Preface
YOu are probably about to teach or take a "first course in proof
techniques," or maybe you just want to learn more about
mathemat. No matter What the reason. 1 1. 1' 1 l-1
ics. No matter What the reason. a student Who wishes to learn the
) 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
J. At this point, students have an intuitive sense of Why things
-. -. -.,
are true, but not the exposure to the detailed and critical thinking
) out not the exposure to the detailed and critical thinking
necessarV to survive in the mathematical world. WC have written
J
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
J nave for the subject, or even create
some along the way, our book has done What it was intended to do.
R6ading. This book was written for a course we teach to first and
second Vear college students. The stVle is informal. A few problems
J o die is informal. A few problems
.' 1 1'. 1
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
.
XI
Xll Preface
Since we encourage the students to draw pictures, we include many
illustrations 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. Students are expected to solve the exercises
before moving on to the problems. Complete solutions to almost all
of the exercises are provided at the end of each chapter. Problems of
varying degrees of difficulty appear at the end of each chapter. Some
problems 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 problems 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 with "tips" on writing,
ranging from the (what should be) obvious to the insider's preference
("Don't start a sentence with a symbol.").
Proving. How can someone learn to prove mathematical results?
There are many theories on this. We believe that learning mathe-
matics is the same as learning to play an instrument or learning
to succeed at a particular sport. Someone must provide the back-
ground: the tips, information on the basic skills, and the insider's
"know how." Then the student has to practice. Musicians and athletes
practice hours a day, and it's not surprising that most mathemati-
cians do, too. We will provide students with the background; the
exercises and problems are there for practice. The instructor ob-
serves, guides, teaches and, if need be, corrects. As with anything
else, the more a student practices, the better she or he will become
at solving problems.
Using this book. What should be in a book like this one? Even a
quick glance at other texts on this subject will tell you that everyone
agrees on certain topics: logic, quantifiers, basic set theoretic con-
cepts, mathematical induction, and the definition and properties of
functions. The depth of coverage is open to debate, of course. We try
to cover logic and quantifiers fairly quickly, because we believe that
~ ~ ~
Preface Xlll
students can only fully appreciate the fundamentals of mathematics
when they are applied to interesting problems.
What is also apparent is that after these essential concepts, ev-
eryone disagrees on what should be included. Even we prefer to vary
our approach depending on our students. We have tried to provide
enough material for a flexible approach.
~ The Minimal Approach. Ifyou need only the basics, cover Chapters
1-17. (If you assume the well ordering principle, or decide to
accept the principle of mathematical induction without proof,
you can also omit Chapter 12.)
~ The Usual Approach. This approach includes Chapters 1-17 and
Chapters 20-22. (This is easily doable in a standard semester, if
the class meets three hours per week.)
~ The Algebra Approach. For an algebraic slant to the course, cover
Chapters 1-17 and Chapters 25 and 26.
~ The Analysis Approach. For a slant towards analysis, cover Chap-
ters 1-22. (This is what we usually cover in our course.) Include as
much material from Chapters 23 and 24 as time allows. Students
usually enjoy an introduction to metric spaces.
~ Projects. We have included projects intended to let students
demonstrate what they can do when they are on their own. We in-
dicate prerequisites for each project, and have tried to vary them
enough that they can be assigned throughout the semester. The
results in these projects come from different areas that we find
particularly interesting. Students can be guided to a project at
their level. Since there are open-ended parts in each project, stu-
dents can take these projects as far as they want to. We usually
encourage the students to work on these in groups.
~ Notation. A word about some of our symbols is in order here. In an
attempt to make this book user-friendly, we indicate the end of a
proof with the well-known symbol I. The end of an example or
exercise is designated by O. If a problem is used later in the text,
we designate it by Problem~. We also have a fair number of "non-
proofs." These are proofs that are questionable, and students are
asked to find the error. We conclude such proofs with the symbol
~. Every other symbol will be defined when we introduce you to
xiv Preface
it. Definitions are incorporated in the text for ease of reading and
the terms defined are given in bold-face type.
Presenting. We also hope that students will make the transition
to thinking of themselves as members of a mathematical commu-
nity. We encourage the students we have in this class to attend talks,
give talks, go to conferences, read mathematical books, watch math-
ematical 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 start-
ing 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.
We hope that through reading, writing, proving, and present-
ing mathematics, we can produce students who will make good
colleagues in every sense of the word.
Acknowledgments. Writing a book is a long process, and we wish
to express our gratitude to those who have helped us along the way.
We are, of course, grateful to the students at Bucknell University
who suffered through the early versions of the manuscript, as well
as those who used later versions. Their comments, suggestions, and
detection of errors are most appreciated. We thank Andrew Shaffer
for help with the illustrations. We also wish to express our thanks to
our colleagues and friends, Gregory Adams, Thomas Cassidy, David
Farmer, and Paul McGuire for helpful conversations. We are partic-
ularly grateful to Raymond Mortini for his willingness to carefully
read (and criticize) the entire text. The book is surely better for it. We
also wish to thank our (former) student editor, Brad Parker. We sim-
ply cannot overstate the value of Brad's careful reading, insightful
comments, and his suggestions for better prose. We thank Univer-
sit,it Bern, Switzerland, for support provided during our sabbaticals.
Finally, we thank Hannes and Madeleine Daepp for putting up with
infinitely many dinner conversations about this text.
Preface XV
We plan to maintain a website with additional material, correc-
tions, and other documentation at
http://www, fac staff, b u ckn ell. edu/u da e pp / re adwrite prove /
Ulrich Daepp and Pamela Gorkin
Lewisburg, Pennsylvania 2003
udaepp~bucknell, edu
pgorkin~bucknell, edu
