Covers All 8 Chapters
SOLUTIONS MANUAL
,Contents
Preface v
1 The Real Numbers 1
√
1.1 Discussion: The Irrationality of 2 ......................................................................... 1
1.2 Some Preliminaries ............................................................................................... 4
1.3 The Axiom of Completeness ............................................................................... 13
1.4 Consequences of Completeness ......................................................................... 18
1.5 Cantor’s Theorem................................................................................................ 29
1.6 Epilogue ............................................................................................................. 33
2 Sequences and Series 35
2.1 Discussion: Rearrangements of Infinite Series...................................................... 35
2.2 The Limit of a Sequence ..................................................................................... 38
2.3 The Algebraic and Order Limit Theorems............................................................. 44
2.4 The Monotone Convergence Theorem and a First Look at
Infinite Series ...................................................................................................... 50
2.5 Subsequences and the Bolzano–Weierstrass Theorem ......................................... 55
2.6 The Cauchy Criterion........................................................................................... 58
2.7 Properties of Infinite Series ................................................................................. 62
2.8 Double Summations and Products of Infinite Series .............................................. 69
2.9 Epilogue ............................................................................................................. 73
3 Basic Topology of R 75
3.1 Discussion: The Cantor Set................................................................................. 75
3.2 Open and Closed Sets ........................................................................................ 78
3.3 Compact Sets . .................................................................................................. 84
3.4 Perfect Sets and Connected Sets ........................................................................ 89
3.5 Baire’s Theorem.................................................................................................. 94
3.6 Epilogue ............................................................................................................. 96
4 Functional Limits and Continuity 99
4.1 Discussion: Examples of Dirichlet and Thomae .................................................... 99
4.2 Functional Limits................................................................................................ 103
4.3 Combinations of Continuous Functions ...................................................................... 109
xi
,xii Contents
4.4 Continuous Functions on Compact Sets ..............................................................114
4.5 The Intermediate Value Theorem........................................................................120
4.6 Sets of Discontinuity.....................................................................................................125
4.7 Epilogue ........................................................................................................................................ 127
5 The Derivative 129
5.1 Discussion: Are Derivatives Continuous?.....................................................................129
5.2 Derivatives and the Intermediate Value Property ..................................................131
5.3 The Mean Value Theorem..................................................................................137
5.4 A Continuous Nowhere-Differentiable Function ...................................................................144
5.5 Epilogue ........................................................................................................................................ 148
6 Sequences and Series of Functions 151
6.1 Discussion: Branching Processes .......................................................................151
6.2 Uniform Convergence of a Sequence of Functions .....................................................154
6.3 Uniform Convergence and Differentiation ....................................................................164
6.4 Series of Functions.......................................................................................................167
6.5 Power Series .................................................................................................................169
6.6 Taylor Series .....................................................................................................176
6.7 Epilogue ........................................................................................................................................ 181
7 The Riemann Integral 183
7.1 Discussion: How Should Integration be Defined? ..................................................183
7.2 The Definition of the Riemann Integral................................................................186
7.3 Integrating Functions with Discontinuities .............................................................191
7.4 Properties of the Integral ....................................................................................195
7.5 The Fundamental Theorem of Calculus...............................................................199
7.6 Lebesgue’s Criterion for Riemann Integrability .....................................................203
7.7 Epilogue ........................................................................................................................................ 210
8 Additional Topics 213
8.1 The Generalized Riemann Integral......................................................................213
8.2 Metric Spaces and the Baire Category Theorem ................................................222
8.3 Fourier Series ....................................................................................................228
8.4 A Construction of R From Q ..............................................................................243
Bibliography 251
Index 253
,Chapter 1
The Real Numbers
√
1.1 Discussion: The Irrationality of 2
Toward the end of his distinguished career, the renowned British mathematician
G.H. Hardy eloquently laid out a justification for a life of studying mathematics in A
Mathematician’s Apology, an essay first published in 1940. At the center of Hardy’s defense
is the thesis that mathematics is an aesthetic discipline. For Hardy, the applied mathematics of
engineers and economists held little charm. “Real mathematics,” as he referred to it, “must be
justified as art if it can be justified at all.”
