A friendly introduction to analysis single and multivariable 2nd ed

A second edition of Advanced Calculus: A Friendly Approach has been renamed to reflect moreaccurately both the content and intent of the book, which follows a thorough introduction to th

A Friendly Introduction

to Analysis

Single and

Multivariable

Upper Saddle River, New Jersey 07458

Library of Congress Cataloging-in-Publications Data

Kosmala, Witold A.J.

A friendly introduction to analysis; single and multivariable.

2nd ed.IWitold A.J Kosmala

p cm.

Includes bibliographical references and index,

ISBN 0-13-045796-5

1 Calculus

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To my wife, Vanessa,

and my children,

Konrad, Doria, Alma, and Henryk

1.6* Finite and Infinite Sets 40

1.7* Ordered Field and a Real Number System 43

I.8* Some Properties of a Real Number 48

1.9* Review 55

1.10* Projects 57

Part 1 Fibonacci Numbers 57

Part 2 Lucas Numbers 61

Part 3 Mean of Real Numbers 61

Part 1 The Transcendental Number e 113

Part 2 Summable Sequences 115

*Optional sections are marked with an asterisk.

vii

xi

Part 1 Monotone Functions 144

Part 2 Continued Fractions 145

Part 1 Compact Sets 179

Part 2 Multiplicative, Subadditive, and Additive Functions 180

5.1 Derivative of a Function 184

5.2 Properties of Differentiable Functions 193

5.3 Mean Value Theorems 201

5.4 Higher Order Derivatives 209

55* L' Hopital's Rules 220

5.6* Review 228

5.7* Projects 232

Part 1 Approximation of Derivatives 232

Part 2 Lipschitz Condition 234

6 Integration Zao6.1 Riemann Integral 241

6.2 Integrable Functions 246

6.3 Properties of the Riemann Integral 250

6.4 Integration in Relation to Differentiation 256

6.5 Improper Integral 265

6.6* Special Functions 274

6.7* Review 283

6.8* Projects 286

Contents ix

Part 1

Part 2

Wallis's Formula 286Euler's Summation Formula 287Part 3

Part 4

Laplace Transforms 289Inverse Laplace Transforms 292

7 Infinite Series 2947.1 Convergence 294

7.2 Tests for Convergence 303

7.3 Ratio and Root Tests 311

7.4 Absolute and Conditional Convergence 317

7.5* Review 325

7.6* Projects 327

Part 1 Summation by Parts 327

Part 2 Multiplication of Series 329

Part 3 Infinite Products 330

Part 4 Cantor Set 332

8 Sequences and Series of Functions 3348.1 Pointwise Convergence 335

8.2 Uniform Convergence 341

8.3 Properties of Uniform Convergence 347

8.4 Pointwise and Uniform Convergence of Series 351

Part 3 An Everywhere Continuous but

Nowhere Differentiable Function 381Part 4 Equicontinuity 381

x Contents

Part 2 Polar Coordinates 426

Part 3 Cantor Function 431

Part 1 Operator Method for Solving Differential Equations

Part 2 Separable and Homogeneous First-Order

11.5 Vector Fields and Work Integrals 500

11.6 Gradient Vector Field 505

11.7 Green's Theorem 511

11.8* Stokes's and Gauss's Theorem 518

11.9* Review 522

11.10* Projects 525

Part 1 Change of Variables for Double Integrals 525

Part 2 Exact Equations 528

12* Fourier Series Not intext, see Instructor's Supplement

Hints and Solutions to Selected Exercises 533

Greek Alphabet 560

Index of Symbols 561

Index - 564

A second edition of Advanced Calculus: A Friendly Approach has been renamed to reflect moreaccurately both the content and intent of the book, which follows a thorough introduction to thetheory of real analysis for single and multivariable functions

Although designed for a two-semester or three-quarter course in the theory of calculus, thefollowing material may be easily subdivided to suit a course of any length entitled Introduction

to Real Analysis, Theoretical Calculus, Honors Calculus, Advanced Calculus, or the lent Intended for undergraduate juniors or seniors or beginning graduate students who havecompleted the calculus sequence, linear algebra, and preferably differential equations, this bookaids in the pursuit of careers in science, statistics, engineering, computer science, and business.Since these areas possess roots in real analysis, a solid background in undergraduate analysis

equiva-is essential and equiva-is the purpose of thequiva-is book Although analysequiva-is equiva-is an important building block,

it is almost universally recognized as one of the toughest undergraduate courses in ics Students are generally not prepared to deal with mathematical proofs when entering thesecourses Thus, a new crop of analysis texts have arisen which teach methods of proof togetherwith theory Unfortunately, such books usually cover only single-variable calculus This is thefirst and only text to cover thoroughly both single- and multivariable calculus at a friendly level

mathemat-A large portion of the material covered should sound somewhat familiar to students from theirstudy of elementary calculus Here, however, theory and deeper understanding are stressed.Through clear, accurate, and in-depth explanations, ideas are built upon, and the reader is ad-

equately prepared for subsequent new material A number of goals are achieved Included

with nearly every proof are thorough explanations of what is being done and the desired goal,

an essential item in beginning chapters As the reader develops skill with proofs, the sion of proof strategies becomes shorter Curiosity is stimulated through the question "why?"Questioning encourages the reader to think about what has been said and either promotes logic

discus-or requires the reader to look back at previously covered material The omission of details atappointed places forces the student to read with pencil and paper

A Friendly introduction to Analysis: Single and Multi variable, works by pointing out goals,

outlining procedures, and giving good intuitive reasons Detailed and complete graphs are

used where necessary, as well as cross-referencing throughout the book Further goals are tointroduce the reader to some important basic concepts and their different terminologies andvarious uses of material in other areas of mathematics This helps the reader to identify with

material when reading other books The reader is not expected to provide a proof for every

xii Preface

result mentioned Without taking away a student's mathematical security, a respect for highermathematics is taught Historical notes add enrichment and, occasionally, motivation Also,expressions such as "it can be shown," "obviously," and "clearly" are used not to intimidate thereader, but to point out that what follows should be easily grasped by the person reading thetext If not, then review of the necessary material is in order A success in mathematics requiresstrong discipline, abundance of patience, and deep concentration

