Introduction to Mathematical Structures and Proofs

410 Pages·2012·2.63 MB·English

by  Larry J. Gerstein (auth.)

Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics

Series Editors:

Sheldon Axler
San Francisco State University

Kenneth Ribet
University of California, Berkeley

Advisory Board:

Colin C. Adams, Williams College
Alejandro Adem, University of British Columbia
Ruth Charney, Brandeis University
Irene M. Gamba, The University of Texas at Austin
Roger E. Howe, Yale University
David Jerison, Massachusetts Institute of Technology
Jeffrey C. Lagarias, University of Michigan
Jill Pipher, Brown University
Fadil Santosa, University of Minnesota
Amie Wilkinson, University of Chicago

Undergraduate Texts in Mathematics are generally aimed at third- and fourth-
year undergraduate mathematics students at North American universities. These
texts strive to provide students and teachers with new perspectives and novel
approaches. The books include motivation that guides the reader to an
appreciation of interrelations among different aspects of the subject. They feature
examples that illustrate key concepts as well as exercises that strengthen
understanding.
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Larry J. Gerstein
Introduction to
Mathematical Structures
and Proofs
Second Edition
Larry J. Gerstein
University of California, Santa Barbara
Santa Barbara, CA
USA

ISBN 978-1-4614-4264-6 ISBN 978-1-4614-4265-3 (eBook)
DOI 10.1007/978-1-4614-4265-3
Springer New York Heidelberg Dordrecht London

