Preface You are probably about to teach or take a"first course in proof techniques,"or maybe you just want to learn more about mathematics.No matter what the reason,a student who wishes to learn the material in this book likes mathematics and we hope to keep it that way.At this point,students have an intuitive sense of why things are true,but not the exposure to the detailed and critical thinking necessary to survive in the mathematical world.We have written this book to bridge this gap. In our experience,students beginning this course have little training in rigorous mathematical reasoning;they need guidance.At the end,they are where they should be;on their own.Our aim is to teach the students to read,write,and do mathematics independently,and to do it with clarity,precision,and care.If we can maintain the enthusiasm they have for the subject,or even create some along the way,our book has done what it was intended to do. Reading.This book was written for a course we teach to first-and second-year college students.The style is informal.A few problems require calculus,but these are identified as such.Students will also need to participate while reading proofs, prodded by questions (such as,"Why?").Many detailed examples are provided in each chapter.Since we encourage the students to draw pictures,we include many il- lustrations as well.Exercises,designed to teach certain concepts,are also included. These can be used as a basis for class discussion,or preparation for the class.Stu- dents are expected to solve the exercises before moving on to the problems.Com- plete solutions to all of the exercises are provided at the end of each chapter.Prob- lems of varying degrees of difficulty appear at the end of each chapter.Some prob- lems are simply proofs of theorems that students are asked to read and summarize; others supply details to statements in the text.Though many of the remaining prob- lems are standard,we hope that students will solve some of the unique problems presented in each chapter. Writing.The bad news is that it is not easy to write a proof well.The good news is that with proper instruction,students quickly learn the basics of writing.We try to write in a way that we hope is worthy of imitation,but we also provide students viiPreface You are probably about to teach or take a “first course in proof techniques,” or maybe you just want to learn more about mathematics. No matter what the reason, a student who wishes to learn the material in this book likes mathematics and we hope to keep it that way. At this point, students have an intuitive sense of why things are true, but not the exposure to the detailed and critical thinking necessary to survive in the mathematical world. We have written this book to bridge this gap. In our experience, students beginning this course have little training in rigorous mathematical reasoning; they need guidance. At the end, they are where they should be; on their own. Our aim is to teach the students to read, write, and do mathematics independently, and to do it with clarity, precision, and care. If we can maintain the enthusiasm they have for the subject, or even create some along the way, our book has done what it was intended to do. Reading. This book was written for a course we teach to first- and second-year college students. The style is informal. A few problems require calculus, but these are identified as such. Students will also need to participate while reading proofs, prodded by questions (such as, “Why?”). Many detailed examples are provided in each chapter. Since we encourage the students to draw pictures, we include many il￾lustrations as well. Exercises, designed to teach certain concepts, are also included. These can be used as a basis for class discussion, or preparation for the class. Stu￾dents are expected to solve the exercises before moving on to the problems. Com￾plete solutions to all of the exercises are provided at the end of each chapter. Prob￾lems of varying degrees of difficulty appear at the end of each chapter. Some prob￾lems are simply proofs of theorems that students are asked to read and summarize; others supply details to statements in the text. Though many of the remaining prob￾lems are standard, we hope that students will solve some of the unique problems presented in each chapter. Writing. The bad news is that it is not easy to write a proof well. The good news is that with proper instruction, students quickly learn the basics of writing. We try to write in a way that we hope is worthy of imitation, but we also provide students vii