Each section in this book is followed by a set of exercises usually arranged in the order

of the associated text material, and of varied difficulty They are not arranged from easiest tohardest Besides, who is to say what the easiest or hardest exercise in a set might be? Exercisesrange from routine to creative and innovative mixing theory and applications In addition, everychapter is followed by a set of review problems of true/false nature Often, knowing what istrue or false is more important than being able to prove a theorem True statements need to beproven A counterexample is requested for false statements, and a change is requested for thosefalse statements that become true with a little help Such problems foster deeper understanding

of the material and stimulate curiosity Students learn to interrelate ideas in this way The

knowledge of what is true and what is false makes it easier for the student to construct proofs.Each review section is followed by optional projects designed to improve students' creativity aswell as to reinforce ideas covered in previous chapters The projects either continue the study

of real analysis or bring out analysis in other areas of mathematics

For quick reference, all definitions, examples, theorems, corollaries, and remarks are bered in succession as chapter number, section number, and position in the sequence withineach section Hints and solutions to selected exercises are located at the back of the book Butsince most exercises may be completed in a variety of ways, viewing this part of the book afterexercises have been attempted is recommended A convenient index of symbols included inthe back of the book lists the page where each symbol appears for the first time The Greekalphabet follows, allowing the student to read Greek symbols correctly A very extensive indexconcludes the book

num-2 Design and Organization

Chapter names closely reflect the content of each chapter In Chapter 1 we provide basicnecessary terminology and proof techniques The main purpose for Chapter 1 is to present

an abundance of material so that the book will be self-contained Some instructors may wish

to skip all of Chapter 1, while others may wish to cover only certain sections or portions ofcertain sections Some parts of Chapter 1 may be referred to only when needed while coveringlater chapters A thorough development of the number system has been omitted, due to boththe length and nature of the course In Chapter 2 we begin a thorough study of real analysisthrough a detailed presentation of sequences Chapter 3 begins with a smooth transition from

sequences to more general functions More general concepts of a limit are also covered In

Chapter 4 we introduce uniform continuity, a new term for the majority of students at this level.Thus, Section 4.4 will require time to gel Chapter S involves the theory of differentiation andsome of its applications It is a fun chapter and extremely useful The first four sections

are a must to cover Mastery of integration techniques from elementary calculus is helpful in

Chapter 6, where theory and development of the integral are stressed Proofs of theoremspresented should not be skipped, although they may seem unexciting at times Section 6.5,

one some time A quick reference back to this section is always a possibility The rest ofChapter 10 presents the theory of functions in two variables The transition to functions of

variables is an easy step, and thus omitted A computer for graphing is highly recommended.Chapter 11 follows a standard development of the double integral Line integrals and Green'stheorem are essential parts of this chapter Section 11.8 shows how material from Chapter 11may be generalized into higher dimensions

Chapters 1 through 8 with optional sections omitted are recommended in a quickly ing one-semester single-variable course Chapters 1 through 8, with optional sections coveredunder the discretion of the instructor, are recommended for a thorough single-variable courseconsisting of two semesters Coverage of the entire textbook with the omission of some optionalsections is recommended for a two-semester typical advanced calculus class As mentioned inthe next subsection, A Friendly Introduction to Analysis: Single and Multivariable is accompa-nied by an Instructor's Supplement which covers additional topics Additional topics provided

mov-by the Instructor's Supplement, as well as project sections in the textbook, supplement the terial as well as provide learning skills for students In addition, instructors are moved into adifferent teaching mode Honors classes and classes with guided discovery approaches maywish especially to incorporate into their courses either the project sections or additional topicsfrom the Instructor's Supplement

ma-3 Supplements

Two handy supplements to the text are an instructor's Solutions Manual and an Instructor's

Supplement The Instructor's Solutions Manual contains solutions to most exercises from the

text This manual can be downloaded from www.prenhall.com In order to do that, select

"browse our catalog," then click on "Mathematics," select your course and choose your text.Under "resources," on the left side, select "instructor." A one-time registration will be requiredbefore you can complete this process

Chosen from the many wonderful topics that one can study, only the classical ones appear

in the textbook itself In order for the text not to be too thick, less common topics were put

into the instructor's Supplement This accompaniment to the book is available to everyone andcan easily be downloaded from the author's Web page, which can be found under the address:http:llwww.mathsci.appstate.eduh-'wak Topics that appear in the Instructor's Supplement arelisted below

Right- and Left-Hand Derivatives

Fixed Points

Newton-Raphson and Secant Methods

Natural Logarithmic Function

Thorough Study of Parabola

Extrema of Functions in Jig

Applications of Fourier Series to Partial Differential Equations

Lagrange Multipliers

Elliptic Equations

There is also a whole chapter on Fourier Series which includes sections on

Convergence

Fourier Cosine and Fourier Sine Series

Differentiation and Integration

Approximation of Fourier Series and Bessel's Inequality

Fourier Integral and Gibbs' Phenomenon

Series of Orthogonal Functions

differen-edition A list of sections and a short introduction are both given at the beginning of each

chapter

Chapters 1 and 5 have both been reorganized fairly extensively Other chapters acquired

a lot of enhancements, such as improvements in material presentations Some examples andexplanations have been changed, and many illustrations have been sharpened Sections havebeen made more equal in length by shuffling some material as well as exercises Mistakes andmisprints have been corrected Clarity, readability, and friendliness have been improved Someexercises have been reworded, renumbered, and many instructions made more precise Only

a few exercises have been revised or removed entirely Answers to true/false review exerciseshave been moved to the Solutions Manual

Special features enjoyed by readers, such as very extensive exercise sets, true/false reviewsections, involved projects, cross-referencing, historical notes, the Solutions Manual, and theInstructor's Supplement, have been modified only a little, with the exception of material onFourier series being added to the Instructor's Supplement

at Appalachian State University; and Vanessa Kosmala In addition, I wish to thank AletheaVitray for her precise illustrations that appeared in the first edition.

