tucson, Arizona, USA. He has taught a junior level course introducing students to the idea of a rigorous proof based on real analysis almost every semester since 1990. Dr. Madden is the winner of the 2015 Southwest Section of the Mathematical Association of America Distinguished Teacher Award.</p> <p><b>Jason A. Aubrey, PhD, </b>is Assistant Professor of Mathematics and Director, Mathematics Center of the University of Arizona. </p>
<p>List of Figures xiii</p> <p>Preface xv</p> <p>Introduction xvii</p> <p><b>Part I A First Pass at Defining </b><b>ℝ 97</b></p> <p><b>1 Beginnings 3</b></p> <p>1.1 A naive approach to the natural numbers 3</p> <p>1.1.1 Preschool: foundations of the natural numbers 3</p> <p>1.1.2 Kindergarten: addition and subtraction 5</p> <p>1.1.3 Grade school: multiplication and division 8</p> <p>1.1.4 Natural numbers: basic properties and theorems 11</p> <p>1.2 First steps in proof 12</p> <p>1.2.1 A direct proof 12</p> <p>1.2.2 Mathematical induction 14</p> <p>1.3 Problems 17</p> <p><b>2 The Algebra of the Natural Numbers 19</b></p> <p>2.1 A more sophisticated look at the basics 19</p> <p>2.1.1 An algebraic approach 21</p> <p>2.2 Mathematical induction 22</p> <p>2.2.1 The theorem of induction 24</p> <p>2.3 Division 27</p> <p>2.3.1 The division algorithm 27</p> <p>2.3.2 Odds and evens 30</p> <p>2.4 Problems 34</p> <p><b>3 Integers 37</b></p> <p>3.1 The algebraic properties of ℕ 37</p> <p>3.1.1 The algebraic definition of the integers 40</p> <p>3.1.2 Simple results about integers 42</p> <p>3.1.3 The relationship between ℕ and ℤ 45</p> <p>3.2 Problems 47</p> <p><b>4 Rational Numbers 49</b></p> <p>4.1 The algebra 49</p> <p>4.1.1 Surveying the algebraic properties of ℤ 49</p> <p>4.1.2 Defining an ordered field 50</p> <p>4.1.3 Properties of ordered fields 51</p> <p>4.2 Fractions versus rational numbers 53</p> <p>4.2.1 In some ways they are different 53</p> <p>4.2.2 In some ways they are the same 56</p> <p>4.3 The rational numbers 58</p> <p>4.3.1 Operations are well defined 58</p> <p>4.3.2 ℚ is an ordered field 63</p> <p>4.4 The rational numbers are not enough 67</p> <p>4.4.1 √2 is irrational 67</p> <p>4.5 Problems 70</p> <p><b>5 Ordered Fields 73</b></p> <p>5.1 Other ordered fields 73</p> <p>5.2 Properties of ordered fields 74</p> <p>5.2.1 The average theorem 74</p> <p>5.2.2 Absolute values 75</p> <p>5.2.3 Picturing number systems 78</p> <p>5.3 Problems 79</p> <p><b>6 The Real Numbers 81</b></p> <p>6.1 Completeness 81</p> <p>6.1.1 Greatest lower bounds 81</p> <p>6.1.2 So what is complete? 82</p> <p>6.1.3 An alternate version of completeness 84</p> <p>6.2 Gaps and caps 86</p> <p>6.2.1 The Archimedean principle 86</p> <p>6.2.2 The density theorem 87</p> <p>6.3 Problems 90</p> <p>6.4 Appendix 93</p> <p><b>Part II Logic, Sets, and Other Basics 97</b></p> <p><b>7 Logic 99</b></p> <p>7.1 Propositional logic 99</p> <p>7.1.1 Logical statements 99</p> <p>7.1.2 Logical connectives 100</p> <p>7.1.3 Logical equivalence 104</p> <p>7.2 Implication 105</p> <p>7.3 Quantifiers 107</p> <p>7.3.1 Specification 108</p> <p>7.3.2 Existence 108</p> <p>7.3.3 Universal 109</p> <p>7.3.4 Multiple quantifiers 110</p> <p>7.4 An application to mathematical definitions 111</p> <p>7.5 Logic versus English 114</p> <p>7.6 Problems 116</p> <p>7.7 Epilogue 118</p> <p><b>8 Advice for Constructing Proofs 121</b></p> <p>8.1 The structure of a proof 121</p> <p>8.1.1 Syllogisms 121</p> <p>8.1.2 The outline of a proof 123</p> <p>8.2 Methods of proof 127</p> <p>8.2.1 Direct methods 127</p> <p>8.2.1.1 Understand how to start 127</p> <p>8.2.1.2 Parsing logical statements 129</p> <p>8.2.1.3 Mathematical statements to be proved 131</p> <p>8.2.1.4 Mathematical statements that are assumed to be true 135</p> <p>8.2.1.5 What do we know and what do we want? 138</p> <p>8.2.1.6 Construction of a direct proof 138</p> <p>8.2.1.7 Compound hypothesis and conclusions 139</p> <p>8.2.2 Alternate methods of proof 139</p> <p>8.2.2.1 Contrapositive 139</p> <p>8.2.2.2 Contradiction 142</p> <p>8.3 An example of a complicated proof 