Introduction to Proof in Abstract Mathematics

411 Pages·2014·13.84 MB·English

by  Andrew Wohlgemuth

Introduction to Proof in
Abstract Mathematics
Andrew Wohlgemuth
University of Maine
Dover Publications, Inc.
Mineola, New York
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Copyright
Copyright © 1990, 2011 by Andrew Wohlgemuth
All rights reserved.
Bibliographical Note
This Dover edition, first published in 2011, is an unabridged republication of the
work originally published in 1990 by Saunders College Publishing, Philadelphia. The
author has provided a new Introduction for this edition.
Library of Congress Cataloging-in-Publication Data
Wohlgemuth, Andrew.
Introduction to proof in abstract mathematics / Andrew Wohlgemuth. — Dover
ed.
p. cm.
Originally published: Philadelphia : Saunders College Pub., c1990.
Includes index.
eISBN-13: 978-0-486-14168-8
1. Proof theory. I. Title.
QA9.54.W64 2011
511.3'6—dc22
20010043415
Manufactured in the United States by Courier Corporation
47854802 2014
www.doverpublications.com
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Introduction to the Dover Edition
The most effective thing that I have been able to do over the years for
students learning to do proofs has been to make things more explicit. The
ultimate end of a process of making things explicit is, of course, a formal
system—which this text contains. Some people have thought that it makes
proof too easy. My own view is that it does not trivialize anything important;
it merely exposes the truly trivial for what it is. It shrinks it to its proper size,
rather than allowing it to be an insurmountable hurdle for the average student.
The system in this text is based on a number of formal inference rules that
model what a mathematician would do naturally to prove certain sorts of
statements. Although the rules resemble those of formal logic, they were
developed solely to help students struggling with proof—without any input
from formal logic. The rules make explicit the logic used implicitly by
mathematicians. After experience is gained, the explicit use of the formal
rules is replaced by implicit reference. Thus, in our bottom-up approach, the
explicit precedes the implicit. The initial, formal step-by-step format (which
allows for the explicit reference to the rules) is replaced by a narrative format
—where only critical things need to be mentioned. Thus the student is led up
to the sort of narrative proofs traditionally found in textbooks. At every stage
in the process, the student is always aware of what is and what is not a proof
—and has specific guidance in the form of a “step discovery procedure” that
leads to a proof outline. The inference rules, and the general method, have
been used in two of my texts (available online) intended for students different
from the intended readers of this text:
(1) Outlining Proofs in Calculus has been used as a supplement in a third-
semester calculus course, to take the mystery out of proofs that a student will
have seen in the calculus sequence.
(2) Deductive Mathematics—An Introduction to Proof and Discovery for
Mathematics Education has been used in courses for elementary education
majors and mathematics specialists.
Andrew Wohlgemuth
August, 2010
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Preface
This text is for a course with the primary purpose of teaching students to do
mathematical proofs. Proof is taught “syntactically”. A student with the
ability to write computer programs can learn to do straightforward proofs
using the method of this text. Our approach leads to proofs of routine
problems and, more importantly, to the identification of exactly what is
needed in proofs requiring creative insights.
The first aim of the text is to convey the idea of proof in such a way that
the student will know what constitutes an acceptable proof. This is
accomplished with the use of very strict inference rules that define the precise
syntax for an argument. A proof is a sequence of steps that follow from
previous steps in ways specifically allowed by the inference rules. In Chapter
1 these rules are introduced as the mathematical material is developed. When
the material gets to the point where strict adherence to the rules makes proofs
long and tedious, certain legitimate shortcuts, or abbreviations, are
introduced. In Chapters 2 and 3 the process of proof abbreviation continues as
the mathematics becomes more complex. The development of the idea of
proof starts with proofs made up of numbered steps with explicit inference
rules and ends with paragraph proofs. An acceptable proof at any stage in this
process is by definition an argument that one could, if necessary, rewrite in
terms of a previous stage, that is, without the conventions and shortcuts.
The second aim of the text is to develop the students’ ability to do proofs.
First, a distinction is drawn between formal mathematical statements and
statements made in ordinary English. The former statements make up what is
called our language. It is these language statements that appear as steps in
proofs and for which precise rules of inference are given. Language
statements are printed in boldface type. Statements in ordinary English are
considered to be in our metalanguage and may contain language statements.
