Proofs and Fundamentals: A First Course in Abstract Mathematics

444 Pages·2000·14.62 MB·English

by  Ethan D. Bloch

Ethan 0. Bloch
Proofs and Fundamentals
A
First Course in Abstract Mathematics
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Ethan D. Bloch
Depaament of Mathematics
and Computer Science ~ ~ 9 . 85547
Bard Colleee ~mu~rnR*RY
Library of Congress Cataloglngia-Publl~atloD~a ta
Bloch, Elhan D., 1956-
Pmofs and fundamentals: a fist caune in abslract mathematics1 Elhan D. Bloch.
p. cm.
Includes bibliographical references and indexes.
ISBN 0-8176-4111-4 (alk. paper) -ISBN 3-7643-4111-4 (alk. paper)
1. Prmf theory. 2. Set theory. I. Title.
QA9.54.BS7 2CCQ
511.3-dc21 00-023309
CIP
Math Subject Classifications 2000: Primary WA35; Secondary W-01
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Contents
Introduction ix
To the Student xiii
To the Instructor xix
Part1 PROOFS 1
1 Informal Logic 3
1.1 Introduction . . . . . . . . . . . -.C . . . . . . . . . . . . . 3
1.2 Statements. . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Relations Behveen Statements . . . . . . . . . . . . . . . 18
1.4 Valid Arguments . . . . . . . . . . . . . . . . . . . . . . 31
1.5 Quantifiers. . . . . . . . . . . . . . . . . . . . . . . . . . 41
2 Strategies for Proofs 55
-
2.1 Mathematical Proofs What They Are and
Why We Need Them . . . . . . . . . . . . . . . . . . . . 55
2.2 Direct Proofs . . . . . . . . . . . . . . . . . . . . . . . . 63
. . . . . . . .
2.3 Proofs by Contrapositive and Contradiction 67
2.4 Cases, and If and Only If . . . . . . . . . . . . . . . . . . 74
vi Contents
. . . . . . . . . . . . . . . . . .
2.5 Quantifiers in Theorems 81
. . . . . . . . . . . . . . . . . . . .
2.6 Writing Mathematics 93
Part I1 FUNDAMENTALS 105
3 Sets 107
. . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Introduction 107
. . . . . . . . . . . . . . . . . .
3.2 Sets -Basic Definitions 109
. . . . . . . . . . . . . . . . . . . . . . . .
3.3 Set Operations 119
. . . . . . . . . . . . . . . . . .
3.4 Indexed Families of Sets 129
4 Functions 135
. . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Functions 135
. . . . . . . . . . . . . . . . . . .
4.2 Image and Inverse Image 145
. . . . . . . . . . . . .
4.3 Composition and Inverse Functions 152
. . . . . . . . . . .
4.4 Injectivity. Suj ectivity and Bijectivity 161
. . . . . . . . . . . . . . . . . . . . . .
4.5 Sets of Functions 170
5 . Relations 177
. . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Relations 177
. . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Congruence 184
. . . . . . . . . . . . . . . . . . . .
5.3 Equivalence Relations 191
6 Infinite and Finite Sets 203
. . . . . . . . . . . . . . . . . . . . .
6.1 Cardinality of Sets 203
-..
. . . . . . . . . . . . .
6.2 Cardinality of the Number Systems 218
. . . . . . . . . . . . . . . . . . .
6.3 Mathematical Induction 226
. . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Recursion 236
Part I11 EXTRAS 249
7 Selected Topics 251
. . . . . . . . . . . . . . . . . . . . . .
7.1 Binary Operations 251
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Groups 258
. . . . . . . . . . . .
7.3 Homomorphisms and Isomorphisms 267
. . . . . . . . . . . . . . . . . . . .
7.4 Partially Ordered Sets 273
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Lattices 284
. . . . . . . . . . . . . . . .
7.6 Counting: Products and Sums 294
. . . . . . . . .
7.7 Counting: Permutations and Combinations 304
8 Number Systems 323
. . . . . . . . . . . . . . . . . . .
8.1 Back to the Beginning 323
. . . . . . . . . . . . . . . . . . . .
8.2 The Natural Numbers 324
. . . . . . . . .
8.3 Further Properties of the Natural Numbers 333
. . . . . . . . . . . . . . . . . . . . . . . . .