145</p> <p>8.4 Problems 149</p> <p><b>9 Sets 151</b></p> <p>9.1 Defining sets 151</p> <p>9.2 Starting definitions 153</p> <p>9.3 Set operations 154</p> <p>9.3.1 Families of sets 157</p> <p>9.4 Special sets 160</p> <p>9.4.1 The empty set 160</p> <p>9.4.2 Intervals 162</p> <p>9.5 Problems 168</p> <p>9.6 Epilogue 171</p> <p><b>10 Relations 175</b></p> <p>10.1 Ordered pairs 175</p> <p>10.1.1 Relations between and on sets 176</p> <p>10.2 A total order on a set 179</p> <p>10.2.1 Definition 179</p> <p>10.2.2 Definitions that use a total order 179</p> <p>10.3 Equivalence relations 182</p> <p>10.3.1 Definitions 182</p> <p>10.3.2 Equivalence classes 184</p> <p>10.3.3 Equivalence partitions 185</p> <p>10.3.3.1 Well defined 187</p> <p>10.4 Problems 188</p> <p><b>11 Functions 193</b></p> <p>11.1 Definitions 193</p> <p>11.1.1 Preliminary ideas 193</p> <p>11.1.2 The technical definition 194</p> <p>11.1.2.1 A word about notation 197</p> <p>11.2 Visualizing functions 202</p> <p>11.2.1 Graphs in ℝ<sup>2 </sup>202</p> <p>11.2.2 Tables and arrow graphs 202</p> <p>11.2.3 Generic functions 203</p> <p>11.3 Composition 204</p> <p>11.3.1 Definitions and basic results 204</p> <p>11.4 Inverses 206</p> <p>11.5 Problems 210</p> <p><b>12 Images and preimages 215</b></p> <p>12.1 Functions acting on sets 215</p> <p>12.1.1 Definition of image 215</p> <p>12.1.2 Examples 217</p> <p>12.1.3 Definition of preimage 218</p> <p>12.1.4 Examples 220</p> <p>12.2 Theorems about images and preimages 222</p> <p>12.2.1 Basics 222</p> <p>12.2.2 Unions and intersections 228</p> <p>12.3 Problems 231</p> <p><b>13 Final Basic Notions 235</b></p> <p>13.1 Binary operations 235</p> <p>13.2 Finite and infinite sets 236</p> <p>13.2.1 Objectives of this discussion 236</p> <p>13.2.2 Why the fuss? 237</p> <p>13.2.3 Finite sets 239</p> <p>13.2.4 Intuitive properties of infinite sets 240</p> <p>13.2.5 Counting finite sets 241</p> <p>13.2.6 Finite sets in a set with a total order 243</p> <p>13.3 Summary 246</p> <p>13.4 Problems 246</p> <p>13.5 Appendix 248</p> <p>13.6 Epilogue 257</p> <p><b>Part III A Second Pass at Defining </b><b>ℝ 261</b></p> <p><b>14 </b><b>ℕ, </b><b>ℤ, and </b><b>ℚ 263</b></p> <p>14.0.1 Basic properties of the natural numbers 263</p> <p>14.0.2 Theorems about the natural numbers 266</p> <p>14.1 The integers 267</p> <p>14.1.1 An algebraic definition 267</p> <p>14.1.2 Results about the integers 268</p> <p>14.1.3 The relationship between natural numbers and integers 270</p> <p>14.2 The rational numbers 272</p> <p>14.3 Problems 279</p> <p><b>15 Ordered Fields and the Real Numbers 281</b></p> <p>15.1 Ordered fields 281</p> <p>15.2 The real numbers 284</p> <p>15.3 Problems 289</p> <p>15.4 Epilogue 290</p> <p>15.4.1 Constructing the real numbers 290</p> <p><b>16 Topology 293</b></p> <p>16.1 Introduction 293</p> <p>16.1.1 Preliminaries 293</p> <p>16.1.2 Neighborhoods 295</p> <p>16.1.3 Interior, exterior, and boundary 298</p> <p>16.1.4 Isolated points and accumulation points 300</p> <p>16.1.5 The closure 303</p> <p>16.2 Examples 305</p> <p>16.3 Open and closed sets 311</p> <p>16.3.1 Definitions 311</p> <p>16.3.2 Examples 315</p> <p>16.4 Problems 316</p> <p><b>17 Theorems in Topology 319</b></p> <p>17.1 Summary of basic topology 319</p> <p>17.2 New results 321</p> <p>17.2.1 Unions and intersections 321</p> <p>17.2.2 Open intervals are open 325</p> <p>17.2.3 Open subsets are in the interior 327</p> <p>17.2.4 Topology and completeness 328</p> <p>17.3 Accumulation points 329</p> <p>17.3.1 Accumulation points are aptly named 329</p> <p>17.3.2 For all A ⊆ ℝ, <i>A′</i> is closed 333</p> <p>17.4 Problems 341</p> <p><b>18 Compact Sets 345</b></p> <p>18.1 Closed and bounded sets 345</p> <p>18.1.1 Maximums and minimums 345</p> <p>18.2 Closed intervals are special 354</p> <p>18.3 Problems 356</p> <p><b>19 Continuous Functions 359</b></p> <p>19.1 First semester calculus 359</p> <p>19.1.1 An intuitive idea of a continuous function 359</p> <p>19.1.2 The calculus definition of continuity 360</p> <p>19.1.3 The official mathematical definition of