BookOfProof bomba, Provas de Matemática

Baixe BookOfProof bomba e outras Provas em PDF para Matemática, somente na Docsity!

Book of Proof

Richard Hammack

Virginia Commonwealth University

Richard Hammack (publisher) Department of Mathematics & Applied Mathematics P.O. Box 842014 Virginia Commonwealth University Richmond, Virginia, 23284

Book of Proof

Edition 2.

© 2013 by Richard Hammack

This work is licensed under the Creative Commons Attribution-No Derivative Works 3. License

Typeset in 11pt TEX Gyre Schola using PDFLATEX

Contents

Preface vii

Introduction viii

v

Preface

I

n writing this book I have been motivated by the desire to create a

high-quality textbook that costs almost nothing.

The book is available on my web page for free, and the paperback

version (produced through an on-demand press) costs considerably less

than comparable traditional textbooks. Any revisions or new editions

will be issued solely for the purpose of correcting mistakes and clarifying

exposition. New exercises may be added, but the existing ones will not be

unnecessarily changed or renumbered.

This text is an expansion and refinement of lecture notes I developed

while teaching proofs courses over the past fourteen years at Virginia

Commonwealth University (a large state university) and Randolph-Macon

College (a small liberal arts college). I found the needs of these two

audiences to be nearly identical, and I wrote this book for them. But I am

mindful of a larger audience. I believe this book is suitable for almost any

undergraduate mathematics program.

This second edition incorporates many minor corrections and additions

that were suggested by readers around the world. In addition, several

new examples and exercises have been added, and a section on the Cantor-

Bernstein-Schröeder theorem has been added to Chapter 13.

Richard Hammack Richmond, Virginia

May 25, 2013

Introduction

T

his is a book about how to prove theorems.

Until this point in your education, mathematics has probably been

presented as a primarily computational discipline. You have learned to

solve equations, compute derivatives and integrals, multiply matrices

and find determinants; and you have seen how these things can answer

practical questions about the real world. In this setting, your primary goal

in using mathematics has been to compute answers.

But there is another side of mathematics that is more theoretical than

computational. Here the primary goal is to understand mathematical

structures, to prove mathematical statements, and even to invent or

discover new mathematical theorems and theories. The mathematical

techniques and procedures that you have learned and used up until now

are founded on this theoretical side of mathematics. For example, in

computing the area under a curve, you use the fundamental theorem of

calculus. It is because this theorem is true that your answer is correct.

However, in learning calculus you were probably far more concerned with

how that theorem could be applied than in understanding why it is true.

But how do we know it is true? How can we convince ourselves or others

of its validity? Questions of this nature belong to the theoretical realm of

mathematics. This book is an introduction to that realm.

This book will initiate you into an esoteric world. You will learn and

apply the methods of thought that mathematicians use to verify theorems,

explore mathematical truth and create new mathematical theories. This

will prepare you for advanced mathematics courses, for you will be better

able to understand proofs, write your own proofs and think critically and

inquisitively about mathematics.

x Introduction

To the instructor. The book is designed for a three credit course. Here

is a possible timetable for a fourteen-week semester.

Week Monday Wednesday Friday

1 Section 1.1 Section 1.2 Sections 1.3, 1.

2 Sections 1.5, 1.6, 1.7 Section 1.8 Sections 1.9∗, 2.

3 Section 2.2 Sections 2.3, 2.4 Sections 2.5, 2.

4 Section 2.7 Sections 2.8∗, 2.9 Sections 2.10, 2.11∗, 2.12∗

5 Sections 3.1, 3.2 Section 3.3 Sections 3.4, 3.5∗

6 EXAM Sections 4.1, 4.2, 4.3 Sections 4.3, 4.4, 4. ∗

7 Sections 5.1, 5.2, 5.3∗^ Section 6.1 Sections 6.2 6.3∗

8 Sections 7.1, 7.2∗, 7.3 Sections 8.1, 8.2 Section 8.

9 Section 8.4 Sections 9.1, 9.2, 9. ∗ Section 10.

10 Sections 10.0, 10.3∗^ Sections 10.1, 10.2 EXAM

11 Sections 11.0, 11.1 Sections 11.2, 11.3 Sections 11.4, 11.

12 Section 12.1 Section 12.2 Section 12.

13 Sections 12.3, 12.4∗^ Section 12.5 Sections 12.5, 12.6∗

14 Section 13.1 Section 13.2 Sections 13.3, 13.4∗

Sections marked with ∗ may require only the briefest mention in class, or

may be best left for the students to digest on their own. Some instructors

may prefer to omit Chapter 3.