Furthermore, I also wish to express my gratitude to those who reviewed all or part of thesecond edition manuscript: John Tolle of Carnegie Mellon University, Aimo I-Iinkkanen of theUniversity of Illinois at Urbana-Champaign, Bradford Crain of Portland State University, Mar-cel Finan of Arkansas Tech University, Warren Este of Montana State University, Greg Rhoads

of Appalachian State University, and a number of readers of the first edition who took the

time to write to me with their ideas and suggestions for the new edition Also, I would like tothank George Lobe!!, my friendly acquisitions editor, as well as the staff at Prentice Hall, Inc.,and members of the Department of Mathematical Sciences at Appalachian State University fortheir assistance and support in the development and production of this manuscript I wish tothank Amr Aboelmagd and his typesetting company LaTeX-Type Incorporation (TypesettingService) for converting my Microsoft Word manuscript to LaTeX and revising all illustrationsfrom the first edition, and Matthew Mellon (mmellon@appaltex.com) for typesetting correc-tions for subsequent printings

Witold A J Kosmalawak@math.appstate.edu

1.6* Finite and Infinite Sets

1.7* Ordered Field and a Real Number System

1.8* Some Properties of Real Numbers

1.9* Review

1.10* Projects

Part 1 Fibonacci Numbers

Part 2 Lucas Numbers

Part 3 Mean of Real Numbers

Introduction to analysis, sometimes referred to as advanced calculus, is a preview to a beautifularea of mathematics called real analysis In real analysis we study topics on real numbers For

this very reason we begin our journey by reviewing sets, in particular sets of real numbers,

properties of a real number system, and functions with the domain and range lying in the realnumbers This chapter sets the stage for the rest of the book and reviews basic terminology that

is probably familiar to the reader The amount of material in this chapter is abundant and shouldserve mostly as a reference when covering future chapters Based on the students' background,the instructor should choose which part of the chapter to cover and which to skip Dwelling

on the material in this chapter too long is not advisable Coverage of Sections 1.3 and 1.4,

involving methods of proof, would be helpful since many students often do not have a good

background in this area In this textbook we attempt to prove many results Writing proofs

takes practice, so without any further delay let us begin by proving statements in elementary setalgebra

2 _ _ Chapter 1 Introduction

members of the set, and by well-defined we mean that there is a definite way of determiningwhether or not a given element belongs to the set To write a set it is customary to use braces{ }, with elements of the set listed or described inside them Lowercase letters are generallyused to represent the elements, whereas capital letters denote sets themselves If an element r

belongs to the set A, then we write r E A If r is not an element of the set A, then we write

r A Thus, if A = { 1, 2, 3}, then 1 E A but 4 O A

At times it is difficult to list all members of a set For example, if the set B consists of all ofthe counting numbers smaller than 100, we usually write

B = {x f x is counting number smaller than 100}

The second of these is read as: B is the set of all elements x such that x is a counting numbersmaller than 100 Thus, we described what is in the set instead of listing the elements When

using the descriptive method, we have to be sure that what we want to include in the set is

indeed what we describe and is regarded by everyone in the same way For example, the set

C = {all young men in the United States} is not a well-defined set since the word "young" hasdifferent meanings to different people

There are many ways of describing any one particular set For example, the set D = {2, 3}can also be written as

x2-5x+6 =0},

t x is a prime number less than 4}, or

E x and x + 1, or x 1 are prime numbers},

where prime numbers are defined below Sets of numbers used throughout this book are

N = set of all natural (counting) numbers = (1, 2, 3,

When the value of x is not specified, assume that x E 9i Real numbers that are not rational

are called irrational Elements in+ and elements in J- are called positive and negative

real numbers, respectively Elements in {x E fl I x ? 0} and {x E 9 x 0} are callednonnegative and nonpositive real numbers, respectively Prime numbers are natural numbersdivisible by exactly two distinct natural numbers, 1 and the natural number itself (Definitions

of divisibility and related ideas are presented formally in Section 1.3.) An integer, p, is even

if p can be written as p = 2s for some integer, s; and an integer, q, is odd if it can be written

_

=

Sec 1.1 * Algebra of Sets 3

as q = 2t + I for some integer, t In Sections 1.7 and 1.8 we discuss real numbers in greater

depth

The following sets denote intervals, where a and b are real numbers with a <b

[a, b] _ {x I a x b} (a, b) _ {x j a < x <b}

[a, b) _ {x f a < x <b} (a, b] _ {x a < x b}

[a, oo) = f x I x ? a } (a, oo) = {x l x > a }

(-oo,a]={x Ix Via} ( oo,a)={x I x <a}.

The symbols oo and -oo are called (plus) infinity and minus infinity, respectively These

sym-bols do not represent real numbers Often, oo is written as +oo We can write 9 _ (-ox, x)and iW _ (0, x) It is customary to identify with points on a line, sometimes called an axis,

or a one-dimensional space Our entire study of mathematical analysis in this textbook will bebased on the fact that we are in the real number system

Definition 1.1.1 If A and B are sets, then A equals B, written as A = B, if and only if both

sets consist of exactly the same elements

The expression if and only if, commonly written by mathematicians as "if," means that twostatements are equivalent This expression always applies to definitions, although it is often not

written that way Equivalently, A = B if and only if whenever x E A, then x E B, and

element, say x, exists such that x E A but x B, or x E B but x A.

Definition 1.1.2 If A and B are sets such that every element of A is also an element of B,

then A is a subset of B, and is denoted by A c B If A is a subset of B but A B, then A iscalled a proper subset2 of B, and is denoted by A C B

The expression A c B means that every element in A is an element in B, and there is an

element in B that is not in A Using subset notation we can say that A = B if and only if

denoted by q5 The set is a subset of every set A because the statement "if x E c, then x E A"

is always true In general, in logic theory, a statement "if p then q" is always true when p isfalse We are now ready to make new sets from given ones

Definition 1.1.3 If A and B are sets, then

(a) A intersection B, denoted by A f1 B, is the set of all elements that belong to both A and

B That is,Af1B={x Ix EAandx E B}.

(b) A union B, denoted by A U B, is the set of all elements that belong to either A or B Thatis,AUB = {x Ix E Aorx e B}.

(c) the complement of B in A, also referred to as the complement of B relative to A, or

A minus B, denoted A \ B, is the set of all elements in A that are not in B That is,

A\B={x }x EAandx 0 B).

