免费证明教程：Book of Proof 第三版

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"Book of Proof 3rd edition 是 Richard Hammack 在2018年出版的一本关于如何证明定理的免费书籍。它分为四个部分：基础、证明条件语句、更深入的证明、关系、函数和基数。" 本书是数学证明理论领域的重要资源，适合想要学习或提升证明技巧的读者。作者Richard Hammack是数学与应用数学系的教授，他将多年的教学经验融入书中，旨在帮助读者掌握严谨的数学证明方法。第一部分“基础”涵盖了证明的基本概念。1.1章节介绍了集合的定义和性质，这是理解所有后续概念的基础。1.2章节讲解了笛卡尔积，它是构建复合关系和函数的关键。1.3章节讨论子集的概念，1.4章节则涉及幂集，这两个概念对于理解集合的结构和操作至关重要。1.5至1.7章节分别讲述了并集、交集、差集以及补集，这些都是处理集合之间关系的基本运算。1.8章节介绍了索引集合，这对于处理有序或结构化的集合特别有用。1.9章节探讨了那些作为数系的集合，如自然数、整数、有理数等。最后，1.10章节通过罗素悖论引入了集合论中的一个重要问题，这有助于读者理解集合论的深度和复杂性。第二部分“证明条件语句”可能涉及逻辑推理、蕴含、等价关系以及条件证明策略等。这部分通常会讲解如何构造和分析复杂的逻辑表达式，并学会在证明中正确使用它们。第三部分“更深入的证明”可能会涵盖更高级的证明技术，如归纳法、反证法、构造性证明等，这些技术是解决更复杂问题时不可或缺的工具。第四部分“关系、函数和基数”将深入到数学的核心概念，包括函数的性质、映射、关系的性质（如对称性、传递性和反身性）以及基数的概念，这涉及到集合大小的比较和无穷的分类。 "Book of Proof 3rd edition" 是一本全面的数学证明教程，不仅适合初学者入门，也适合有一定基础的读者巩固和深化理解。通过阅读和实践书中的例子和习题，读者可以逐步建立强大的数学思维能力和证明技巧。

4 Sets

Some sets are so signiﬁcant 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 =

1,2,3,4,5,6,7,...

The set of integers

Z =

...,−3,−2,−1,0,1,2,3,4,...

is another fundamental set. The symbol

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 =

(0,0),(1,0),(0,1),(1,1)

has as elements the four corner points

of a square on the

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 =

2,3

2,4

ªª

, which has three elements: the

number

, the set

2,3

and the set

2,4

. Thus

1 ∈ E

and

2,3

∈ E

and

2,4

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

Consider the set

M =

©£

0 0

1 0

0 1

1 0

1 1

¤ª

of three two-by-two matrices. We

have

0 0

∈ M

, but

1 1

0 1

∉ M

. Letters can serve as symbols denoting a set’s

elements: If a =

0 0

, b =

1 0

0 1

and c =

1 0

1 1

, then M =

a, b, c

is a ﬁnite set, its

cardinality

size

is the number of elements

it has, and this number is denoted as

. Thus for the sets above,

=2,

=5,

=4,

=3 and

=3.

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

empty

set

is the set

©ª

that has no elements. We denote it as

;

, so

©ª

. Whenever

you see the symbol

;

, it stands for

©ª

. Observe that

;

. 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,

2,4,6,8

is a “box” containing four

numbers. The empty set

©ª

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=

;

Richard Hammack Book of Proof

Introduction to Sets 5

This box analogy can help us think about sets. The set

F =

;

©©

;

ªªª

may look strange but it is really very simple. Think of it as a box containing

three things: an empty box, a box containing an empty box, and a box

containing a box containing an empty box. Thus

. The set

G =

N,Z

is a box containing two boxes, the box of natural numbers and the box of

integers. Thus

=2.

A special notation called

set-builder notation

is used to describe sets

that are too big or complex to list between braces. Consider the inﬁnite set

of even integers

E =

...,−6,−4,−2,0,2,4,6,...

. In set-builder notation this

set is written as

E =

2n : n ∈Z

We read the ﬁrst brace as “the set of all things of form,” and the colon as

“such that.” So the expression

E =

2n : n ∈Z

reads as “

equals the set of

all things of form

, such that

is an element of

.” The idea is that

consists of all possible values of 2n, where n takes on all values in Z.

In general, a set X written with set-builder notation has the syntax

X =

expression : rule

where the elements of

are understood to be all values of “expression” that

are speciﬁed by “rule.” For example, above

is the set of all values of the

expression

that satisfy the rule

n ∈Z

. There can be many ways to express

the same set. For example,

E =

2n : n ∈ Z

n : n is an even integer

n : n =2 k, k ∈Z

. Another common way of writing it is

E =

n ∈Z : n is even

read “

is the set of all

such that

is even.” Some writers use a bar

instead of a colon; for example, E =

n ∈Z |n is even

. We use the colon.

Example 1.1

Here are some further illustrations of set-builder notation.

n : n is a prime number

2,3,5,7,11,13,17,...

n ∈N : n is prime

2,3,5,7,11,13,17,...