Acknowledgments. I thank my students in VCU’s MATH 300 courses

for offering feedback as they read the first edition of this book. Thanks

especially to Cory Colbert and Lauren Pace for rooting out typographical

mistakes and inconsistencies. I am especially indebted to Cory for reading

early drafts of each chapter and catching numerous mistakes before I

posted the final draft on my web page. Cory also created the index,

suggested some interesting exercises, and wrote some solutions. Thanks

to Andy Lewis and Sean Cox for suggesting many improvements while

teaching from the book. I am indebted to Lon Mitchell, whose expertise

with typesetting and on-demand publishing made the print version of this

book a reality.

And thanks to countless readers all over the world who contacted me

concerning errors and omissions. Because of you, this is a better book.

Part I

Fundamentals

CHAPTER 1

Sets

A

ll of mathematics can be described with sets. This becomes more and

more apparent the deeper into mathematics you go. It will be apparent

in most of your upper level courses, and certainly in this course. The

theory of sets is a language that is perfectly suited to describing and

explaining all types of mathematical structures.

1.1 Introduction to Sets

A set is a collection of things. The things in the collection are called

elements of the set. We are mainly concerned with sets whose elements

are mathematical entities, such as numbers, points, functions, etc.

A set is often expressed by listing its elements between commas, en-

closed by braces. For example, the collection

is a set which has

four elements, the numbers 2 , 4 , 6 and 8. Some sets have infinitely many

elements. For example, consider the collection of all integers,

Here the dots indicate a pattern of numbers that continues forever in both

the positive and negative directions. A set is called an infinite set if it

has infinitely many elements; otherwise it is called a finite set.

Two sets are equal if they contain exactly the same elements. Thus

because even though they are listed in a different

order, the elements are identical; but

. Also

We often let uppercase letters stand for sets. In discussing the set

we might declare A =

and then use A to stand for

. To express that 2 is an element of the set A, we write 2 ∈ A, and

read this as “ 2 is an element of A,” or “ 2 is in A,” or just “ 2 in A.” We also

have 4 ∈ A, 6 ∈ A and 8 ∈ A, but 5 ∉ A. We read this last expression as “ 5 is

not an element of A,” or “ 5 not in A.” Expressions like 6 , 2 ∈ A or 2 , 4 , 8 ∈ A

are used to indicate that several things are in a set.

4 Sets

Some sets are so significant and prevalent that we reserve special

symbols for them. The set of natural numbers (i.e., the positive whole

numbers) is denoted by N, that is,

N =

The set of integers

Z =

is another fundamental set. The symbol R stands for the set of all real

numbers , a set that is undoubtedly familiar to you from calculus. Other

special sets will be listed later in this section.

Sets need not have just numbers as elements. The set B =

T, F

consists

of two letters, perhaps representing the values “true” and “false.” The set

C =

a, e, i, o, u

consists of the lowercase vowels in the English alphabet.

The set D =

has as elements the four corner points

of a square on the x-y coordinate plane. Thus (0, 0) ∈ D, (1, 0) ∈ D, etc., but

(1, 2) ∉ D (for instance). It is even possible for a set to have other sets

as elements. Consider E =

, which has three elements: the

number 1 , the set

and the set

. Thus 1 ∈ E and

∈ E and

∈ E. But note that 2 ∉ E, 3 ∉ E and 4 ∉ E.

Consider the set M =

{ [

0 0 0 0

]

[

1 0 0 1

]

[

1 0 1 1

] }

of three two-by-two matrices.

We have

[

0 0 0 0

]

∈ M, but

[

1 1 0 1

]

∉ M. Letters can serve as symbols denoting a

set’s elements: If a =

[

0 0 0 0

]

, b =

[

1 0 0 1

]

and c =

[

1 0 1 1

]

, then M =

a, b, c

If X is a finite set, its cardinality or size is the number of elements

it has, and this number is denoted as |X |. Thus for the sets above, |A| = 4 ,

|B| = 2 , |C| = 5 , |D| = 4 , |E| = 3 and |M| = 3.