2Idea of a proper subset is important in many areas of mathematics In algebra we study proper groups, characters, ideals, and congruence In linear algebra we have proper values and propervectors In topology, proper cyclic elements and proper faces In this book one important concept

sub-of a proper subset is revealed in Section 1.6.

c

q5

4 Chapter 1 Introduction(d) A and B are disjoint if they have no elements in common That is, A fl B = b.

The shaded regions in Figure 1.1.1 represent Definition 1.1.3, parts (a)-(c).3

The symbol 0 is used to indicate that a proof, answer, or remark is complete Also,

observe that x A \ B does not mean that x A and x E B Why?

THEOREM 1.1.5 If A, B, and C are sets, then

(a) A fl A = A and A U A = A (Idempotent property)

(b) A fl B= B fl A and A U B= B U A (Commutative property)

(c) (A fl B) fl C = A fl (B fl C) and (AU B) U C = A U (B U C) (Associative property)

(d) A fl (B U C) = (A fl B) U (A fl C) and A U (B fl C) = (A U B) fl(A U C). (Distributiveproperty)

Morgan's4 laws)

Proof of f rst equality in part (d) Since we are to prove that two sets are equal, we need toshow that one set is a subset of the other, and vice versa To prove that A fl (B U C) c(A fl B) U (A fl C), pick an arbitrary element, say x E A fl (B U C) We will show that

x E (A fl B) U (A fl C) By Definition 1.1.3, part (a), x E A and x E B U C Thus, x E A and

X E B, or x E A and x e C Hence, x E A fl B or x E A fl C, and by Definition 1.1.3, part (b),

x E (A f1 B) U (A fl C)

3The illustrations in Figure 1.1.1 are known as Venn diagrams, named after John Venn (1834 1923).Venn was an English ordained priest who is recognized for his work in logic and probability theory Later in life Venn became a historian and demonstrated skill in building high-quality machines 4Augustus De Morgan (1806-1871), an English mathematician and logician born in India, is recognized for the development of the foundations of algebra, arithmetic, probability theory, and calculus.

Sec 1.1 * Algebra of Sets 5

We now need to prove conversely that if y E (A fl B) U (A n C), then y e A fl (B U C) Since

yE(AfB)U(AfC),thenyEAfBoryEAfC.Thus,yEAandyEB,oryEAand

y E C In other words, y E A, and further, y E B or y E C Therefore, y E A and y E B U C.Hence, y E A fl (B U C), and the proof of the first equality in part (d) is complete

Proof of first equality in part (e) Again, we need to show that one set is a subset of the other,

and vice versa, to give equality of the two sets Let x E A \ (B U C) We need to show that

x E (A \ B) fl (A \ C) Now since x E A \ (B U C), by Definition 1.1.3, part (c), x E A and

x B U C Hence, x E A, x B, and x C Thus, x E A\ B and x e A\ C Therefore,

x E (A\B)n(A \C),soA\(BUC) c (A\B)n(A \C).

To prove inclusion in the other direction, pick an arbitrary element x e (A \ B) n (A \ C)

Then, x E A\ B and x e A\ C Therefore, x E A, x B, and x C, so x E A and x B U C

Hence, x E A \ (B U C) Thus, (A \ B) n (A \ C) C A \ (B U C) and the proofof the firstequality in part (e) is complete The shaded region in Figure 1.1.2 represents A \ (B U C)

Figure 1.1.2

Figure 1.1.2 only illustrates the result and does not constitute a proof, but it may help thereader understand the proof In Exercise 3, you are asked to prove other statements in Theo-

rem 1.1.5 In view of Theorem 1.1.5, part (c), we can write A n (B n C) = A n B fl C and

A U (B U C) = A U B U C Furthermore, if A 1, A2, , A1z are n sets, where n is some fixednatural number, then

k=1

In words, the intersection above is the set of elements that belong to all of the sets in thecollection, and the union is the set of all elements that belong to at least one of the sets in

the collection

6 Chapter 1 Introduction

Definition 1.1.6 If A and B are two nonempty sets, then the Cartesian product A x B is

the set of all ordered pairs (a, b) such that a E A and b E B That is,

A = B = Ill, then SR x 91 = t(a, b) a, b E JI}, which is often denoted by 912

Remark 1.1.7 Just as we identified 91 with points on a line, we can identify 912 with points

in a plane (a two-dimensional space), which leads to the Cartesian (rectangular) coordinatesystem, named after Descartes.5 To do this, form an x y plane by intersecting horizontal andvertical lines The horizontal line is called x-axis and the vertical line the y-axis They intersect

at a point called the origin The positive direction on the x-axis is to the right of the origin,

and the positive direction on the y-axis is above the origin To associate an ordered pair (a, b),

an element of 912, with a point P in the plane, we first locate the value (i.e., the number) a on

the x-axis and then move vertically to the value b The point P is also denoted by P (a, b).The point P (a, b) is a visualization of ordered pair (a, b) in a plane The number a is called

the x-coordinate (or first coordinate), whereas the value b is called the y-coordinate (or secondcoordinate) Often, the number a is also referred to as a point In this case a is a point in 91 and(a, b) is a point in 912 Use the context to determine whether (a, b) represents a point in 912 or

an interval in 911 The point (0, 0) is the origin In general, an ordered pair (b, a) is not the same

as an ordered pair (a, b) Two ordered pairs (a, b) and (c, d) are equal if and only if a = c and

b = d Thus, (a, b) = (b, a) if and only if a = b Hence, the point represented by this ordered

pair must lie on a line passing through the origin and rising at 450.

One of the best known results in mathematics is the Pythagorean6 theorem

THEOREM 1.1.8 (Pythagorean Theorem) A triangle with legs of length a and b and

hypotenuse of length c is a right triangle if and only if a2 + b2 = c2

In other words, if one angle in a triangle has a measure of 90°, that is, radians,7 then

2a2 + b2 = c2 The converse is also true That is, if a2 + b2 = c2, then the angle opposite the

side of length c must be a right angle; that is, it must be of 90° For a generalization of the

Pythagorean theorem, see the law of cosines in Section 9.3 Proof of the Pythagrean theorem isrequested in Exercise 7

5 Rene Descartes (1596-1550), a French mathematician and philosopher, is known in Latin as Cartesius.Descartes founded analytic geometry, introduced the symbols of equality and square and cuberoots, proved how roots of an equation are related to sign changes within it, and did some work in physics.