To help make his point, Hardy includes two theorems from classical Greek mathematics,
which, in his opinion, possess an elusive kind of beauty that, although difficult to define, is
easy to recognize. The first of these results is Euclid’s proof that there are an infinite number
of prime numbers. The second result is the discovery, attributed to the school of Pythagoras
from around √ 500
B.C., that 2 is irrational. It is this second theorem that demands our attention.
(A course in number theory would focus on the first.) The argument uses only arithmetic, but
its depth and importance cannot be overstated. As Hardy says, “[It] is a ‘simple’ theorem,
simple both in idea and execution, but there is no doubt at all about [it being] of the highest
class. [It] is as fresh and significant as when it was discovered—two thousand years have not
written a wrinkle on [it].”
Theorem 1.1.1. There is no rational number whose square is 2.
Proof. A rational number is any number that can be expressed in the form p/q, where p and q
are integers. Thus, what the theorem asserts is that no matter 2
how p and q are chosen, it is never the case that (p/q) = 2. The line of attack
is indirect, using a type of argument referred to as a proof by contradiction. The idea is to
assume that there is a rational number whose square is 2 and then proceed along logical
lines until we reach a conclusion that is unacceptable. At this point, we will be forced to retrace
our steps and reject the erroneous
1
,
,2 Chapter 1. The Real Numbers
assumption that some rational number squared is equal to 2. In short, we will prove that the
theorem is true by demonstrating that it cannot be false.
And so assume, for contradiction, that there exist integers p and q satisfying
2
(1) = 2.
p
q
We may also assume that p and q have no common factor, because, if they had one, we could
simply cancel it out and rewrite the fraction in lowest terms. Now, equation (1) implies
(2) p2 = 2q2.
From this, we can see that the integer p2 is an even number (it is divisible by 2), and hence
p must be even as well because the square of an odd number is odd. This allows us to write
p = 2r, where r is also an integer. If we substitute 2r for p in equation (2), then a little algebra
yields the relationship
2r2 = q 2.
But now the absurdity is at hand. This last equation implies that q2 is even, and hence q
must also be even. Thus, we have shown that p and q are both even (i.e., divisible by 2)
when they were originally assumed to have no common factor. From this logical impasse, we can
only conclude that equation (1) cannot hold for any integers p and q, and thus the theorem is
proved.
A component of Hardy’s definition of beauty in a mathematical theorem is that the
result have lasting and serious implications for a network of other mathematical ideas. In this
case, the ideas under assault were the Greeks’ un- derstanding of the relationship between
geometric length and arithmetic number. Prior to the preceding discovery, it was an assumed
and commonly used fact that, given two line segments AB and CD, it would always be
possible to find a third line segment whose length divides evenly into the first two. In modern
terminology, this is equivalent to asserting that the length of CD is a rational multiple of the
length of AB. Looking at the diagonal of a unit square (Fig. 1.1), it now followed (using the
Pythagorean Theorem) that this was not always the case. Because the Pythagoreans implicitly
interpreted number to mean ra- tional number, they were forced to accept that number was a
strictly weaker notion than length.
Rather than abandoning arithmetic in favor of geometry (as the Greeks seem to have done),
our resolution to this limitation is to strengthen the concept of number by moving from the
rational numbers to a larger number system. From a modern point of view, this should seem like
a familiar and somewhat natural phenomenon. We begin with the natural numbers
N = {1, 2, 3, 4, 5, .. . }.
