Transition to Advanced Mathematics PDF
Preview Transition to Advanced Mathematics
Transition to Advanced Mathematics Textbooks in Mathematics Series editors: Al Boggess, Kenneth H. Rosen An Introduction to Analysis, Third Edition James R. Kirkwood Student Solutions Manual for Gallian’s Contemporary Abstract Algebra, Tenth Edition Joseph A. Gallian Elementary Number Theory Gove Effinger, Gary L. Mullen Philosophy of Mathematics Classic and Contemporary Studies Ahmet Cevik An Introduction to Complex Analysis and the Laplace Transform Vladimir Eiderman An Invitation to Abstract Algebra Steven J. Rosenberg Numerical Analysis and Scientific Computation Jeffery J. Leader Introduction to Linear Algebra Computation, Application and Theory Mark J. DeBonis The Elements of Advanced Mathematics, Fifth Edition Steven G. Krantz Differential Equations Theory, Technique, and Practice, Third Edition Steven G. Krantz Real Analysis and Foundations, Fifth Edition Steven G. Krantz Geometry and Its Applications, Third Edition Walter J. Meyer Transition to Advanced Mathematics Danilo R. Diedrichs and Stephen Lovett Modeling Change and Uncertainty Machine Learning and Other Techniques William P. Fox and Robert E. Burks Abstract Algebra A First Course, Second Edition Stephen Lovett Multiplicative Differential Calculus Svetlin Georgiev, Khaled Zennir Applied Differential Equations The Primary Course Vladimir A. Dobrushkin Introduction to Computational Mathematics: An Outline William C. Bauldry Mathematical Modeling the Life Sciences Numerical Recipes in Python and MATLABTM N. G. Cogan https://www.routledge.com/Textbooks-in-Mathematics/book-series/CANDHTEXBOOMTH Transition to Advanced Mathematics Danilo R. Diedrichs Wheaton College, USA Stephen Lovett Wheaton College, USA First edition published 2022 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2022 Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and pub- lisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information stor- age or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copy- right.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Diedrichs, Danilo R., author. | Lovett, Stephen (Stephen T.), author. Title: Transition to advanced mathematics / Danilo R. Diedrichs, Wheaton College, USA, Stephen Lovett, Wheaton College, USA. Description: First edition. | Boca Raton : Chapman & Hall/CRC Press, 2022.| Series: Textbooks in mathematics | Includes bibliographical references and index. Identifiers: LCCN 2021055607 (print) | LCCN 2021055608 (ebook) |ISBN 9780367494445 (hardback) | ISBN 9781032261003 (paperback) |ISBN 9781003046202 (ebook) Subjects: LCSH: Mathematics--Study and teaching. Classification: LCC QA11.2 .D535 2022 (print) | LCC QA11.2 (ebook) | DDC 510--dc23/eng20220301 record available at https://lccn.loc.gov/2021055607 LC ebook record ∆available at https://lccn.loc.gov/2021055608 ISBN: 978-0-367-49444-5 (hbk) ISBN: 978-1-032-26100-3 (pbk) ISBN: 978-1-003-04620-2 (ebk) DOI: 10.1201/9781003046202 Typeset in CMR10 font by KnowledgeWorks Global Ltd. Contents Preface ix I Introduction to Proofs 1 1 Logic and Sets 3 1.1 Logic and Propositions . . . . . . . . . . . . . . . . . . 3 1.2 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Logical Equivalences . . . . . . . . . . . . . . . . . . . 23 1.4 Operations on Sets . . . . . . . . . . . . . . . . . . . . 31 1.5 Predicates and Quantifiers . . . . . . . . . . . . . . . . 37 1.6 Nested Quantifiers . . . . . . . . . . . . . . . . . . . . 45 2 Arguments and Proofs 57 2.1 Constructing Valid Arguments . . . . . . . . . . . . . 57 2.2 First Proof Strategies . . . . . . . . . . . . . . . . . . 72 2.3 Proof Strategies . . . . . . . . . . . . . . . . . . . . . . 83 2.4 Generalized Unions and Intersections . . . . . . . . . . 93 3 Functions 99 3.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.2 Properties of Functions . . . . . . . . . . . . . . . . . 111 3.3 Choice Functions; The Axiom of Choice . . . . . . . . 121 4 Properties of the Integers 125 4.1 A Definition of the Integers . . . . . . . . . . . . . . . 125 4.2 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . 132 4.3 Greatest Common Divisor; Least Common Multiple . 137 4.4 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . 145 4.5 Induction . . . . . . . . . . . . . . . . . . . . . . . . . 152 4.6 Modular Arithmetic . . . . . . . . . . . . . . . . . . . 162 vi Contents 5 Counting and Combinatorial Arguments 171 5.1 Counting Techniques . . . . . . . . . . . . . . . . . . . 