Principles of Mathematics E-Book

Vladimir Lepetic

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 656

Veröffentlichungsjahr: 2015

Beschreibung

Presents a uniquely balanced approach that bridges introductory and advanced topics in modern mathematics

An accessible treatment of the fundamentals of modern mathematics, Principles of Mathematics: A Primer provides a unique approach to introductory andadvanced mathematical topics. The book features six main subjects, whichcan be studied independently or in conjunction with each other including: settheory; mathematical logic; proof theory; group theory; theory of functions; andlinear algebra.

The author begins with comprehensive coverage of the necessary building blocks in mathematics and emphasizes the need to think abstractly and develop an appreciation for mathematical thinking. Maintaining a useful balance of introductory coverage and mathematical rigor, Principles of Mathematics: A Primer features:

• Detailed explanations of important theorems and their applications
• Hundreds of completely solved problems throughout each chapter
• Numerous exercises at the end of each chapter to encourage further exploration
• Discussions of interesting and provocative issues that spark readers’ curiosity and facilitate a better understanding and appreciation of the field of mathematics

Principles of Mathematics: A Primer is an ideal textbook for upper-undergraduate courses in the foundations of mathematics and mathematical logic as well as for graduate-level courses related to physics, engineering, and computer science. The book is also a useful reference for readers interested in pursuing careers in mathematics and the sciences.