The influential German mathematician Leopold Kronecker (1823–1891) once asserted that
“The natural numbers are the work of God. All of the rest is
, √
1.1. Discussion: The Irrationality of 2 3
D
•
√ 2
1
C
• •
A 1 B
√
Figure 1.1: 2 exists as a geometric length.
the work of mankind.” Debating the validity of this claim is an interesting
conversation for another time. For the moment, it at least provides us with a
place to start. If we restrict our attention to the natural numbers N, then we can
perform addition perfectly well, but we must extend our system to the integers
Z = {.. . , −3, −2, −1, 0, 1, 2, 3, . . . }
if we want to have an additive identity (zero) and the additive inverses necessary to
define subtraction. The next issue is multiplication and division. The number 1 acts
as the multiplicative identity, but in order to define division we need to have
multiplicative
Q = inverses. Thus, we extend our system again to the rational numbers
p where p and q are integers with q = 0 .
fractions
all
Taken together, the propertiesq of Q discussed in the previous paragraph
essentially make up the definition of what is called a field. More formally stated, a
field is any set where addition and multiplication are well-defined operations that
are commutative, associative, and obey the familiar distributive property a(b + c) =
ab + ac. There must be an additive identity, and every element must have an additive
inverse. Finally, there must be a multiplicative identity, and multiplicative inverses
must exist for all nonzero elements of the field. Neither Z nor N is a field. The
finite set {0, 1, 2, 3, 4} is a field when addition and multiplication are computed
modulo 5. This is not immediately obvious but makes an interesting exercise
(Exercise 1.3.1).
The set Q also has a natural order defined on it. Given any two rational
numbers r and s, exactly one of the following is true:
r < s, r = s, or r > s.
This ordering is transitive in the sense that if r < s and s < t, then r < t, so we
are conveniently led to a mental picture of the rational numbers as being laid out
from left to right along a number line. Unlike Z, there are no intervals of empty
space. Given any two rational numbers r < s, the rational number
,4 Chapter 1. The Real Numbers
√
2
↓
1 ↑ 1.5 2
1.414
√
Figure 1.2: Approximating 2 with rational numbers.
(r+s)/2 sits halfway in between, implying that the rational numbers are densely nestled together.
With the field properties of Q allowing us to safely carry out the algebraic operations of
addition, subtraction, multiplication, and division, let’s remind ourselves just what it is that
Q is lacking. By Theorem 1.1.1, it is apparent that we cannot always take square roots.
The problem, however, is actually more fundamental than this. Using only rational
numbers, it is√possible to
approximate 2 quite well (Fig. 1.2). For instance, 1.4142 = 1.999396. By √
adding
but, more
even so, decimal places
we are now welltoaware
our approximation,
that there is awe“hole”
can get
in even closer to a value for 2,
√
the rational number line where 2 ought to be. Of course, there are quite a few other
√ √
holes—at 3 and 5, for example. Returning to the dilemma of the
ancient Greek mathematicians, if we want every length along the number line to correspond to an
actual number, then another extension to our number system is in order. Thus, to the chain N
⊆ Z ⊆ Q we append the real numbers R.
The question of how to actually construct R from Q is rather complicated
business. It is discussed in Section 1.3, and then again in more detail in Section
8.4. For the moment, it is not too inaccurate to say that R is obtained by filling in the
gaps in Q. Wherever there is a hole, a new irrational number is defined and placed into the
ordering that already exists on Q. The real numbers are then the union of these irrational
numbers together with the more familiar rational ones. What properties does the set of
irrational numbers have? How do the sets of rational and irrational numbers fit together? Is
there a kind of symmetry between the rationals and the irrationals, or is there some sense in
which we can argue that one type of real number is more common than the other? The one
method we have seen so far for generating examples of irrational √
numbers is through square roots. Not too surprisingly, other roots such as 3 2
√
5
or 3 are most often irrational. Can all irrational numbers be expressed as
algebraic combinations of nth roots and rational numbers, or are there still other irrational
numbers beyond those of this form?
1.2 Some Preliminaries
The vocabulary necessary for the ensuing development comes from set theory and the theory of
functions. This should be familiar territory, but a brief review
,1.2. Some Preliminaries 5
of the terminology is probably a good idea, if only to establish some agreed-upon notation.