The primary distinction between metalanguage and language is that, in the
former, interpretation of statements (based on context, education, and so
forth) is essential. The workings of language, on the other hand, are designed
to be mechanical and independent of any interpretation. The
language/metalanguage distinction serves to clarify and smooth the transition
from beginning, formal proofs to proofs in narrative style. It begins to atrophy
naturally in Chapter 3. Presently almost all mathematics students will have
had experience in computer programming, in which they have become
comfortable operating on at least two language levels—in, say, the operating
system and a programming language. The language/metalanguage distinction
takes advantage of this.
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Language statements are categorized by their form: for example, “if …,
then …” or “… and …”. For each form, two inference rules are given, one
for proving and one for using statements of that form. The inference rules are
designed to do two jobs at once. First, they form the basis for training in the
logic of the arguments generally used in mathematics. Second, they serve to
guide the development of a proof. Previous steps needed to establish a given
step in a proof are dictated by the inference rule for proving statements
having the form of the given step. Inference rules and theorems are
introduced in a sequence that ensures that early in the course there will be
only one possible logical proof of a theorem. The discovery of this proof is
accomplished by a routine process fitting the typical student’s previous
orientation toward mathematics and thus easing the transition from
computational to deductive mathematics. Henry Kissinger put the matter
simply:
The absence of alternatives clears the mind marvelously.
Theorems in the text are given in metalanguage and contain language
statements. The first task in developing a proof is to decide which language
statements are to be assumed for the sake of proof (the hypotheses) and which
one is to be proved (the conclusion). Although there is no routine procedure
for doing this, students have no trouble with it.
Our approach is a compromise between the formal, which is precise but
unwieldy, and the informal, which may, especially for beginning students, be
ambiguous. A completely formal approach would involve stating theorems in
our language. This approach would necessitate providing too many
definitions and rules of inference before presenting the first theorem for proof
by students. Our approach enables students to build on each proof idea as it is
introduced. Understanding of standard informal mathematical style, which we
have called metalanguage, is conditioned by a gradual transition from a
formal foundation. Our gradual transition contrasts with the traditional
juxtaposition of the formal and informal in which an introductory section
from classical logic is followed abruptly by narrative style involving language
that has not been so precisely defined.
Our syntactical method has worked well in practice. A much greater
percentage of our students can give correct proofs to theorems that are new to
them since the method was adopted. The value of a student’s ability to
function in an ill-defined environment has been replaced by the value of
doing a lot of hard work. Students no longer become lost in an environment
with which they cannot cope or in which their only hope lies in
memorization.
Chapters 1, 2, and 3 cover material generally considered to be core
material. In Chapters 1 and 2, illustrative examples use computational
properties of the real numbers, which are now introduced very early in the
school curriculum. These properties are given in Appendix 1 and can be
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either used implicitly throughout or introduced at some stage deemed
appropriate by the instructor. Chapters 4, 5, and 6 are written in increasingly
informal style. Except for the treatment of free variables in Chapter 4, these
chapters are logically independent and material from them can be used at the
discretion of the instructor. Chapter 4 contains introductory material on
sequences and continuous functions of a real variable. Chapter 5 contains
material on the cardinality of familiar sets, and Chapter 6 is an introduction to
an axiomatically defined algebraic structure in the form of some beginning
group theory. These chapters illustrate how the proof techniques developed in
Chapters 1 through 3 apply to material in abstract algebra and advanced
calculus.
The only way to learn how to do proofs is by proving theorems. The text
proofs provided in Chapters 1, 2, and 3 serve mainly to (1) illustrate the use
of inference rules, (2) demonstrate some basic idea on the nature of proof or
some specific technique, or (3) exemplify the rules of the game for doing the
proofs given as exercises. This is a departure from standard mathematical
exposition in which the student is a spectator to the main development and
many computational examples, and easy results are “left” for exercises. Thus,
in standard exposition, the organization and presentation of definitions and
theorems have the goal of facilitating proofs given in the text or illustrating
mathematical concepts. In our text, the definitions, theorems, and rules of
inference—and the sequence in which they are presented—have the goal of
organizing the theorems to be proved by the student. The text is therefore a
compromise between text-free teaching methods, in which organization sets
up student proofs of theorems but in which there are no illustrative proofs or
proof methods, and standard exposition, in which organization sets up proofs
done by the author and student exercises are secondary.
