Preface ix of"nonproofs."These are "proofs"with errors,gaps,or both;the students are asked to find the flaw and to fix it.We conclude such "proofs"with the symbol Every other symbol will be defined when we introduce you to it.Definitions are incorporated in the text for ease of reading and the terms defined are given in boldface type. Presenting.We also hope that students will make the transition to thinking of themselves as members of a mathematical community.We encourage the students we have in this class to attend talks,give talks,go to conferences,read mathemat- ical books,watch mathematical movies,read journal articles,and talk with their colleagues about the things in this course that interest them.Our (incomplete,but lengthy)list of references should serve a student well as a starting point.Each of the projects works well as the basis of a talk for students,and we have included some background material in each section.We begin the chapter on projects with some tips on speaking about mathematics. What's new in this edition.We have made many changes to the first edition.First, all exercises now have solutions and every chapter,except for the first,has at least twenty problems of varying difficulty.As a result,the text has now roughly twice as many problems than before.As in the first edition,definitions are incorporated in the text.In this edition,all definitions newly introduced in a chapter appear again in a section with formal statements of the new definitions.We have included a detailed description of definitions by recursion and a recursion theorem.We've added axioms of set theory to the appendix.We have included new projects:one on the axiom of choice and one on complex numbers.We have added some interesting pieces to two projects,Picture Proofs and The Best Number of All(and Some Other Pretty Good Ones). Some chapters have been changed or added.The first edition's Chapter 12,which required more of students than previous chapters,has been broken into two chapters, now enumerated Chapters 12 and 13.If the instructor wishes,it is possible to simply assume the results in Chapter 13 and omit the chapter.We have also included a new chapter,Chapter 24,on the Cantor-Schroder-Bernstein theorem.We feel that this is the proper culmination to Chapters 21-23 and a wonderful way to end the course, but be forewarned that it is not an easy chapter. Thanks to many of you who used the text,we were able to pinpoint areas where we could improve many of our explanations,provide more motivation,or present a different perspective.Our goal was to find simpler,more precise explanations,and we hope that we have been successful.One new feature of this text that may interest instructors of the course:We have written solutions to every third problem.These are available on our website (see below). Of course,we have updated our reference list,made corrections to errors that appeared in the first version,and,most likely,introduced new errors in the second version.We hope you will send us corrections to errors that you find in the text,as well as any suggestions you have for improvement. We hope that through reading,writing,proving,and presenting mathematics,we can produce students who will make good colleagues in every sense of the word.Preface ix of “nonproofs.” These are “proofs” with errors, gaps, or both; the students are asked to find the flaw and to fix it. We conclude such “proofs” with the symbol ut? . Every other symbol will be defined when we introduce you to it. Definitions are incorporated in the text for ease of reading and the terms defined are given in boldface type. Presenting. We also hope that students will make the transition to thinking of themselves as members of a mathematical community. We encourage the students we have in this class to attend talks, give talks, go to conferences, read mathemat￾ical books, watch mathematical movies, read journal articles, and talk with their colleagues about the things in this course that interest them. Our (incomplete, but lengthy) list of references should serve a student well as a starting point. Each of the projects works well as the basis of a talk for students, and we have included some background material in each section. We begin the chapter on projects with some tips on speaking about mathematics. What’s new in this edition. We have made many changes to the first edition. First, all exercises now have solutions and every chapter, except for the first, has at least twenty problems of varying difficulty. As a result, the text has now roughly twice as many problems than before. As in the first edition, definitions are incorporated in the text. In this edition, all definitions newly introduced in a chapter appear again in a section with formal statements of the new definitions. We have included a detailed description of definitions by recursion and a recursion theorem. We’ve added axioms of set theory to the appendix. We have included new projects: one on the axiom of choice and one on complex numbers. We have added some interesting pieces to two projects, Picture Proofs and The Best Number of All (and Some Other Pretty Good Ones). Some chapters have been changed or added. The first edition’s Chapter 12, which required more of students than previous chapters, has been broken into two chapters, now enumerated Chapters 12 and 13. If the instructor wishes, it is possible to simply assume the results in Chapter 13 and omit the chapter. We have also included a new chapter, Chapter 24, on the Cantor–Schroder–Bernstein theorem. We feel that this ¨ is the proper culmination to Chapters 21–23 and a wonderful way to end the course, but be forewarned that it is not an easy chapter. Thanks to many of you who used the text, we were able to pinpoint areas where we could improve many of our explanations, provide more motivation, or present a different perspective. Our goal was to find simpler, more precise explanations, and we hope that we have been successful. One new feature of this text that may interest instructors of the course: We have written solutions to every third problem. These are available on our website (see below). Of course, we have updated our reference list, made corrections to errors that appeared in the first version, and, most likely, introduced new errors in the second version. We hope you will send us corrections to errors that you find in the text, as well as any suggestions you have for improvement. We hope that through reading, writing, proving, and presenting mathematics, we can produce students who will make good colleagues in every sense of the word