Sets
Intuitively speaking, a set is any collection of objects. These objects are referred to as the
elements of the set. For our purposes, the sets in question will most often be sets of real
numbers, although we will also encounter sets of functions and, on a few rare occasions, sets
whose elements are other sets.
Given a set A, we write x ∈ A if x (whatever it may be) is an element of A. If x is not an
element of A, then we write x ∈ / A. Given two sets A and B, the union is written A ∪ B and is
defined by asserting that
x ∈A∪B provided that x ∈A or x ∈B (or potentially both).
The intersection A ∩B is the set defined by the rule
x ∈A∩B provided x ∈A and x ∈B.
Example 1.2.1. (i) There are many acceptable ways to assert the contents of a set. In the
previous section, the set of natural numbers was defined by listing the elements: N = {1, 2, 3, . . .
}.
(ii) Sets can also be described in words. For instance, we can define the set
E to be the collection of even natural numbers.
(iii) Sometimes it is more efficient to provide a kind of rule or algorithm for determining the
elements of a set. As an example, let
S = {r ∈Q : r2 < 2}.
Read aloud, the definition of S says, “Let S be the set of all rational numbers whose squares
are less than 2.” It follows that 1 ∈S, 4/3 ∈S, but 3/2 ∈ / S because 9/4 ≥2.
Using the previously defined sets to illustrate the operations of intersection and union, we
observe that
N ∪E = N, N ∩E = E, N ∩S = {1}, and E ∩S = ∅.
The set ∅is called the empty set and is understood to be the set that contains no elements. An
equivalent statement would be to say that E and S are disjoint.
A word about the equality of two sets is in order (since we have just used the notion).
The inclusion relationship A ⊆ B or B ⊇ A is used to indicate that every element of A is also an
element of B. In this case, we say A is a subset of B, or B contains A. To assert that A = B
means that A ⊆B and B ⊆A. Put another way, A and B have exactly the same elements.
Quite frequently in the upcoming chapters, we will want to apply the union and intersection
operations to infinite collections of sets.
, 6 Chapter 1. The Real Numbers
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SOLUTIONS MANUAL
,Contents
Preface v
1 The Real Numbers 1
√
1.1 Discussion: The Irrationality of 2 ......................................................................... 1
1.2 Some Preliminaries ............................................................................................... 4
1.3 The Axiom of Completeness ............................................................................... 13
1.4 Consequences of Completeness ......................................................................... 18
1.5 Cantor’s Theorem................................................................................................ 29
1.6 Epilogue ............................................................................................................. 33
2 Sequences and Series 35
2.1 Discussion: Rearrangements of Infinite Series...................................................... 35
2.2 The Limit of a Sequence ..................................................................................... 38
2.3 The Algebraic and Order Limit Theorems............................................................. 44
2.4 The Monotone Convergence Theorem and a First Look at
Infinite Series ...................................................................................................... 50
2.5 Subsequences and the Bolzano–Weierstrass Theorem ......................................... 55
2.6 The Cauchy Criterion........................................................................................... 58
2.7 Properties of Infinite Series ................................................................................. 62
2.8 Double Summations and Products of Infinite Series .............................................. 69
2.9 Epilogue ............................................................................................................. 73
3 Basic Topology of R 75
3.1 Discussion: The Cantor Set................................................................................. 75
3.2 Open and Closed Sets ........................................................................................ 78
3.3 Compact Sets . .................................................................................................. 84
3.4 Perfect Sets and Connected Sets ........................................................................ 89
3.5 Baire’s Theorem.................................................................................................. 94
3.6 Epilogue ............................................................................................................. 96
4 Functional Limits and Continuity 99
4.1 Discussion: Examples of Dirichlet and Thomae .................................................... 99
4.2 Functional Limits................................................................................................ 103
4.3 Combinations of Continuous Functions ...................................................................... 109
xi
,xii Contents
4.4 Continuous Functions on Compact Sets ..............................................................114
4.5 The Intermediate Value Theorem........................................................................120
4.6 Sets of Discontinuity.....................................................................................................125