Library of Congress Control Number: 2012940655

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PREFACE TO THE SECOND
EDITION
For the second edition, apart from minor changes in the first edition material,
the following sections have been added: 5.9, 6.7–6.10, 7.1, and 7.2.
My feeling is that a basic course should include at least the following sec-
tions
Chapter 1: 1.1–1.5
Chapter 2: All, except perhaps 2.2
Chapter 3: quickly
Chapter 4: 4.1–4.3
Chapter 5: 5.2, 5.3, 5.5–5.8
Chapter 6: 6.3–6.6
Then, if there is time for more material, there are several possibilities. For a
taste of theoretical computer science, there is Section 4.5. For more combi-
natorics there is a brief introduction to graph theory in Section 5.9. Further
topics in number theory can be found in Sections 6.7–6.10 and in the intro-
duction to the Gaussian integers in Section 7.2. Section 7.1 introduces the
field C of complex numbers. Students already minimally acquainted with C
and interested in number theory can go directly to Section 7.2 after Section
6.4, though for that it would be useful to first see the definitions of “ring” and
“field” in Section 6.1.
Because students in this course will have seen functions before at some
level, e.g., in an introductory calculus course, probably Chapter 3 can be done
quickly. But students should be comfortable proving that the composition of
v
vi PREFACE TO THE SECOND EDITION
two bijections is a bijection. (For mysterious psychological reasons, proving
surjectivity seems to be especially difficult for students.)
There is no algorithm available that will guarantee a successful transition
to mathematical maturity, or even a clear definition of what is meant by that
maturity. But to attain it—whatever its definition—students need to be chal-
lenged by sophisticated concepts and acquire the patience and fortitude to
overcome the initial confusion that those concepts generate. My experience
has been that working with the power set P(S) of a set S is especially useful
for this purpose, perhaps because the student needs to be comfortable viewing
an element of P(S) as both a thing and as a collection of things. Then, if
students grasp Cantor’s theorem saying that the cardinality of S is less than
that of P(S), and that consequently there are infinitely many different levels
ofinfinity, theycome toappreciatehowthecarefulexpression ofmathematical
ideas can lead us beyond what can be achieved by gut instinct alone. More-
over, working through Cantor’s argument provides an excellent opportunity
to observe and appreciate genius at work.
Number theory offers an abundance of other displays of mathematical ge-
nius. For example, consider Euclid’s startlingly short and simple proof that
there are infinitely many prime numbers. And then there are the dramati-
cally different proofs of that fact by Polya (using Fermat numbers) and Euler
(using infinite series), demonstrating the remarkable diversity of perspectives
from which a mathematical notion can be explored. Finally, number the-
ory provides an abundance of easily stated unsolved problems, some of which
have been with us for centuries, and this helps the students appreciate that
mathematics is a living science, not just a body of facts. This too is part of
mathematical sophistication.
IthankJessicaWirtzforherfineworkpreparinganassortmentofdiagrams.
And I thank Debbie Ceder, a keyboard virtuoso, for her patience and for her
extraordinary efforts dealing with the wide range of complexities that arose
in handling an assortment of files to produce a coherent final document.
Finally, I dedicate this book with love to my wife Susan and my sons David
and Ben. They continue to astonish and inspire me.
Larry J. Gerstein
PREFACE TO THE FIRST EDITION
This is a textbook for a one-term course whose goal is to ease the transition
from lower-division calculus courses to upper-division courses in linear and
abstract algebra, real and complex analysis, number theory, topology, com-
binatorics, and so on. Without such a “bridge” course, most upper-division
instructors feel the need to start their courses with the rudiments of logic,
set theory, equivalence relations, and other basic mathematical raw materials
before getting on with the subject at hand. Students who are new to higher
mathematics are often startled to discover that mathematics is a subject of
ideas, and not just formulaic rituals, and that they are now expected to under-
stand and create mathematical proofs. Mastery of an assortment of technical
tricks may have carried the students through calculus, but it is no longer a
guarantee of academic success.
Students need experience in working with abstract ideas at a nontrivial
level if they are to achieve the sophisticated blend of knowledge, discipline,
and creativity that we call “mathematical maturity.” I don’t believe that
“theorem-proving” can be taught any more than “question-answering” can be
taught. Nevertheless, I have found that it is possible to guide students gen-
tly into the process of mathematical proof in such a way that they become
comfortable with the experience and begin asking themselves questions that
will lead them in the right direction. As with learning to swim or ride a bi-
cycle, there are usually anxieties to be overcome; and, especially in view of
the “cookbook” experience of many calculus courses, it takes a while for stu-
dents to come to believe that they may be capable of solving a problem even
when no instantaneous solution presents itself. But in time students become
vii
viii PREFACE TO THE FIRST EDITION
familiar with the process by which we prove most theorems: a thoughtful os-
cillation between what we know to be true and what we want to show to be
true, until the gap between the two has been closed. Sometimes this involves
a gradual buildup of knowledge, coupled with a breakup of the objective into
more manageable parts. Sometimes computational experiments lead to new
understanding of the structures under investigation. Sometimes sudden in-
sights come after a period of quiet reflection. There is no section in the book
called “How to Prove a Theorem,” because it would be dishonest to pretend
that mechanical rituals can replace creative thinking when we are doing sub-
stantial mathematics. However, after going through the material in this book,
the student who is asked to prove something will not feel like a stranger in a
hostile land.
Part of thetransitionto mathematical maturity involves learning to use the
language of mathematics. Having convinced ourselves that we have solved a
difficult problem, we need to write up the solution in a way that will convince
thepossiblyskepticalreader. Thistaskcanbemadeeasierbythejudicioususe
of the notation and terminology that have been developed for the purpose of
presenting mathematics in a clear and efficient fashion. We will spend a good
deal of time exploring this mode of expression, because mastery of language
is an important step toward the mastery of ideas.
In writing this book, beyond introducing fundamental mathematical struc-
tures and exploring techniques of proof, I have tried to convey some of the
excitement and delightful confusion that a professional mathematician expe-
riences when confronting the unknown. It is important that students develop
an awareness of mathematics as an independent science, and not just as a
collection of tools. Like any science, mathematics is a thriving wonderland of
research, with many mysteries to keep us humble despite the subject’s many
remarkable achievements. The final chapter, on number theory, includes the
statement of several problems whose solutions have so far eluded mathemati-
cians, in some cases for centuries.
Strictly speaking, there are no college-level prerequisites for the material
to be found here; indeed, this book could be used as a source of special topics
for talented high school students. But in fact I have assumed that this is
not the student’s first encounter with college mathematics, and that some
“seasoning” from, say, ayear ofcalculus hasalreadyoccurred (thoughcalculus
is not a prerequisite for anything here). I have also assumed that the student
is prepared to pursue ideas with considerable intensity.
Because this is an introductory text, I have made every effort to give stu-
dents a broad view of the mathematical experience. Accordingly, the book
PREFACE TO THE FIRST EDITION ix
includes a wide-ranging assortment of examples and imagery to motivate the
material and to enhance the underlying intuitions. I have tried to strike a bal-
ance between rigor and informality, not by operating in some middle region
but by using both styles in what I think is a reasonably balanced way. Also,
I have not hesitated to consider a given topic from more than one perspective
or at more than one level of rigor.
Mostexercisesetsincludeatleastsomeroutineexercisesthatcheckwhether
the student has mastered the meaning of terminology and notation. But the
majority oftheexercises aremoresubstantial andwillrequire somecogitation,
experimentation, review of definitions, clarification of goals (“What do I have
to show?” “What am I after?”), and perhaps some struggle. While re-reading
some or all of the section (and especially the definitions) may be helpful in
solving an exercise, it will usually be futile to search for a worked example
in a section that is identical to the exercise except for a trivial change. My
goal throughout has been to encourage the flexible and original thinking that
characterizes creative mathematical activity, not to serve as a drill sergeant.
In some cases a complicating issue will arise in an exercise that will not be
completely resolved until later in the book, though that later material will not
be needed in order to solve the exercise.
Throughout my work on this project I have been sustained by the cheerful
affection I have received from my family. I thank my sons, Ben and David, for
the inspiration I have derived from their creativity; and I thank David again
for his three drawings. Finally, I want to express my unbounded appreciation
to my wife Susan, the Lone Ranger of mathematics copyediting, who has come
to the rescue again and again, her red pencils ablaze in the moonlight. Where
she has found confusion she has brought clarity; where she has found despair
she has brought hope; where she has found sadness she has brought joy.
I dedicate this book to my remarkable family.
Larry J. Gerstein