171 5.2 Concept of a Combinatorial Proof . . . . . . . . . . . 183 5.3 Pigeonhole Principle . . . . . . . . . . . . . . . . . . . 189 5.4 Countability and Cardinality . . . . . . . . . . . . . . 195 6 Relations 205 6.1 Relations . . . . . . . . . . . . . . . . . . . . . . . . . 205 6.2 Partial Orders. . . . . . . . . . . . . . . . . . . . . . . 213 6.3 Equivalence Relations . . . . . . . . . . . . . . . . . . 224 6.4 Quotient Sets . . . . . . . . . . . . . . . . . . . . . . . 231 II Culture, History, Reading, and Writing 239 7 Mathematical Culture, Vocation, and Careers 241 7.1 21st Century Mathematics . . . . . . . . . . . . . . . 242 7.2 Collaboration, Associations, and Conferences . . . . . 259 7.3 Studying Upper-Level Mathematics . . . . . . . . . . 275 7.4 Mathematical Vocations . . . . . . . . . . . . . . . . . 290 8 History and Philosophy of Mathematics 311 8.1 History of Mathematics Before the Scientific Revolu- tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 8.2 Mathematics and Science . . . . . . . . . . . . . . . . 325 8.3 The Axiomatic Method . . . . . . . . . . . . . . . . . 339 8.4 History of Modern Mathematics . . . . . . . . . . . . 363 8.5 Philosophical Issues in Mathematics . . . . . . . . . . 388 9 Reading and Researching Mathematics 401 9.1 Journals . . . . . . . . . . . . . . . . . . . . . . . . . 402 9.2 Original Research Articles . . . . . . . . . . . . . . . 415 9.3 Reading and Expositing Original Research Articles . 427 9.4 Researching Primary and Secondary Sources . . . . . 435 10 Writing and Presenting Mathematics 445 10.1 Mathematical Writing . . . . . . . . . . . . . . . . . . 446 10.2 Project Reports . . . . . . . . . . . . . . . . . . . . . 464 10.3 Mathematical Typesetting . . . . . . . . . . . . . . . 471 10.4 Advanced Typesetting . . . . . . . . . . . . . . . . . . 483 10.5 Oral Presentations . . . . . . . . . . . . . . . . . . . . 496 Contents vii A Rubric for Assessing Proofs 509 A.1 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 A.2 Understanding / Terminology . . . . . . . . . . . . . . 511 A.3 Creativity . . . . . . . . . . . . . . . . . . . . . . . . . 513 A.4 Communication . . . . . . . . . . . . . . . . . . . . . . 514 B Index of Theorems and Definitions from Calculus and Linear Algebra 517 B.1 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 517 B.2 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . 519 Bibliography 521 Index 527 Preface Purpose of this Book This book prepares students for entrance into the discipline of math- ematics, including its way of thinking and its way of communicating. Every contemporary person with some formal education has en- countered mathematical concepts. Surprisingly, when asked what mathematics is, few can offer a concise definition of mathematics. Some high school graduates might reply that mathematics is the studyofpropertiesofnumbersandofspace,butthisanswerisneither comprehensive nor cohesive. Some people contend that there is no generally accepted definition for mathematics. Some authors define mathematics as the science of patterns of numbers (arithmetic and algebra), of space (geometry), and so on. (See [24].) Though this definition offers a cohesive description of the topics usually accepted under the umbrella of mathematics, what unifies mathematics is the manner of thought. Mathematical thought is precise. Perhaps because of its precision, for millennia mathematics has provided epistemological strength to countless areas such as agricul- ture,architecture,engineering,naturalsciences,socialsciences,foren- sics, and many more. In his famous 1960 article, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Nobel Prize laureate physicist Eugene Wigner dubbed mathematics a “wonderful gift [77].” Despite the abstract nature of mathematics and the man- ner in which much of mathematics develops by working on problems internal to itself, interaction between mathematics and other disci- plines often leads to new areas of investigation within mathematics and offers fruitful results in other sciences. Another aspect about mathematics that surprises some outsiders isthatitremainsaveryactivefieldofinquiry. Mathematicsdoesen- joy a timeless nature. What Euclid or Archimedes rigorously proved 2000 years ago remains valid today. And yet every day students and researchers prove new theorems and push the boundaries of what weknow. ThemathematicsArXiV(https://arxiv.org/archive/math) listsresultsproducedbypeopletoday,rightnow. Anyonecanengage
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