6Pythagoras of Samos (c.580-c.500 B.c.), a Greek geometer and philosopher, interpreted all things usingnumbers Pythagoras also founded a school, considered a cult, that was political, philosophical,and religious The cult, also known as the secret society, was closely knit, regulated the diet andway of life of its members, and had a common method of education.

7 Suppose that A O B is an angle, where D is at the origin of the xy-plane and A is on the x-axis one unitaway from Q Suppose that as the initial side OA is rotated to the terminal side OB, the point Atravels s units along the arc of the circle Then, the radian measure 0 of the angle A O B is given

by 9 = s if the rotation is counterclockwise, or 8 = -s if the rotation is clockwise The number

Tr, called pi, is the ratio of the circumference of a circle to its diameter.

Sec 1.1* Algebra of Sets 7

Definition 1.1.9 (Euclidean8 Distance Formula) If Pi (x1, Y1) and P2 (x2, y2) are two points

in the xy-plane, then the distance d (Pi , P2) between them is given by

4 If A, B, and C are sets, prove that

(a) A B if and only if A n B= A

(b) Afl B = A\(A\B).

(c) (A \ B) U (B \ A.) _ (A U B) \ (AnB)

(d) (A fl B) x C = (A x C) fl (B x c).

(e) A l B and A \ B are disjoint, and A = (A f1 B) U (A \ B)

5 If the sets A, B, C, and D represent the intervals [1, 3), (1, 4), (2, 5], and [3, 5], tively, shade the region in SY12 that represents

(a) only Time?

(b) none of these magazines?

8Euclid (c.365 B c.-300 B c.), a great mathematician of ancient Greece, received mathematical trainingfrom the students of Plato (427-347 B.C.) and founded a school in Alexandria in Egypt Euclid'sbook Elements, in 13 volumes, compiled the most important mathematical facts available at thattime, including a complete solution to the study of Pythagorean triples For over two thousandyears, the first six of Euclid's volumes were students' introduction to geometry Euclid's Elements were first printed in 1482 and had more than one thousand editions thereafter.

1.2* Relations and Functions

In this lengthy section we define what is meant by a function and review a lot of terminology

We begin with relations and then decide which relation is a function

Definition 1.2.1 A relation is a set of ordered pairs The visualization of these ordered

pairs in a plane is called the graph of a relation, also called a curve Two relations are equal ifthey have the same graph

Example 1.2.2 Sets

S5 = { (x ,y) I x 2 + y2 = 1 }, and Sb = { (x , y) I x, y E Z and y x}

are all examples of relations Figure 1.2.1 shows the graphs of these relations Note that 53 can

be written as 53 = }(x, 2x + 1) I x E fl} Lowercase letters are real numbers unless otherwise

Definition 1.2.3 A function f, often called a mapping, is a relation in which no two

differ-ent ordered pairs have the same first coordinate The graph of a function f is denoted by graph(f) The domain of a function f, denoted by D1, is defined by

relation x2 + y2 = r2 represents a circle with center (0, 0) and radius r > 0 If r = 0, the

circle is degenerate and consists of only one point

Remark 1.2.4.

(a) If f is a function, for each element a e D f there is exactly one element b E Rf such that(a, b) E f The value a is an independent variable The value b is a dependent variablesince it is the result of "applying f to a" The value b is often written as f (a)

S4 = {(x,y)Ix

Sec 1.2 * Relations and Functions 9

10 Chapter 1 Introduction

f : A - B Using different words, f : A B means that f maps the set A to B In

this notation we use perhaps a more general set B rather than R1 because R1 can at times

be very hard to find A function can be thought of as a rule, transformation, mapping, ormachine that takes all of the values from D1 and assigns to them corresponding values.See Figure 1.2.2

A= D1

Figure 1.2.2

B

(c The set of all first coordinates of an ordered pair in f is the domain of f The range of

f is the set of all second coordinates, that is, all resulting functional values The range of

f can be written as {f(a) E a E D1}, or simply as f (A) In Chapters 1-8 we considerthose functions whose domain and range form a subset of fit Thus, functions will bereal-valued functions of a real variable

(d) According to Definition 1.2.3, we say that the function f : A - B is indeed welt-defined

if for (a, c) E f and (a, d) E f, we have c = d Thus, xl = x2 E A implies thatf(x1) = f (x2) Equivalently, if f (x 1) f (x2 ), then xl x2 This also means that f issingle-valued

(e) If C B, the set {x I f (x) E C} is denoted by f -1(C) and is called the inverse image

of C under f.

(f) If f : A -* B and if for some xo E A we have f (xo) = yo, then yo is called theimage

of xo and xp is called a preimage, or inverse image, of yo Can yo have more than one

preimage? If f (xo) = yo, then we can also say that f maps xo to yo Furthermore, if

xo E A, then xo is in the domain of the function f Thus, f is defined at xo and f (xo)

exists If xo is not in the domain off, then we say that f is not defined at xa, or that f (xo)does not exist Importantly, f is the name of a function, whereas f (x) is the y-coordinatecorresponding to some value x Figure 1.2.3 shows the graph of some function f Notethe dark line segments that represent D1 and R1 A dot on a graph means that the point

is on the graph, whereas a circle means that the point is excluded

(g) Every function f must have a domain D1 Either D1 is understood or it is specified IfD1 is understood, then D f consists of all values that can have an image under f In such

a case D f is called the natural domain Clearly, the natural domain can be specified If

Df does not contain all values from the natural domain, then the domain is restricted.Recall that two functions are the same only if they have the same graph Thus, changing

C

Sec 1.2 * Relations and Functions 11

Answer Since we are able to take the square root only of nonnegative real numbers to produce

a real number in R1, we must ensure that x -1 > 0 Thus, Df = {x I x 1) After substituting

these values for x in the equation y = x - 1, we find that R f = {y I y ? 0} Recall that

because the square root of a number is never negative, Rl could not possibly include negativevalues

Remark 1.2.6 The function f from Example 1.2.5 can be stated in a different and more

popular fashion That is, f (x) = x - 1.