Leseprobe

Table of Contents

COVER

TITLE PAGE

COPYRIGHT

DEDICATION

PREFACE

CHAPTER 1: SET THEORY

1.1 INTRODUCTION

1.2 SET THEORY – DEFINITIONS, NOTATION, AND TERMINOLOGY – WHAT IS A SET?

1.3 SETS GIVEN BY A DEFINING PROPERTY

1.4 THE ALGEBRA OF SETS

1.5 THE POWER SET

1.6 THE CARTESIAN PRODUCT

1.7 THE SETS

N

,

Z

, AND

Q

1.8 THE SET R – REAL NUMBERS I

1.9 A SHORT MUSING ON TRANSFINITE ARITHMETIC

1.10 THE SET

R

– REAL NUMBERS II

CHAPTER 2: LOGIC

2.1 INTRODUCTION

2.2 PROPOSITIONAL CALCULUS

12

2.3 ARGUMENTS I

PREDICATE CALCULUS

2.4 ARGUMENTS II

2.5 A SHORT REVISIT TO SET THEORY

2.6 BOOLEAN ALGEBRA

2.7 SUPPLEMENTARY PROBLEMS

CHAPTER 3: PROOFS

3.1 INTRODUCTION

3.2 DIRECT PROOF

3.3 INDIRECT PROOF

3.4 MATHEMATICAL INDUCTION

3.5 SUPPLEMENTARY PROBLEMS

CHAPTER 4: FUNCTIONS

4.1 INTRODUCTION

4.2 RELATIONS

4.3 FUNCTIONS

CHAPTER 5: GROUP THEORY

5.1 INTRODUCTION

5.2 FUNDAMENTAL CONCEPTS OF GROUP THEORY

5.3 SUBGROUPS

5.4 CYCLIC GROUPS

5.5 HOMOMORPHISMS AND ISOMORPHISMS

5.6 NORMAL SUBGROUPS

5.7 CENTRALIZER, NORMALIZER, STABILIZER

5.8 QUOTIENT GROUP

5.9 THE ISOMORPHISM THEOREMS

5.10 DIRECT PRODUCT OF GROUPS

CHAPTER 6: LINEAR ALGEBRA

6.1 INTRODUCTION

6.2 VECTOR SPACE

6.3 LINEAR DEPENDENCE AND INDEPENDENCE

6.4 BASIS AND DIMENSION OF A VECTOR SPACE

6.5 SUBSPACES

6.6 LINEAR TRANSFORMATIONS – LINEAR OPERATORS

6.7 ISOMORPHISM OF LINEAR SPACES

6.8 LINEAR TRANSFORMATIONS AND MATRICES

6.9 LINEAR SPACE

M

6.10 MATRIX MULTIPLICATION

6.11 SOME MORE SPECIAL MATRICES. GENERAL LINEAR GROUP

6.12 RANK OF A MATRIX

6.13 DETERMINANTS

6.14 THE INVERSE AND THE RANK OF A MATRIX REVISITED

6.15 MORE ON LINEAR OPERATORS

6.16 SYSTEMS OF LINEAR EQUATIONS I

6.17 SYSTEMS OF LINEAR EQUATIONS II

6.18 THE BASICS OF EIGENVALUE AND EIGENVECTOR THEORY

INDEX

END USER LICENSE AGREEMENT

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Guide

Cover

Table of Contents

Preface

Begin Reading

List of Illustrations

CHAPTER 1: SET THEORY

Figure 1.1 Venn diagram

Figure 1.2 Sets

X, Y,

and

Z

Figure 1.3 Subset

Figure 1.4

Figure 1.5

Figure 1.6 The concept of cardinality for a four-element set A à la Cantor

Figure 1.7

Figure 1.8 intersection

Figure 1.9 .

Figure 1.10 Partition

P

(

A

)

Figure 1.11

Figure 1.12

Figure 1.13

Figure 1.14 Function .

Figure 1.15 Function

Figure 1.16 One-to-one function

Figure 1.17 Surjection

Figure 1.18 Bijection (one-to-one correspondence)

Figure 1.20

Figure 1.21

Figure 1.22

Figure 1.23

Figure 1.24

Figure 1.25

Figure 1.26

Figure 1.27

Figure 1.28

Figure 1.29

Figure 1.30

CHAPTER 2: LOGIC

Figure 2.1 A parsing (ancestral) tree for a formula

CHAPTER 4: FUNCTIONS

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13

Figure 4.14 Partition

Figure 4.15

Figure 4.16

Figure 4.17 R

Figure 4.18 .

Figure 4.19 Function

Figure 4.20 Domain and range of

f

Figure 4.21 The function

f

Figure 4.22 Function

g

Figure 4.23

Figure 4.24

Figure 4.25

Figure 4.26 Image and preimage

Figure 4.27

Figure 4.28 ;

Figure 4.29 Identity function

Figure 4.30 Injection

Figure 4.31 Surjection

Figure 4.32 Bijection

Figure 4.33 Inverse function

Figure 4.34

Figure 4.35

Figure 4.36

Figure 4.37 Composition of functions

Figure 4.38

Figure 4.39

Figure 4.40 Associativity of composition

Figure 4.41

Figure 4.42

Figure 4.43

Figure 4.44

Figure 4.45

CHAPTER 5: GROUP THEORY

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 5.5

Figure 5.6 Subgroup diagram for

Figure 5.7

Figure 5.8

Figure 5.9 Isomorphism

Figure 5.10 .

Figure 5.11 .

Figure 5.12 Normal subgroup

Figure 5.13 Orbit of

in

G

Figure 5.14

CHAPTER 6: LINEAR ALGEBRA

Figure 6.1

Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5

Figure 6.6

Figure 6.7

Figure 6.8

Figure 6.9

Figure 6.10

Figure 6.11

Figure 6.12

Figure 6.13

Figure 6.14

List of Tables

CHAPTER 2: LOGIC

Table 2.1 Truth Table for ∼

p

Table 2.2 Truth Table for

p

∧

q

Table 2.3 Truth Table for

p

∨

q

Table 2.4 Truth Table for

Table 2.5 Truth Table for

p

⊕

q

Table 2.6 Truth Table for

p

→

q

Table 2.7 Truth Table for

p

↔

q

Table 2.8 Tautology and Contradiction

Table 2.9 Truth Table for

Table 2.10 Truth Table for

Table 2.11 Truth Table for ((∼

p

) → (

p

∧

q

)) ∨ (

p

∧ (

q

↔

r

))

Table 2.12 Truth Table for ∼ (

p

↔ (∼(

q

∧ ∼

r

)))

Table 2.13 Truth Table for

Table 2.14

Table 2.15 Table for

Table 2.16 Table for

Table 2.17 Truth Table for

Table 2.18 Table for

Table 2.19 Table for

Table 2.20

Table 2.21

Table 2.22 Modus Ponens

Table 2.23 Modus Tollens

Table 2.24

Table 2.25

PRINCIPLES OF MATHEMATICS

A Primer

VLADIMIR LEPETIC

DePaul University

Copyright © 2016 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

Lepetic, Vladimir, 1950- author.