8.4 The Integers 338
. . . . . . . . . . . . . . . . . . .
8.5 The Rational Numbers 348
. . . . . . .
8.6 The Real Numbers and the Complex Numbers 352
. . . . . . . . . . . . .
8.7 Appendix: Proof of Theorem 8.2.1 361
9 Explorations 363
. . . . . . . . . . . . . . . . . . . . . . . . .
9.1 Introduction 363
. . . . . . . . . . . . . . . . .
9.2 Greatest Common Divisors 364
. . . . . . . . . . . . . . . . . . . . . .
9.3 Divisibility Tests 366
. . . . . . . . . . . . . . . . . . .
9.4 Real-Valued Functions 367
. . . . . . . . . . . . . . . . . . .
9.5 Iterations of Functions 368
. . . . . . . . . .
9.6 Fibonacci Numbers and Lucas Numbers 369
. . . . . . . . . . . . . . . . . . . . . . . . . .
9.7 Fuzzy Sets 371
Appendix: properties of Numbers 375
Hints for Selected Exercises 379
References 405
Index 413

Introduction
In an effort to make advanced mathematics accessible to a wide variety
of students, and to give even the most mathematically inclined students
a solid basis upon which to build their continuing study of mathematics,
there has been a tendency in recent years to introduce students to the for-
mulation and writing of rigorous mathematical proofs, and to teach topics
such as sets,.functions, relations and countability, in a "transition" course,
rather than in traditional courses such as linear algebra. A transition course
functions as a bridge between computational courses such as Calculus, and
more theoretical courses such as linear algebra and abstract algebra.
This text contains core topics that I believe any transition course should
cover, as well as some optional material intended to give the instructor
some flexibility in designing a course. The presentation is straightforward
and focuses on the essentials, without being too elementary, too exces-
sively pedagogical, and too full to distractions.
Some of features of this text are the following:
(1) Symbolic logic and the use of logical notation are kept to a minimum.
-
We discuss only what is absolutely necessary as is the case in most
advanced mathematics courses that are not focused on logic per se.
(2) We distinguish between truly general techniques (for example, direct
proof and proof by contradiction) and specialized techniques, such as math-
ematical induction, which are particular mathematical tools rather than
general proof techniques.
x Introduction
(3) We avoid an overemphasis on "fun" topics such as number theory, com-
binatorics or computer science related topics, since they are not as central
as a thorough treatment of sets, functions and relations for core mathe-
matics courses such as linear algebra, abstract algebra and real analysis.
Even the hvo sections on combinatorics in Chapter 7 were written with a
focus on reinforcing the use of sets, functions and relations, rather than
emphasizing clever counting arguments,
(4) The material is presented in the way that mathematicians actually use
it rather than in the most axiomatically direct way. For example, a function
is a special type of a relation, and from a strictly axiomatic point of view,
it would make sense to treat relations first, and then develop functions as
a special case of relations. I believe most mathematicians do not think of
functions in this way (except perhaps for some combinatorialists), and we
cover functions before relations, offering clearer treatments of each. topic.
(5) A section devoted to the proper writing of mathematics has been in-
cluded, to help remind students and instructors of the importance of good
writing.
Outline of the text
The hook is divided into three parts: Proofs, Fundamentals and Extras, re-
spectively. At the end of the book is a brief appendix summarizing a few
basic properties of the standard number systems (integers, rationaI num-
bers, real numbers) that we use, a section of hints for selected exercises,
an index and a bibliography. The core material in this text, which should
be included in any course, consists of Parts I and I1 (Chapters 1-6). A one-
semester course can comfortably include all the core material, together
with a small amount of material from Part 111, chosen according to the
taste of the instructor.
Part I, Proofs, consists of Chapters 1-2, covering informal logic and
proof techniques, respectively. These two chapters discuss theohow" of
modern mathematics, namely the methodology of rigorous proofs as is
currently practiced by mathematicians. Chapter 1i s a precursor to rigorous
proofs, and is not about mathematical proofs per se. The exercises in this
chapter are all informal, in contrast to the rest of the book. Chapter 2, while
including some real proofs, also has a good bit of informal discussion.
Part 11, Fundamentals, consists of Chapters 3-6, covering sets, functions,
relations and cardinality. This material is basic to all of modern mathemat-
Introduction xi
ics. In contrast to Part I, this material is written in a more straightforward
definition/theorem/proof style, as is found in most contemporary advanced
mathematics texts.
