Reading, Writing, and Proving

1. Front Matter

   Pages i-xvi

   PDF

2. The How, When, and Why of Mathematics

   Pages 1-15

3. Logically Speaking

   Pages 17-29

4. Introducing the Contrapositive and Converse

   Pages 31-38

5. Set Notation and Quantifiers

   Pages 39-52

6. Proof Techniques

   Pages 53-62

7. Sets

   Pages 63-77

8. Operations on Sets

   Pages 79-87

9. More on Operations on Sets

   Pages 89-96

10. The Power Set and the Cartesian Product

    Pages 97-108

11. Relations

    Pages 109-117

12. Partitions

    Pages 119-127

13. Order in the Reals

    Pages 129-145

14. Functions, Domain, and Range

    Pages 147-161

15. Functions, One-to-One, and Onto

    Pages 163-173

16. Inverses

    Pages 175-189

17. Images and Inverse Images

    Pages 191-205

18. Mathematical Induction

    Pages 207-221

19. Sequences

    Pages 223-235

20. Convergence of Sequences of Real Numbers

    Pages 237-250

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.

• Algebra
• Arithmetic
• Cantor
• Cauchy–Bunyakovsky–Schwarz inequality
• Countable set
• Finite
• Problem solving
• Problem-solving
• calculus
• function
• mathematical induction
• mathematics
• proof
• theorem

From the reviews:

U. Daepp and P. Gorkin

Reading, Writing, and Proving

A Closer Look at Mathematics

"Aids students in their transition from calculus (or precalculus) to higher-level mathematics . . . The authors have included a wide variety of examples, exercises with solutions, problems, and over 40 illustrations."

—L'ENSEIGNEMENT MATHEMATIQUE

"Daepp and Gorkin (both, Bucknell Univ.) offer another in the growing genre of books designed to teach mathematics students the rigor required to write valid proofs … . The book is well written and should be easy for a first- or second- year college mathematics student to read. There are many ‘tips’ offered throughout, along with many examples and exercises … . A book worthy of serious consideration for courses whose goal is to prepare students for upper-division mathematics courses. Summing Up: Highly recommended." (J.R. Burke, CHOICE, 2003)

"The book Reading, Writing, and Proving … provides a fresh, interesting, and readable approach to the often-dreaded ‘Introduction to Proof’ class. … RWP contains more than enough material for a one-semester course … . I was charmed by this book and found it quite enticing. … My students found the overall style, the abundance of solved exercises, and the wealth of additional historical information and advice in the book exceptionally useful. … well-conceived, solidly executed, and very useful textbook." (Maria G. Fung, MAA online, December, 2004)

"The book is intended for undergraduate students beginning their mathematical career or attending their first course in calculus. … Throughout the book … students are encouraged to 1) learn to understand the problem, 2) devise a plan to solve the problem, 3) carry out that plan, and 4) look back and check what the results told them. This concept is very valuable. … The book is written in an informal way, which will pleasethe beginner and not offend the more experienced reader." (EMS Newsletter, December, 2005)

• Department of Mathematics, Bucknell University, Lewisburg, USA

  Ulrich Daepp, Pamela Gorkin

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