4.7 Epilogue ........................................................................................................................................ 127
5 The Derivative 129
5.1 Discussion: Are Derivatives Continuous?.....................................................................129
5.2 Derivatives and the Intermediate Value Property ..................................................131
5.3 The Mean Value Theorem..................................................................................137
5.4 A Continuous Nowhere-Differentiable Function ...................................................................144
5.5 Epilogue ........................................................................................................................................ 148
6 Sequences and Series of Functions 151
6.1 Discussion: Branching Processes .......................................................................151
6.2 Uniform Convergence of a Sequence of Functions .....................................................154
6.3 Uniform Convergence and Differentiation ....................................................................164
6.4 Series of Functions.......................................................................................................167
6.5 Power Series .................................................................................................................169
6.6 Taylor Series .....................................................................................................176
6.7 Epilogue ........................................................................................................................................ 181
7 The Riemann Integral 183
7.1 Discussion: How Should Integration be Defined? ..................................................183
7.2 The Definition of the Riemann Integral................................................................186
7.3 Integrating Functions with Discontinuities .............................................................191
7.4 Properties of the Integral ....................................................................................195
7.5 The Fundamental Theorem of Calculus...............................................................199
7.6 Lebesgue’s Criterion for Riemann Integrability .....................................................203
7.7 Epilogue ........................................................................................................................................ 210
8 Additional Topics 213
8.1 The Generalized Riemann Integral......................................................................213
8.2 Metric Spaces and the Baire Category Theorem ................................................222
8.3 Fourier Series ....................................................................................................228
8.4 A Construction of R From Q ..............................................................................243
Bibliography 251
Index 253
,Chapter 1
The Real Numbers
√
1.1 Discussion: The Irrationality of 2
Toward the end of his distinguished career, the renowned British mathematician
G.H. Hardy eloquently laid out a justification for a life of studying mathematics in A
Mathematician’s Apology, an essay first published in 1940. At the center of Hardy’s defense
is the thesis that mathematics is an aesthetic discipline. For Hardy, the applied mathematics of
engineers and economists held little charm. “Real mathematics,” as he referred to it, “must be
justified as art if it can be justified at all.”
To help make his point, Hardy includes two theorems from classical Greek mathematics,
which, in his opinion, possess an elusive kind of beauty that, although difficult to define, is
easy to recognize. The first of these results is Euclid’s proof that there are an infinite number
of prime numbers. The second result is the discovery, attributed to the school of Pythagoras
from around √ 500
B.C., that 2 is irrational. It is this second theorem that demands our attention.
(A course in number theory would focus on the first.) The argument uses only arithmetic, but
its depth and importance cannot be overstated. As Hardy says, “[It] is a ‘simple’ theorem,
simple both in idea and execution, but there is no doubt at all about [it being] of the highest
class. [It] is as fresh and significant as when it was discovered—two thousand years have not
written a wrinkle on [it].”
Theorem 1.1.1. There is no rational number whose square is 2.
Proof. A rational number is any number that can be expressed in the form p/q, where p and q
are integers. Thus, what the theorem asserts is that no matter 2
how p and q are chosen, it is never the case that (p/q) = 2. The line of attack
is indirect, using a type of argument referred to as a proof by contradiction. The idea is to
assume that there is a rational number whose square is 2 and then proceed along logical
lines until we reach a conclusion that is unacceptable. At this point, we will be forced to retrace
our steps and reject the erroneous
1
,
,2 Chapter 1. The Real Numbers
assumption that some rational number squared is equal to 2. In short, we will prove that the
theorem is true by demonstrating that it cannot be false.
And so assume, for contradiction, that there exist integers p and q satisfying
2
(1) = 2.
p
q
We may also assume that p and q have no common factor, because, if they had one, we could
simply cancel it out and rewrite the fraction in lowest terms. Now, equation (1) implies
(2) p2 = 2q2.