: n ∈Z

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

x ∈R : x

−2 =0

2,−

x ∈Z : x

−2 =0

x ∈Z :

−3,−2,−1,0,1,2,3

2x : x ∈Z,

−6,−4,−2,0,2,4,6

x ∈Z :

−1,0,1

Free PDF version

6 Sets

Items 6–8 above highlight a conﬂict of notation that we must always

be alert to. The expression

means absolute value if

is a number

and cardinality if

is a set. The distinction should always be clear from

context. Consider

x ∈Z :

in Example 1.1 (6) above. Here

x ∈Z

, so

is a number (not a set), and thus the bars in

must mean absolute value,

not cardinality. On the other hand, suppose

A =

©©

1,2

3,4,5,6

ªª

and

B =

X ∈ A :

. The elements of

are sets (not numbers), so the

the expression for B must mean cardinality. Therefore B =

©©

1,2

ªª

Example 1.2 Describe the set A =

7a +3b : a, b ∈Z

Solution:

This set contains all numbers of form

7a +3b

, where

and

are integers. Each such number

7a +3b

is an integer, so

contains only

integers. But which integers? If

is any integer, then

n = 7n +3(−2n)

, so

n = 7 a +3b

where

a = 7n

and

b = −2n

. Therefore

n ∈ A

. We’ve now shown

that

contains only integers, and also that every integer is an element

of A. Consequently A =Z.

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

that are so common that they are given special names and symbols.

• The empty set: ;=

©ª

• The natural numbers: N =

1,2,3,4,5,...

• The integers: Z =

...,−3,−2,−1,0,1,2,3,4,5,...

• The rational numbers: Q =

x : x =

, where m, n ∈Z and n 6=0

• The real numbers: R

We visualize the set

of real numbers is as an inﬁnitely long number line.

1 2

4−4

−3

−2 −1

Notice that

is the set of all numbers in

that can be expressed as a

fraction of two integers. You may be aware that

Q 6=R

, as

2 ∉Q

but

2 ∈R

(If not, this point will be addressed in Chapter 6.)

In calculus you encountered intervals on the number line. Like

, these

too are inﬁnite sets of numbers. Any two numbers

a, b ∈R

with

a < b

give

rise to various intervals. Graphically, they are represented by a darkened

segment on the number line between

and

. A solid circle at an endpoint

indicates that that number is included in the interval. A hollow circle

indicates a point that is not included in the interval.

Richard Hammack Book of Proof

8 Sets

C. Find the following cardinalities.

29.

©©

3,4

ªª

30.

©©

1,4

, a, b,

©©

3,4

ªª

;

ªª

31.

©©©

3,4

ªª

32.

©©©

1,4

, a, b,

©©

3,4

ªª

;

ªªª

33.

x ∈Z :

<10

34.

x ∈N :

<10

35.

x ∈Z : x

<10

36.

x ∈N : x

<10

37.

x ∈N : x

38.

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]

40.

(x, y) : x ∈[0, 1], y ∈[1,2]

41.

(x, y) : x ∈[−1, 1], y =1

42.

(x, y) : x =2, y ∈[0,1]

43.

(x, y) :

=2, y ∈[0, 1]

44.

(x, x

) : x ∈R

45.

(x, y) : x, y ∈R, x

+ y

46.

(x, y) : x, y ∈R, x

+ y

≤1

47.

(x, y) : x, y ∈R, y ≥ x

−1

48.

(x, y) : x, y ∈R, x >1

49.

(x, x + y) : x ∈R, y ∈Z

50.

(x,

) : x ∈R, y ∈N

51.

(x, y) ∈R

: (y −x)(y +x) =0

52.

(x, y) ∈R

: (y −x

)(y +x

) =0

1.2 The Cartesian Product

Given two sets

and

, it is possible to “multiply” them to produce a new

set denoted as

A ×B

. This operation is called the Cartesian product. To

understand it, we must ﬁrst understand the idea of an ordered pair.

Deﬁnition 1.1

ordered pair

is a list

(x, y)

of two things

and

enclosed in parentheses and separated by a comma.

For example,

(2,4)

is an ordered pair, as is

(4,2)

. These ordered pairs are

diﬀerent because even though they have the same things in them, the order

is diﬀerent. We write

(2,4) 6=(4,2)

. Right away you can see that ordered pairs

can be used to describe points on the plane, as was done in calculus, but

they are not limited to just that. The things in an ordered pair don’t have

to be numbers. You can have ordered pairs of letters, such as

(`, m)

, ordered

pairs of sets such as

¡©

2,5

3,2

ª¢

, even ordered pairs of ordered pairs like

(2,4),(4,2)

. The following are also ordered pairs:

1,2,3

ª¢

R,(0,0)

. Any

list of two things enclosed by parentheses is an ordered pair. Now we are

ready to deﬁne the Cartesian product.

Deﬁnition 1.2

The

Cartesian product

of two sets

and

is another

set, denoted as A ×B and deﬁned as A ×B =

(a, b) : a ∈ A, b ∈B

Richard Hammack Book of Proof

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