Definition 1.2.7 Consider a function f which maps set A into set B

x l = x2 If f is one-to-one, then f is called an injection, or an infective function

(b) If B = R f, that is, B = f (A), we say that f is a function from A onto B and callit asurjection, or a surjective function That is, f takes A onto B if for each b E B, there is

an a E A such that f (a) = b

(c) If f is both one-to-one and onto, then f is called a bijection, or a bijective function

Compare the one-to-one property with what is meant by the function being well defined

Remark 1.2.8 Suppose that a function f maps set A into set B

(a) If f is an injection, then x l , x2 E A with x x2 implies that f (x i) f (x2) Inaddition, any horizontal line drawn through the graph of f will intersect it at most once.Hence, this horizontal line test is used to determine whether or not a function is one-to-one

(b) If f is a surjection, then every value in B has at least one preimage in A That is, if

yo E B, then there exists xo E A such that f (xo) = yo

(a) f is increasing if and only if f(x1) f (x2)

(b) f is strictly increasing if and only if f(x1) < f (x2).

(c) f is decreasing if and only if f (xi) f (x2)

(d) f is strictly decreasing if and only if f (x 1) > f (x)

(e) f is constant if and only if f(x1) = f (x2).

If a function is either increasing or decreasing, then it is monotone Similarly, a strictly

monotone function is either strictly increasing or strictly decreasing

Definition 1.2.11 Consider a function f that maps set A into set B

(a) f is said to be an even function if and only if f (-x) = f (x) for all x A.

(b) f is said to be an odd function if and only if f (-x) = -f(x) for all x E A

It should be noted that the set A in Definition 1.2.11 must have a property that if x E A,

then -x E A as well

Example 1.2.12.

x 2 = f (x) However, f : [-1, 3) - 91, defined by f (x) = x 2, is not even, since

another example of an even function Recall that trigonometric functions are functions

that can be written in terms of sin x and cos x Geometrically, the graph of any even

function is symmetric with respect to the vertical axis

:

Sec 1.2 * Relations and Functions 13

y

x

-x3 = - f (x) Geometrically, the graph of any odd function is symmetric with respect

to the origin That is, if a straight line drawn through the point (o, 0) intersects the graph

of the function at a point (a, b), then it must also intersect the graph of the function at thepoint ( -a, -b) See Figure 1.2.5 Functions sin x and arctan x on l i are other examples

of odd functions In addition, if x = 0 is in the domain of an odd function f, then

The absolute value function is f (x) = lx J with x E 91

For example, 31 = 3 and - 31 = 3 Therefore, the domain for the absolute value function

is the set of all real numbers, whereas the range is the set of all nonnegative real numbers SeeFigure 1.2.6 Note that x l = x2 for any real number x See Exercise 14(c) in Section 1.8 formore properties of the absolute value function

Definition 1.2.14 Suppose that a function f : A -> B Then f is bounded (on A) if and

only if there exists a real number M1 such that J f (x) l M1 for all x E A The number M1 is

called a bound for f If no such M1 exists, then f is unbounded Furthermore, f is bounded

above if and only if there exists a real number M2 such that f (x) M2 for all x E A; and

f is bounded below if and only if there exists a real number M3 such that f (x) ? M3 for all

x E A The numbers M2 and M3 are called upper bounds and lower bounds of f, respectively.The smallest of all upper bounds, if an upper bound exists, is called the least upper bound, or

14 Chapter 1 Introduction

supremum, of f and is denoted by sup f = supXEA f (x) The largest of all lower bounds, if

a lower bound exists, is called the greatest lower bound, or infimum, of f and is denoted byinf f = f (x)

A function f is bounded if and only if If 1 is bounded above Observe that since for anyfunction f : A -k B, f (A) is a set of real numbers, Definition 1.2.14 applies to any set S C 91

A nonempty set S C 91 is bounded above if and only if there exists a real number M1 such that

x M1 for all x E S A nonempty set S is bounded below if and only if there exists M2 E 91such that x M2 for all x E S A nonempty set S is bounded if and only if it is bounded aboveand bounded below The supremum and infimum of a nonempty set are defined the same way

as the supremum and infimum of a function See Theorem 1.7.7 for more about the supremum

of a set

Definition 1.2.15 Consider a function f : A -* B.

(a) A point x4 E A is a root of f if and only if f (xo) = 0 That is a value where thegraph

of f intersects the horizontal axis We say that f vanishes at xo E A if and only if xp is a

root off In addition, f is identically zero, denoted by 1(x) = 0, if and only if f (x) = 0

for all x E A If f (x) - 0, then f is called a zero function

(b) The function f has an absolute (global) maximum, (or simply maximum), at a valuex1 E A if and only if f (x) f(x1) for all x E A The value f(x1) is the absolute

(global) maximum, (or simply maximum), of f and is denoted by maxXEA f (x) or by

max f

(c) The function f has an absolute (global) minimum (or simply minimum) at a value x2 E A

if and only if f (x) ? f (x2) for all x E A The value f (x2) is the absolute (global)

minimum (or simply minimum) of f and is denoted by minXEA f (x) or by min f

(d) If B = A and f (x) = x for all x E A, then f is called an identity function

A function f has an extremum at x = x1 if and only if it has a maximum or a minimum at

x = x 1 If max f exists and equals a value M, then sup f = M (see Exercise 5)

yourself that g is bounded, sup g = max g = 1, inf g = 0, and min g does not exist In

Definition 1.2.17.