There is a special set that, although small, plays a big role. The

empty set is the set

that has no elements. We denote it as ;, so ; =

Whenever you see the symbol ;, it stands for

. Observe that |;| = 0. The

empty set is the only set whose cardinality is zero.

Be careful in writing the empty set. Don’t write

when you mean ;.

These sets can’t be equal because ; contains nothing while

contains

one thing, namely the empty set. If this is confusing, think of a set as a

box with things in it, so, for example,

is a “box” containing four

numbers. The empty set ; =

is an empty box. By contrast,

is a box

with an empty box inside it. Obviously, there’s a difference: An empty box

is not the same as a box with an empty box inside it. Thus ; 6 =

. (You

might also note |;| = 0 and

∣ = 1 as additional evidence that ; 6 =

6 Sets

These last three examples highlight a conflict of notation that we must

always be alert to. The expression |X | means absolute value if X is a number

and cardinality if X is a set. The distinction should always be clear from

context. Consider

x ∈ Z : |x| < 4

in Example 1.1 (6) above. Here x ∈ Z, so x

is a number (not a set), and thus the bars in |x| must mean absolute value,

not cardinality. On the other hand, suppose A =

and

B =

X ∈ A : |X | < 3

. The elements of A are sets (not numbers), so the |X |

in the expression for B must mean cardinality. Therefore B =

We close this section with a summary of special sets. These are sets or

types of sets that come up so often that they are given special names and

symbols.

• The empty set: ; =

• The natural numbers: N =

• The integers: Z =

• The rational numbers: Q =

x : x =

m

n

, where m, n ∈ Z and n 6 = 0

• The real numbers: R (the set of all real numbers on the number line)

Notice that Q is the set of all numbers that can be expressed as a fraction

of two integers. You are surely aware that Q 6 = R, as

p

2 ∉ Q but

p

2 ∈ R.

Following are some other special sets that you will recall from your

study of calculus. Given two numbers a, b ∈ R with a < b, we can form

various intervals on the number line.

• Closed interval: [a, b] =

x ∈ R : a ≤ x ≤ b

• Half open interval: (a, b] =

x ∈ R : a < x ≤ b

• Half open interval: [a, b) =

x ∈ R : a ≤ x < b

• Open interval: (a, b) =

x ∈ R : a < x < b

• Infinite interval: (a, ∞) =

x ∈ R : a < x

• Infinite interval: [a, ∞) =

x ∈ R : a ≤ x

• Infinite interval: (−∞, b) =

x ∈ R : x < b

• Infinite interval: (−∞, b] =

x ∈ R : x ≤ b

Remember that these are intervals on the number line, so they have in-

finitely many elements. The set (0. 1 , 0 .2) contains infinitely many numbers,

even though the end points may be close together. It is an unfortunate

notational accident that (a, b) can denote both an interval on the line and

a point on the plane. The difference is usually clear from context. In the

next section we will see still another meaning of (a, b).

Introduction to Sets 7

Exercises for Section 1.

A. Write each of the following sets by listing their elements between braces.

1.

{ 5 x − 1 : x ∈ Z

}

{ 3 x + 2 : x ∈ Z

}

{ x ∈ Z : − 2 ≤ x < 7

}

{ x ∈ N : − 2 < x ≤ 7

}

{ x ∈ R : x 2 = 3

}

{ x ∈ R : x 2 = 9

}

{ x ∈ R : x 2

• 5 x = − 6

}

{ x ∈ R : x^3 + 5 x^2 = − 6 x

}

{ x ∈ R : sin π x = 0

}

{ x ∈ R : cos x = 1

}

{ x ∈ Z : |x| < 5

}

{ x ∈ Z : | 2 x| < 5

}

{ x ∈ Z : | 6 x| < 5

}

{ 5 x : x ∈ Z, | 2 x| ≤ 8

}

{ 5 a + 2 b : a, b ∈ Z

}

{ 6 a + 2 b : a, b ∈ Z

}

B. Write each of the following sets in set-builder notation.

17.

{ 2 , 4 , 8 , 16 , 32 , 64...