Andrew Wohlgemuth
Orono, Maine
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Suggestions for Using This Text
Chapters 1 through 3 can be used as a text for a sophomore-level one-
semester course prerequisite for full courses in abstract algebra and advanced
calculus. In our course at Maine, the core consists of students’ proving those
numbered theorems whose proofs are listed as exercises. A sample syllabus is
given on page 347. The entire text can be used for a two-semester course.
“Additional Proof Ideas” at the ends of some sections present proof in various
traditional ways that supplement our basic approach. This material may be
included as taste or emphasis dictates but is not necessary for a basic course.
Problems on this material are identified as “Supplementary Problems” in
pertinent sections. The text’s precise proof syntax, which enables students to
recognize a valid proof, also makes it possible to use undergraduates as
graders (good students who have previously taken the course, for example).
Thus the text’s approach to proof makes possible a teaching environment that
provides quick feedback on many proofs, even in large classes.
Many students arrive in upper-level courses with no clear idea of just what
constitutes a proof. Time is spent dealing not only with new mathematics and
significant problems, but with the idea of proof. Some reviewers have
suggested that our text could be used by students independently to
supplement upper-level texts. Chapter 1 and selected portions of Chapters 2
and 3 could be used to replace the introductory sections on proofs and logic
of advanced texts.
The practice exercises are given as self-tests for understanding of the
inference rules. Answers to practice exercises are given in the text. Solutions
to other problems are given in the Solutions Manual. A few problems,
identified as such, will be very challenging for beginning students. The setup
and initial progress in a proof attempt are possible using our routine
procedure—without hints. Hints are more appropriately given, on an
individual basis, after the student has had time to get stuck. The real joy in
solving mathematical problems comes not from filling in details after being
given a hint, but from thinking of creative steps oneself. Premature or
unneeded hints, infamous for taking the fun out of driving a car, can also take
the fun out of mathematics.
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Acknowledgments
Thanks are due to many people for their contributions to the text. Professor
Chip Snyder patiently listened to ideas in their formative stages. His wisdom
as a teacher and understanding as a mathematician were invaluable. The
reviewers, Professors Charles Biles, Orin Chein, Joel Haack, and Gregory
Passty, are responsible for significant material added to the original
manuscript as well as for stylistic improvements. Professors Robert Franzosa
and Chip Snyder have made helpful suggestions based on their use of the first
three chapters of the text in our course at Maine. Robert Stern, Senior
Mathematics Editor at Saunders, exercised congenial professional control of
the process of text development. Mary Patton, the Project Editor, effectively
blended the needs of a different kind of text with presentable style.
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Contents
Chapter 1 Sets and Rules of Inference
1.1 Definitions
1.2 Proving For All Statements
1.3 Using For All and Or Statements
1.4 Using and Proving Or Statements
1.5 And Statements
1.6 Using Theorems
1.7 Implications
1.8 Proof by Contradiction
1.9 Iff
1.10 There Exists Statements
1.11 Negations
1.12 Index sets
Chapter 2 Functions
2.1 Functions and Sets
2.2 Composition
2.3 One-to-One Functions
2.4 Onto Functions
2.5 Inverses
2.6 Bijections
2.7 Infinite Sets
2.8 Products, Pairs, and Definitions
Chapter 3 Relations, Operations, and the Integers
3.1 Induction
3.2 Equivalence Relations
3.3 Equivalence Classes
3.4 Well-Defined Operations
3.5 Groups and Rings
3.6 Homomorphisms and Closed Subsets of
3.7 Well-Defined Functions
3.8 Ideals of
3.9 Primes
3.10 Partially Ordered Sets
Chapter 4 Proofs in Analysis
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4.1 Sequences
4.2 Functions of a Real Variable
4.3 Continuity
4.4 An Axiom for Sets of Reals
4.5 Some Convergence Conditions for Sequences
4.6 Continuous Functions on Closed Intervals
4.7 Topology of
Chapter 5 Cardinality
5.1 Cantor’s Theorem
5.2 Cardinalities of Sets of Numbers
Chapter 6 Groups
6.1 Subgroups
6.2 Examples
6.3 Subgroups and Cosets
6.4 Normal Subgroups and Factor Groups
6.5 Fundamental Theorems of Group Theory
Appendix 1 Properties of Number Systems
Appendix 2 Truth Tables
Appendix 3 Inference Rules
Appendix 4 Definitions
Appendix 5 Theorems
Appendix 6 A Sample Syllabus
Answers to Practice Exercises
Index
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