Introduction to proof in abstract mathematics

480 Pages·2011·13.62 MB·English

by  Wohlgemuth

Introduction to Proof in Abstract
Mathematics
Andrew Wohlgemuth
University of Maine
Dover Publications, Inc.
Mineola, New York
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
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
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.
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 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
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.
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.
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
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