(a) A function f : 91 - 91 is called a polynomial if and only if it can be written as

f (x) = anxn + an_if ' + + a1x + ao,

where n is some fixed nonnegative integer called the degree (order) of the polynomial,

and a1, for i = 0, 1, 2, , n, is a real number called a coefficient with an 0 The

f

Sec 1.2 * Relations and Functions 15

Some upper bound M2

number an is called the leading coefficient The polynomial f (x) = c, where c is any

real number, is called a constant function, or a polynomial of degree zero See part (e) ofDefinition 1.2.10 If the degree of a polynomial f is 0 or 1, then the function f is a linearfunction, and its graph is a line Part of a line that contains two distinct points and all thepoints in between is called a line segment If the degree of a polynomial f is 2, then the

function f is called a quadratic function, and its graph is a parabola If the degree of

a polynomial f is 3, then the function f is called a cubic function The quotient of twopolynomials is called a rational function

(b) A function f : A - 111, defined by f (x) = [x] = Lxi, is called the greatest integer

function, or integer floor function, and f (x) gives the greatest integer less than or equal

to x (see Figure 1.2.8) The symbol is referred to as a floor bracket The symbol isused more commonly than [-j

(c) A function f : A - 91, defined by f (x) -= fxl, is called the integer ceiling function and

f (x) gives the least integer greater than or equal to x (see Figure 1.2.9)

(d) A function f is said to be algebraic if and only if its formula contains strictly algebraicoperations, that is, addition, subtraction, multiplication, division, and radical extraction.Functions that are not algebraic are called transcendental Trigonometric, logarithmic,and exponential functions are familiar transcendental functions See Remark 3.3.6

(e) If n* is some integer and the set A = {x E N I x ? n *), then a function f : A -* 91 iscalled a sequence Often, f (n) is denoted by an and is called the nth term of the sequence

In number theory, such a function is called arithmetical or a number-theoretic function(see Definition 2.1.1)

An important aspect of a graph of a linear function is its steepness, called the slope Sincetwo distinct points in 912 determine a line, we define the slope9 m of a line by

for any two distinct points P (x l , Yi) and Q (x2, y2) on the line with x1 x2 Observe that if f

is a linear function f (x) = a1x + ao, then a1 = m, the slope of the line Why? If m > 0, then

f is increasing, and if m < 0, then f is decreasing A constant function f (x) = ao has slope

0 A vertical line is not a function and has no slope Why? Parallel lines are defined to be linesthat have the same slope or when the two lines are vertical; and lines are called perpendicular

if they have slopes whose product is -1, except when one of the lines is vertical and other ishorizontal

The greatest integer function is sometimes called a postage or tax function Several of theproperties for [x], x a real number, are the following:

(g) -[-x] is the least integer greater than or equal to x

(h) [x] = 1 if x ? 0; that is, the integer k is incremented, beginning with k = 1,

until k becomes larger than x, and for each such k, a 1 is written Summing up the 1's wehave [x ]

Definition 1.2.1$ If functions f, g : A - B, that is, functions f and g map A into B, then

Sec 1.2 * Relations and Functions 17

In parts (e) and (f) an alternative notation can be used, namely (f vg) (x) - max t f (x) , g (x) }

and (f n g) (x) i min[ f (x) , g (x) } Figure 1.2.10 illustrates the meaning of f v g.'° Often,f2(x) represents [f(x)12 In general, f'1(x) = [ f (x) ] n, but not for n = -1 The symbol

f '.(x) does not represent [ f (x )] -1 The symbol f1 represents the inverse of a function f

see Section 1.5 and x 1 = 1 represents the reciprocal of the function .

f (x)

x

Figure 1.2.10

Definition 1.2.19 Suppose that f : A - 9t and g : B = f (A) a R Then the composition

(composite) function of g on f, g o f, maps A into R, and is defined by (g o f) (x) = g (f (x))

func-Exercises 1.2

1. (a) Give two examples of relations that are not functions

(b) Graph 4x2 + 4y2 - 4x + 12y +9 = 0 [See Exercise 22 in Section 1.8.]

1 °The symbol V is read as "vee" and the symbol n is read as "wedge."

(f g)(

(c) f : JR - J t, where f (x) = 3 x

(d) f : [-1, 1] - [0, 4), where f (x) = x2.

4 Determine whether the given function, f, is increasing, strictly increasing, decreasing,strictly decreasing, bounded, bounded above, or bounded below Also find sup f, inf 1 ,max f, and ruin f, if they exist

(a) f : [a, co) - where f (x) = x

(d) f : + > JR, where f (x) =

x

5 Prove that if a function f has a maximum, then sup f exists and max f = sup f

6 Determine whether the given function f is even, odd, or neither

(a) f(x)=21x1-x2

(b) f(x)=/+1

x3(c) f (x) _ -sin x

2x

(d) f (x) = 3 x - x

7 (a) If f, g : A - 91 are even, prove that f + g is even

(b) If f, g : A - 91 are odd, prove that f g is even

(c) Is there a function f that is both even and odd? If so, give an example of one

8 Graph the given functions

-_

Sec 1,2 * Relations and Functions

(e) f (x) = x - [x j , with -2 <x < 2

(f) f(x) = lx+l1+Ix-21

x+Ix1(g) f (x)=

11 Find two functions, f and g, so that f o g g o f.

12 Suppose that f is an odd function and g is an even function Show that the functions

go f, g o g, and f o g are all even

13. (a) If both functions f : A - B and g : B -* C are injective, prove that go f : A - C

15 Verify that every function defined on a symmetric interval about the origin that is on Jt or

on an interval (-c, c) for some c e JI can be written as the sum of an even and an odd

function

17 Use shifting, reflecting, stretching, and/or compressing of f (x) = x2 to graph g (x) =

-2(x + 3)2 + 1 [See Exercise 22 in Section 1.8.]

18 Suppose that f : X -> Y Prove that

(a) if A c B c X, then f (A) c f (B).

20 Chapter 1 Introduction

20 Suppose that f : X - Y with A, B c Y Prove that

(a) f 1(A U B) = f1(A) U f 1(B).

22 Suppose that f : X - Y Prove that

(a) f (A fl B) = f (A) fl f (B) for all A, B c X if and only if f is injective

(b) f (A\B) = f(A)\f(B) for all A, B c X if and only if f is injective.