Part 111, Extras, consists of Chapters 7-9, and has brief treatments of
a variety of topics, including groups, homomorphisms, partially ordered
sets, lattices and combinatorics, has a lengthier axiomatic treatment of
some of the standard number systems, and concludes with topics for
exploration by the reader.
Some instructors might choose to skip Sections 4.5 and 6.4, the former
because it is very abstract, and the latter because it is viewed as not nec-
essary. Though skipping either or both of these hvo sections is certainly
plausible, I would urge instructors not to do so. Section 4.5 is intended to
help students prepare for dealing with sets of linear maps in linear alge-
bra. Section 6.4 is a topic that is often skipped over in the mathematical
education of many undergraduates, and that is unfortunate, since it pre-
vents the all too common (though incorrect) attempt to define sequences
"by induction!'
Acknowledgments
As with many texts in mathematics, this book developed out of lecture
notes, first used at Bard college in the spring of 1997. The first draft of this
text made partial use of class notes taken by Bard students Todd Krause,
Eloise Michael, Jesse Ross in Math 231 in the spring of 1995. Much of the
inspiration (and work) for Sections 8.2-8.4 comes from the 1993 senior
project at Bard of Neal Brofee.
Thanks go to the following individuals for their valuable assistance,
and extremely helpful comments on various drafts: Robert Cutler, Peter
Dolan, Richard Goldstone, Mark Halsey, Leon Harkleroad, Robert Martin,
Robert McGrail and Lauren Rose. Bard students Leah Bielski, AmyCara
Brosnan, Sean Callanan, Emilie Courage, Urska Dolinsek, Lisa Down-
ward, Brian Duran, Jocelyn Fouri, Jane Gilvin, Shankar Gopalakrishnan,
Maren Holmen, Baseeruddin Khan, Emmanuel Kypraios, JuN~SL aSalle,
Dareth McKenna, Daniel Newsome, Luke Nickerson, Brianna Norton,
Sarah Shapiro, Jaren Smith, Matthew Turgeon, D. Zach Watkinson and
Xiaoyu Zhang found many errors in various drafts, and provided useful
suggestions for improvements.
My appreciation goes to Ann Kostant, Executive Editor of Mathemat-
icsJF'hysics at Birkhauser, for her unflagging support and continual good
xii Introduction
advice, for the second time around; thanks also to Elizabeth hew,
Tom Grasso, Amy Hendrickson and Martin Stock, and to the unnamed
reviewers, who read through the manuscript with eagle eyes. Thanks to the
Mathematics Department at the University of Pennsylvania, for hosting me
during a sabbatical when parts of this book were written. The commuta-
tive diagrams in this text were composed using Paul Taylor's commutative
diagrams package.
It is impossible to acknowledge every source for every idea, theorem or
exercise in this text. Most of the results, and many of the exercises, are
standard; some of these I first encountered as a student, others I learned
from a variety of sources. The following are texts that I consulted regu-
larly. Texts that are similar to this one: [Ave90], [FR90], [FP92], [Ger96],
[Mor87]; texts on logic: [Cop68], [KMM80]; texts on set theory: [Dev93],
[Vau95]; texts on combinatorics: [Bog90], [Epp90], [GW94], [Rob84];
texts on abstract algebra: [Dea66], [Fra94], [GG88], [Blo87]; texts on
posets and lattices: [Bir48], [Bog90], [CD73], [LP98]; texts on writing
mathematics: [KLR89].
Like many mathematicians, I am the product of my education. I would
like to express my appreciation for my mathematics professors at Reed
College 1974-78: Burrowes Hunt, John Leadley, Ray Mayer, Rao Potluri,
Joe Roberts and Thomas Weiting. It was they who first instilled in me many
of the ideas and attitudes seen throughout this book. In particular, I have
been decidedly influenced by the lecture notes of John Leadley for Math
113 (The Real Numbers) and Math 331 (Linear Algebra), the former being
a source for parts of Chapter 8, and the latter being the most abstract linear
algebra text I have seen.
Finally, I wish to thank my mother-in-law Edith Messer, for many visits
during which she took our newborn son Gil for hours at a stretch, allowing
me bits of time for writing between diaper changes; and, especially, my
wife Nancy Messer for her support and encouragement during the time
when this book was written.