Principles of mathematics : a primer / Vladimir Lepetic.

pages cm

Includes index.

ISBN 978-1-119-13164-9 (cloth)

1. Mathematics–Philosophy. I. Title.

QA8.4.L447 2016

510.1–dc23

2015025151

To Ivan and Marija

PREFACE

I suspect that everyone working in academia remembers the days when as a student, being unhappy with the assigned textbook, they promised themselves one day to write the “right” book – one that would be easy and fun to read, have all the necessary material for successfully passing those (pesky) exams, but also be serious enough to incite the reader to dig further. It would, one hoped, open new venues to satisfy readers' curiosity provoked by frequent, but causal, remarks on the themes beyond the scope of an introductory text. Thus, the hope continues, students would be seduced by the intimation of mathematical immensity, sense the “flavor” of mathematical thinking, rather than just learning some “tricks of the trade.” Of course, undertaking such an enterprise might easily be considered by many to be too ambitious at best and pure hubris at worst, for many potential readers are likely aware of the plethora of excellent books already available. Well then, why this book and not another one? First, this book is what its title says – a primer. However, unlike many other introductory texts, it is intended for a wide variety of readers; it (or parts of it) can be used as a starter for college undergraduate courses fulfilling “general education” math requirements, but also as an overture to more serious mathematics for students aspiring for careers in math and science. I am sure that rather smart high school students could also use the book to maintain and enhance their enthusiasm for mathematics and science. In any case, I am of the opinion that “introduction” and “rigor” should not exclude each other. Similarly, I don't think that avoiding discussion of weighty issues necessarily makes a text reader-friendly, in particular when mathematics and the sciences are in question. I do think, however, that many profound issues can be introduced accessibly to a beginner and, most importantly, in a way that provokes intellectual curiosity and consequently leads to a better appreciation of the field in question. Another thing, in my opinion equally significant, that any introductory mathematics text should convey is the importance of recognizing the difference and mutual interconnectedness in “knowing,” “understanding,” and “explaining” (i.e., “being able to explain”) something. Admittedly any book, even the most advanced one, is an introduction in such an endeavor; yet one has to start somewhere so why not with a “primer” like this. So, with such philosophy in mind – which by the way can also serve as an apologia, albeit not a very transparent one – all efforts have been made to meticulously follow well-established mathematical formalism and routines. Incidentally, contrary to some educators, I don't think that the standard mathematical routine “definition–lemma–theorem–proof–corollary” is necessarily a deterrent to learning the subject. The only way one can see the “big picture” is to acquire a unique technique that will empower one to do so. The real beauty reveals itself after years of study and practice. If you want to successfully play a musical instrument you need to learn to read the notes, learn some music theory, and then relentlessly practice until you reach a reasonable command of your instrument. After years of doing so, and if you are lucky enough, you become an artist. The payoff, however, is immense.