}

{ 0 , 4 , 16 , 36 , 64 , 100 ,...

}

{

... , − 6 , − 3 , 0 , 3 , 6 , 9 , 12 , 15 ,...

}

{

... , − 8 , − 3 , 2 , 7 , 12 , 17 ,...

}

{ 0 , 1 , 4 , 9 , 16 , 25 , 36 ,...

}

{ 3 , 6 , 11 , 18 , 27 , 38 ,...

}

{ 3 , 4 , 5 , 6 , 7 , 8

}

{ − 4 , − 3 , − 2 , − 1 , 0 , 1 , 2

}

{

... , 1 8 , 1 4 , 1 2 , 1 , 2 , 4 , 8 ,...

}

{

... , 1 27 ,^

1 9 ,^

1 3 ,^1 ,^3 ,^9 ,^27 ,...^

}

{

... , − π , − π 2 , 0 , π 2 , π , 3 π 2 , 2 π , 5 π 2 ,...

}

{

... , − 3 2 ,^ −^

3 4 ,^0 ,^

3 4 ,^

3 2 ,^

9 4 ,^3 ,^

15 4 ,^

9 2 ,...^

}

C. Find the following cardinalities.

∣ ∣

{{ 1

} ,

{ 2 ,

{ 3 , 4

}} , ;

}∣ ∣

∣ ∣

{{ 1 , 4

} , a, b,

{{ 3 , 4

}} ,

{ ;

}}∣ ∣

∣ ∣

{{{ 1

} ,

{ 2 ,

{ 3 , 4

}} , ;

}}∣ ∣

∣ ∣

{{{ 1 , 4

} , a, b,

{{ 3 , 4

}} ,

{ ;

}}}∣ ∣

∣ ∣

{ x ∈ Z : |x| < 10

}∣ ∣

∣ ∣

{ x ∈ N : |x| < 10

}∣ ∣

∣ ∣

{ x ∈ Z : x 2 < 10

}∣ ∣

∣ ∣

{ x ∈ N : x 2 < 10

}∣ ∣

∣ ∣

{ x ∈ N : x 2 < 0

}∣ ∣

∣ ∣

{ x ∈ N : 5x ≤ 20

}∣ ∣

D. Sketch the following sets of points in the x-y plane.

39.

{ (x, y) : x ∈ [1, 2], y ∈ [1, 2]

}

{ (x, y) : x ∈ [0, 1], y ∈ [1, 2]

}

{ (x, y) : x ∈ [− 1 , 1], y = 1

}

{ (x, y) : x = 2 , y ∈ [0, 1]

}

{ (x, y) : |x| = 2 , y ∈ [0, 1]

}

{ (x, x^2 ) : x ∈ R

}

{ (x, y) : x, y ∈ R, x^2 + y^2 = 1

}

{ (x, y) : x, y ∈ R, x^2 + y^2 ≤ 1

}

{ (x, y) : x, y ∈ R, y ≥ x 2 − 1

}

{ (x, y) : x, y ∈ R, x > 1

}

{ (x, x + y) : x ∈ R, y ∈ Z

}

{ (x, x

2 y ) : x ∈ R, y ∈ N

}

{ (x, y) ∈ R 2 : (y − x)(y + x) = 0

}

{ (x, y) ∈ R^2 : (y − x^2 )(y + x^2 ) = 0

}

The Cartesian Product 9

For another example,

×

. If you are

a visual thinker, you may wish to draw a diagram similar to Figure 1.1.

The rectangular array of such diagrams give us the following general fact.

Fact 1.1 If A and B are finite sets, then |A × B| = |A| · |B|.

The set R × R =

(x, y) : x, y ∈ R

should be very familiar. It can be viewed

as the set of points on the Cartesian plane, and is drawn in Figure 1.2(a).