From this, we can see that the integer p2 is an even number (it is divisible by 2), and hence
p must be even as well because the square of an odd number is odd. This allows us to write
p = 2r, where r is also an integer. If we substitute 2r for p in equation (2), then a little algebra
yields the relationship
2r2 = q 2.
But now the absurdity is at hand. This last equation implies that q2 is even, and hence q
must also be even. Thus, we have shown that p and q are both even (i.e., divisible by 2)
when they were originally assumed to have no common factor. From this logical impasse, we can
only conclude that equation (1) cannot hold for any integers p and q, and thus the theorem is
proved.
A component of Hardy’s definition of beauty in a mathematical theorem is that the
result have lasting and serious implications for a network of other mathematical ideas. In this
case, the ideas under assault were the Greeks’ un- derstanding of the relationship between
geometric length and arithmetic number. Prior to the preceding discovery, it was an assumed
and commonly used fact that, given two line segments AB and CD, it would always be
possible to find a third line segment whose length divides evenly into the first two. In modern
terminology, this is equivalent to asserting that the length of CD is a rational multiple of the
length of AB. Looking at the diagonal of a unit square (Fig. 1.1), it now followed (using the
Pythagorean Theorem) that this was not always the case. Because the Pythagoreans implicitly
interpreted number to mean ra- tional number, they were forced to accept that number was a
strictly weaker notion than length.
Rather than abandoning arithmetic in favor of geometry (as the Greeks seem to have done),
our resolution to this limitation is to strengthen the concept of number by moving from the
rational numbers to a larger number system. From a modern point of view, this should seem like
a familiar and somewhat natural phenomenon. We begin with the natural numbers
N = {1, 2, 3, 4, 5, .. . }.
The influential German mathematician Leopold Kronecker (1823–1891) once asserted that
“The natural numbers are the work of God. All of the rest is
, √
1.1. Discussion: The Irrationality of 2 3
D
•
√ 2
1
C
• •
A 1 B
√
Figure 1.1: 2 exists as a geometric length.
the work of mankind.” Debating the validity of this claim is an interesting
conversation for another time. For the moment, it at least provides us with a
place to start. If we restrict our attention to the natural numbers N, then we can
perform addition perfectly well, but we must extend our system to the integers
Z = {.. . , −3, −2, −1, 0, 1, 2, 3, . . . }
if we want to have an additive identity (zero) and the additive inverses necessary to
define subtraction. The next issue is multiplication and division. The number 1 acts
as the multiplicative identity, but in order to define division we need to have
multiplicative
Q = inverses. Thus, we extend our system again to the rational numbers
p where p and q are integers with q = 0 .
fractions
all
Taken together, the propertiesq of Q discussed in the previous paragraph
essentially make up the definition of what is called a field. More formally stated, a
field is any set where addition and multiplication are well-defined operations that
are commutative, associative, and obey the familiar distributive property a(b + c) =
ab + ac. There must be an additive identity, and every element must have an additive
inverse. Finally, there must be a multiplicative identity, and multiplicative inverses
must exist for all nonzero elements of the field. Neither Z nor N is a field. The
finite set {0, 1, 2, 3, 4} is a field when addition and multiplication are computed
modulo 5. This is not immediately obvious but makes an interesting exercise
(Exercise 1.3.1).
The set Q also has a natural order defined on it. Given any two rational
numbers r and s, exactly one of the following is true:
r < s, r = s, or r > s.
This ordering is transitive in the sense that if r < s and s < t, then r < t, so we
are conveniently led to a mental picture of the rational numbers as being laid out
from left to right along a number line. Unlike Z, there are no intervals of empty
space. Given any two rational numbers r < s, the rational number
,4 Chapter 1. The Real Numbers
√
2
↓
1 ↑ 1.5 2
1.414
√
Figure 1.2: Approximating 2 with rational numbers.
(r+s)/2 sits halfway in between, implying that the rational numbers are densely nestled together.
With the field properties of Q allowing us to safely carry out the algebraic operations of
addition, subtraction, multiplication, and division, let’s remind ourselves just what it is that
Q is lacking. By Theorem 1.1.1, it is apparent that we cannot always take square roots.
