You are probably about to teach or take a “?rst course in proof techniques,� or maybe you just want to learn more about mathem- ics. 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 in- pendently, 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 ?rst and second year college students. The style is informal. A few problems require calculus, but these are identi?ed as such. Students will also needtoparticipatewhilereadingproofs,proddedbyquestions(such as, “Why?�). Many detailed examples are provided in each chapter.