continuity 363</p> <p>19.1.4 Examples 364</p> <p>19.2 Theorems about continuity 374</p> <p>19.2.1 Three specific functions 374</p> <p>19.2.2 Multiplying a continuous function by a constant 377</p> <p>19.2.3 Adding continuous functions 378</p> <p>19.2.4 Multiplying continuous functions 379</p> <p>19.2.5 Polynomial functions 382</p> <p>19.2.6 Composition of continuous functions 382</p> <p>19.2.7 Dividing continuous functions 384</p> <p>19.2.8 Gluing functions together 385</p> <p>19.3 Problems 386</p> <p><b>20 Continuity and Topology 389</b></p> <p>20.1 Preliminaries 389</p> <p>20.1.1 Continuous images mess up topology 389</p> <p>20.2 The topological definitions of continuity 391</p> <p>20.3 Compact images 397</p> <p>20.3.1 The main theorem 397</p> <p>20.3.2 The extreme value theorem 400</p> <p>20.3.3 The intermediate value theorem 401</p> <p>20.4 Problems 404</p> <p><b>21 A Few Final Observations 407</b></p> <p>21.1 Inverses of continuous functions 407</p> <p>21.1.1 A strange example 408</p> <p>21.1.2 The theorem about inverses of continuous functions 409</p> <p>21.2 The intermediate value theorem and continuity 412</p> <p>21.3 Continuity at discrete points 413</p> <p>21.4 Conclusion 413</p> <p>Index 415</p>
<p><strong>Daniel J. Madden, PhD,</strong> is an Associate Professor of Mathematics at The University of Arizona, Tucson, Arizona, USA. He has taught a junior level course introducing students to the idea of a rigorous proof based on real analysis almost every semester since 1990. Dr. Madden is the winner of the 2015 Southwest Section of the Mathematical Association of America Distinguished Teacher Award. <p><strong>Jason A. Aubrey, PhD,</strong> is Assistant Professor of Mathematics and Director, Mathematics Center of the University of Arizona.
<p> <strong>An engaging and accessible introduction to mathematical proof incorporating ideas from real analysis</strong> <p>A mathematical proof is an inferential argument for a mathematical statement. Since the time of the ancient Greek mathematicians, the proof has been a cornerstone of the science of mathematics. The goal of this book is to help students learn to follow and understand the function and structure of mathematical proof and to produce proofs of their own. <p><em>An Introduction to Proof through Real Analysis</em> is based on course material developed and refined over thirty years by Professor Daniel J. Madden and was designed to function as a complete text for both first proofs and first analysis courses. Written in an engaging and accessible narrative style, this book systematically covers the basic techniques of proof writing, beginning with real numbers and progressing to logic, set theory, topology, and continuity. The book proceeds from natural numbers to rational numbers in a familiar way, and justifies the need for a rigorous definition of real numbers. The mathematical climax of the story it tells is the Intermediate Value Theorem, which justifies the notion that the real numbers are sufficient for solving all geometric problems. <ul> <li>Concentrates solely on designing proofs by placing instruction on proof writing on top of discussions of specific mathematical subjects</li> <li>Departs from traditional guides to proofs by incorporating elements of both real analysis and algebraic representation</li> <li>Written in an engaging narrative style to tell the story of proof and its meaning, function, and construction</li> <li>Uses a particular mathematical idea as the focus of each type of proof presented</li> <li>Developed from material that has been class-tested and fine-tuned over thirty years in university introductory courses</li> </ul> <br> <p><em>An Introduction to Proof through Real Analysis</em> is the ideal introductory text to proofs for second and third-year undergraduate mathematics students, especially those who have completed a calculus sequence, students learning real analysis for the first time, and those learning proofs for the first time.

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