(c) f -1(f (A)) = A for all A c X if and only if f is injective

(d) f (f " 1(B)) = B for all B c Y if and only if f is surjective

23 Suppose that f : X - Y with A c X and B c Y Find examples which show that the

given equalities are false

(a) fa'(f(A)) = A

(b) f(.f 1(B)) = B

24 Suppose that f : X - Y with A, B c X Find examples which show that the given

equalities are false

an axiom (or postulate) we mean a statement that is accepted without proof from which otherpropositions can be derived

AXIOM 1.3.1 (Well-ordering Principle) Every nonempty subset of N has a smallest

Sec 1.3 * Mathematical Induction 21

(d) n2 + n + 41 is a prime number for any n E N.11

(e) For every n > 2, there are no nonzero integers x, y, and z such that xn + yn = zn

Our first inclination is to substitute in a few numbers to see whether or not the expression

makes sense If a statement is false for one value, then it is a false statement no matter what

happens at the other values, in which case, no proof is necessary To demonstrate a statement's

invalidity, we simply substitute in a value that makes it false Statement (c) above is false,

since if n = 3, 3 3 + 1 Obtaining true statements after many values leads us to suspect

that statement (a) is true for all specified values However, concluding anything without a valid

proof is incorrect In proving statement (a), we could perhaps start with any two arbitrary

even natural numbers, say n and m, and show that n + m is even Proceeding with the proof,

since n and m are even, natural numbers r and s exist such that n = 2r and m = 2s.Therefore,

n + m = 2r + 2s = 2(r + s), and since 2 is a factor, n +m is divisible by 2 and thus is even The

proof of statement (a) is now complete Statement (d) can be misleading, since after trying the

values 1 through 39, n2 + n + 41 continues to result in a prime number However, substituting

the number 40 into the expression gives 402 + 40 + 41 = 16$1 = 412, which is certainly not

prime, Hence, statement (d) is false From 1537 until 1995, statement (e) was a conjecture

known as Fermat's12 last theorem A conjecture is a statement that mathematicians feel is true

but are unable to prove, as was the case with Fermat's last theorem No one had been able to

verify that a solution of the given expression existed, nor could they prove its impossibility for

all n > 2 Fermat's last theorem finally became a theorem rather than a conjecture in 1995.1 3

Recall that if n = 2, statement (e) reduces to the vast study of Pythagorean triangles Finally,

statement (b), which can be verified in several ways, can be rewritten as

The symbol , called sigma, represents summation Toexpandk_ 1 k, we progressively

in-crement k by 1, beginning with 1 and ending with n, and then add the resulting terms Clearly,

for any two functions f, g : N 91 with m, n E N and 1 m n, k_ 1 cf (k) _

c k,1 f (k), k=1 [f (k)+g(k)J = k-1 f(k)± k,1 g(k), and k_1 f (k) - k , f(k)+

k=m+1 f (k) Example 1.3.3 uses a straightforward, relatively quick and popular method

known as mathematical induction to prove statement (b)

THEOREM 1.3.2 (Principle of Mathematical Induction) If P (n) is a statement for each

n E N such that

" It is worth noting that n2 - 79n + 1601 produces a prime number for 0 n 79, n E N However,

it has been proved that no polynomial with integer coefficients can yield only prime numbers for

integer inputs.

12Pierre de Fermat (1601-1665) was a French lawyer whose hobby was mathematics Fermat solved

many calculus problems and became best known in number theory His work, which inspired

Newton, was not appreciated by other mathematicians during Fermat's life.

13Andrew Wiles, a British mathematician working at Princeton University, presented a "proof" of

Fer-mat's last theorem in 1993, using a 200-page argument He tried to achieve this result by proving

that semistable elliptic curves are modular The "gap" in his paper was closed by Wiles and

Richard Taylor in 1995.

n(n + 1}

22 Chapter 1 Introduction

(a) P(1) is true, and

(b) for each k E N, if P (k) is true, then P (k + 1) is true,

then P (n) is true for all n E N

In part (b), P (k) is called the induction hypothesis The principle of mathematical induction

is often referred to as mathematical induction, or even induction for short The name principle

is often used since mathematical induction is equivalent to the well-ordering principle (SeeExample 1.4.4 and Exercise 7 in Section 1.4.) For now, let us assume that Theorem 1.3.2 is trueand demonstrate its use

induc-tion (See Exercise to for a different proof.)

n n

2 We need to verify thehypotheses of parts (a) and (b) of Theorem 1.3.2 Clearly, P(1) is true, since the left-hand side

of the statement above degenerates to 1 and the right-hand side is also

2

1(1

+ 1) = I Next,assumethat P (k) is true for some k E N We will show that P (k + 1) is true Thus, suppose

k(k + 1)that 1 +2+3 + + k = 2 We need to show that

If in the statement P (n), the smallest value of n is a natural number other than 1, for erality call it r, then condition (a) in Theorem 1.3.2 is replaced by (a)', which says that P (r) istrue

gen-Checking condition (a) or (a)' of Theorem 1.3.2, whichever applies, is essential to the lidity of the induction hypotheses P (k) For example, if P (n) is the statement n = n + 1 for allnatural numbers n > 3, condition (b) of Theorem 1.3.2 would be satisfied, since if k = k + 1,then certainly k + 1 = k + 2, for k > 3 Therefore, if condition (a)' is not checked, we would

va-be led to va-believe things that are false

Example 1.3.4 Prove that for any fixed real number x and all n E N,

Sec 1.3 * Mathematical Induction 23

Proof Although other methods of proof are available, we once again illustrate the use ofmathematical induction Let P (n) be the statement to be proven P(1) says that x 2 - 1 =

(x - 1) (x + 1), which is true Next assume that P (n) is true, that is, that

Therefore, P (k + 1) is true Hence, P (n) is true for all n E N

We say that an integer b is divisible by a nonzero integer a if there exists some integer csuch that b = ac If this is the case, we also say that a is a divisor of b, that a is a factor of b,

or that b is a multiple of a These definitions also apply to polynomials In Example 1.3.4, weproved not only the specified formula but also that x - 1 is a factor of xn+1 - 1 However, itwas not necessary to prove the formula to show that x - 1 is a factor of xn+1 - 1 See Exercise2(e) We often write the proven expression above as the geometric sum

where in the last expression we assumed that x° = 1, even if x = 0 Why?

Definition 1.3.5 If n is a nonnegative integer, then n factorial, denoted by n!, is the number

LEMMA 1.3.6 If k and n are integers such that 0 <k n, prove that