The problem, however, is actually more fundamental than this. Using only rational
numbers, it is√possible to
approximate 2 quite well (Fig. 1.2). For instance, 1.4142 = 1.999396. By √
adding
but, more
even so, decimal places
we are now welltoaware
our approximation,
that there is awe“hole”
can get
in even closer to a value for 2,
√
the rational number line where 2 ought to be. Of course, there are quite a few other
√ √
holes—at 3 and 5, for example. Returning to the dilemma of the
ancient Greek mathematicians, if we want every length along the number line to correspond to an
actual number, then another extension to our number system is in order. Thus, to the chain N
⊆ Z ⊆ Q we append the real numbers R.
The question of how to actually construct R from Q is rather complicated
business. It is discussed in Section 1.3, and then again in more detail in Section
8.4. For the moment, it is not too inaccurate to say that R is obtained by filling in the
gaps in Q. Wherever there is a hole, a new irrational number is defined and placed into the
ordering that already exists on Q. The real numbers are then the union of these irrational
numbers together with the more familiar rational ones. What properties does the set of
irrational numbers have? How do the sets of rational and irrational numbers fit together? Is
there a kind of symmetry between the rationals and the irrationals, or is there some sense in
which we can argue that one type of real number is more common than the other? The one
method we have seen so far for generating examples of irrational √
numbers is through square roots. Not too surprisingly, other roots such as 3 2
√
5
or 3 are most often irrational. Can all irrational numbers be expressed as
algebraic combinations of nth roots and rational numbers, or are there still other irrational
numbers beyond those of this form?
1.2 Some Preliminaries
The vocabulary necessary for the ensuing development comes from set theory and the theory of
functions. This should be familiar territory, but a brief review
,1.2. Some Preliminaries 5
of the terminology is probably a good idea, if only to establish some agreed-upon notation.
Sets
Intuitively speaking, a set is any collection of objects. These objects are referred to as the
elements of the set. For our purposes, the sets in question will most often be sets of real
numbers, although we will also encounter sets of functions and, on a few rare occasions, sets
whose elements are other sets.
Given a set A, we write x ∈ A if x (whatever it may be) is an element of A. If x is not an
element of A, then we write x ∈ / A. Given two sets A and B, the union is written A ∪ B and is
defined by asserting that
x ∈A∪B provided that x ∈A or x ∈B (or potentially both).
The intersection A ∩B is the set defined by the rule
x ∈A∩B provided x ∈A and x ∈B.
Example 1.2.1. (i) There are many acceptable ways to assert the contents of a set. In the
previous section, the set of natural numbers was defined by listing the elements: N = {1, 2, 3, . . .
}.
(ii) Sets can also be described in words. For instance, we can define the set
E to be the collection of even natural numbers.
(iii) Sometimes it is more efficient to provide a kind of rule or algorithm for determining the
elements of a set. As an example, let
S = {r ∈Q : r2 < 2}.
Read aloud, the definition of S says, “Let S be the set of all rational numbers whose squares
are less than 2.” It follows that 1 ∈S, 4/3 ∈S, but 3/2 ∈ / S because 9/4 ≥2.
Using the previously defined sets to illustrate the operations of intersection and union, we
observe that
N ∪E = N, N ∩E = E, N ∩S = {1}, and E ∩S = ∅.
The set ∅is called the empty set and is understood to be the set that contains no elements. An
equivalent statement would be to say that E and S are disjoint.
A word about the equality of two sets is in order (since we have just used the notion).
The inclusion relationship A ⊆ B or B ⊇ A is used to indicate that every element of A is also an
element of B. In this case, we say A is a subset of B, or B contains A. To assert that A = B
means that A ⊆B and B ⊆A. Put another way, A and B have exactly the same elements.
Quite frequently in the upcoming chapters, we will want to apply the union and intersection
operations to infinite collections of sets.
, 6 Chapter 1. The Real Numbers
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