The set R × N =

(x, y) : x ∈ R, y ∈ N

can be regarded as all of the points on

the Cartesian plane whose second coordinate is a natural number. This

is illustrated in Figure 1.2(b), which shows that R × N looks like infinitely

many horizontal lines at integer heights above the x axis. The set N × N

can be visualized as the set of all points on the Cartesian plane whose

coordinates are both natural numbers. It looks like a grid of dots in the

first quadrant, as illustrated in Figure 1.2(c).

x x x

y y y

(a) (b) (c)

R × R R × N N × N

Figure 1.2. Drawings of some Cartesian products

It is even possible for one factor of a Cartesian product to be a Cartesian

product itself, as in R × (N × Z) =

(x, (y, z)) : x ∈ R, (y, z) ∈ N × Z

We can also define Cartesian products of three or more sets by moving

beyond ordered pairs. An ordered triple is a list (x, y, z). The Cartesian

product of the three sets R, N and Z is R×N×Z =

(x, y, z) : x ∈ R, y ∈ N, z ∈ Z

Of course there is no reason to stop with ordered triples. In general,

A 1 × A 2 × · · · × An =

(x 1 , x 2 ,... , xn) : xi ∈ Ai for each i = 1 , 2 ,... , n

Be mindful of parentheses. There is a slight difference between R×(N×Z)

and R × N × Z. The first is a Cartesian product of two sets; its elements are

ordered pairs (x, (y, z)). The second is a Cartesian product of three sets; its

elements look like (x, y, z). To be sure, in many situations there is no harm

in blurring the distinction between expressions like (x, (y, z)) and (x, y, z),

but for now we consider them as different.

10 Sets

We can also take Cartesian powers of sets. For any set A and positive

integer n, the power A

n

is the Cartesian product of A with itself n times:

A

n = A × A × · · · × A =

(x 1 , x 2 ,... , xn) : x 1 , x 2 ,... , xn ∈ A

In this way, R

2

is the familiar Cartesian plane and R

3

is three-dimensional

space. You can visualize how, if R

2

is the plane, then Z

2

(m, n) : m, n ∈ Z

is a grid of points on the plane. Likewise, as R^3 is 3 -dimensional space,

Z

3

(m, n, p) : m, n, p ∈ Z

is a grid of points in space.

In other courses you may encounter sets that are very similar to R

n

but yet have slightly different shades of meaning. Consider, for example,

the set of all two-by-three matrices with entries from R:

M =

{[

u v w x y z

]

: u, v, w, x, y, z ∈ R

This is not really all that different from the set

R

6

(u, v, w, x, y, z) : u, v, w, x, y, z ∈ R

The elements of these sets are merely certain arrangements of six real

numbers. Despite their similarity, we maintain that M 6 = R

6

, for two-by-

three matrices are not the same things as sequences of six numbers.

Exercises for Section 1.

A. Write out the indicated sets by listing their elements between braces.

1. Suppose A =

{ 1 , 2 , 3 , 4

} and B =

{ a, c

} .

(a) A × B

(b) B × A

(c) A × A

(d) B × B

(e) ; × B

(f) (A × B) × B

(g) A × (B × B)

(h) B 3

2. Suppose A =

{ π , e, 0

} and B =

{ 0 , 1

} .

(a) A × B

(b) B × A

(c) A × A

(d) B × B

(e) A × ;

(f) (A × B) × B

(g) A × (B × B)

(h) A × B × B

{ x ∈ R : x 2 = 2

} ×

{ a, c, e

}

{ n ∈ Z : 2 < n < 5

} ×

{ n ∈ Z : |n| = 5

}

{ x ∈ R : x^2 = 2

} ×

{ x ∈ R : |x| = 2

}

{ x ∈ R : x 2 = x

} ×

{ x ∈ N : x 2 = x

}

{ ;

} ×

{ 0 , ;

} ×

{ 0 , 1

}

{ 0 , 1

} 4

B. Sketch these Cartesian products on the x-y plane R 2 (or R 3 for the last two).

9.

{ 1 , 2 , 3

} ×

{ − 1 , 0 , 1

}

{ − 1 , 0 , 1

} ×

{ 1 , 2 , 3

}

11. [0, 1] × [0, 1]

12. [− 1 , 1] × [1, 2]

{ 1 , 1. 5 , 2

} × [1, 2]

14. [1, 2] ×

{ 1 , 1. 5 , 2

}

{ 1

} × [0, 1]

16. [0, 1] ×

{ 1

}

17. N × Z

18. Z × Z

19. [0, 1] × [0, 1] × [0, 1]

{ (x, y) ∈ R 2 : x 2

• y 2 ≤ 1

} × [0, 1]