The book contains six chapters and literally hundreds of solved problems. In addition, readers will find that every chapter ends with a number of supplementary exercises. It is my experience that in a one-semester course with class meeting twice a week, an instructor can leisurely cover two chapters of his/her choice, and they would still have ample time to pick and choose additional topics from other chapters they deem interesting/relevant. The first chapter contains a fairly detailed introduction to Set Theory, and it may be also considered as a conceptual/“philosophical” introduction to everything that follows. Chapters 2 and 3 on Logic and Proofs follow naturally, although I would not have many arguments against those who would prefer to start with Chapters 2 and 3 and subsequently discuss Set Theory. Readers interested only in Functions and/or Group Theory can, after Chapter 1, immediately jump to Chapters 4 and/or 5. Similarly, the last chapter on Linear Algebra can be approached independently provided, of course, the reader has been at least briefly introduced to the necessary preliminaries from Chapters 1, 4, and 5. Finally, the case could be made that a text of this kind, in order to justify an implicitly hinted philosophy, should necessarily have a chapter on Topology and Category Theory. I wholeheartedly agree. However, that would require adding at least another 300 pages to this primer, and the sheer volume of such a book would likely be a deterrent rather than an enticement for a beginner and thus defeat its very purpose. Postponing topology and category theory for some later time hopefully will be just a temporary weakness. It is not unreasonable to expect that after carefully going through Sets and Functions, for instance, the reader will anticipate further subtleties in need for clarification and reach for a book on topology. Similarly, those wondering about a possible theory that would subsume all others might find Category Theory an appropriate venue to reach that goal. In any case, provoking intellectual curiosity and imagination is the main purpose of this text, and the author wishes that the blame for any failure in this endeavor could be put, or at least partially placed, on the shoulders of the course instructor. Alas, the shortcomings are all mine.

I am aware that it is impossible, and no effort would be adequate, to express my gratitude to all of my teachers, colleagues, and students who have over decades influenced my thinking about mathematics. This impossibility notwithstanding, I need to mention Ivan Supek who, at my early age, introduced me to the unique thinking about mathematics, physics, and philosophy, coming directly from Heisenberg, whose assistant and personal friend he had been for years. Vladimir Devidé who discovered for me the world of Gödel and, many years later, my PhD adviser, Louis Kauffman, who put the final touch on those long fermenting ideas.

Lastly, I want to thank Ivan Lepetic, who painstakingly read the whole manuscript, made many corrections and improvements, and drew the illustrations.

VLADIMIR LEPETIC

CHAPTER 1SET THEORY

“The question for the ultimate foundations and the ultimate meaning of mathematics remains open; we do not know in which direction it will find its final solution nor even whether a final objective answer can be expected at all.

“Mathematizing” may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalization.”

1

H. Weyl1

1.1 INTRODUCTION

The fact that you chose to read this book makes it likely that you might have heard of Kurt Gödel,2 the greatest logician since Aristotle,3 whose arguably revolutionary discoveries influenced our views on mathematics, physics, and philosophy, comparable only to the discoveries of quantum mechanics. Well, even if you have not heard of him I want to start by rephrasing his famous theorem:

Mathematics is inexhaustible!

Notwithstanding the lack of a definition of what mathematics is, that still sounds wonderful, doesn't it? At this point, you may not fully understand the meaning of this “theorem” or appreciate its significance for mathematics and philosophy. You may even disagree with it, but I suppose you would agree with me that mathematics is the study of abstract structures with enormous applications to the “real world.” Also, wouldn't you agree that the most impressive features of mathematics are its certainty, abstractness, and precision? That has always been the case, and mathematics continues to be a vibrant, constantly growing, and definitely different discipline from what it used to be. I hope you would also agree (at least after reading this book) that it possesses a unique beauty and elegance recognized from ancient times, and yet revealing its beauty more and more with/to every new generation of mathematicians. Where does it come from? Even if you accept the premise that it is a construct of our mind, you need to wonder how come it represents/reflects reality so faithfully, and in such a precise and elegant way. How come its formalism matches our intuition so neatly? Is that why we “trust” mathematics (not mathematicians) more than any other science; indeed, very often we define truth as a “mathematical truth” without asking for experimental verification of its claims? So, it is definitely reasonable to ask at the very beginning of our journey (and we will ask this question frequently as we go along): Does the world of mathematics exist outside of, and independently of, the physical world and the actions of the human mind? Gödel thought so. In any case, keep this question in mind as you go along – it has been in the minds of mathematicians and philosophers for centuries.