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Appendix Axioms of Number Theory
The axiomatic approach
You can’t prove anything in mathematics without making some initial assumptions.
The ancient Greeks knew this well. Euclid began the Elements with five basic assumptions, called postulates (a term synonymous with axioms), which he used to derive all
of the theorems of plane geometry. Today, mathematicians follow the model provided
by Euclid’s Elements, starting with a list of basic axioms and deriving all results from
these. In this appendix, we will state our axioms for number theory. All of the number
theory in this book, and more, can be built using these simple assumptions.
But how shall we choose our axioms? While it might not be clear exactly how to
do this, we are guided by the following lofty goals:
• Each axiom should be a simple, intuitively true fact about the integers.
• Taken together, the axioms should characterize the integers uniquely.
In other words, the axioms should be self-evident truths that collectively tell the
whole story—any true statement about the integers should be a logical consequence
of the axioms. The axioms provide the organic mathematical flour from which we can
bake all manner of delicious number-theoretic pastries, leavened with the yeast of our
imagination.
While the second goal listed above, to characterize the integers uniquely,
might seem a bit overwhelming, it is easy to make progress on the first goal—just
start making a list of simple, intuitively true statements about Z. Of course, the
integers form a set, but what really makes Z come alive are the operations we can
perform: We can add and multiply integers, and we can compare their sizes. That
is, the set Z is endowed with two binary operations and , and a relation .
The story of Z is the story of these three operators , , and . The axioms we
give will therefore focus on the algebraic properties of Z (properties of and )
and the order properties of Z (properties of ). One very special order property
of Z, the Well-Ordering Principle, is special enough to deserve an axiom of
its own. Looking ahead, then, our axioms will take the following form:
Axiom 1. Algebraic Properties of Z (Properties of ⴙ and ⴢ)
Axiom 2. Order Properties of Z (Properties of ⬍)
Axiom 3. The Well-Ordering Principle
The road not taken
One approach to a rigorous development of number theory is to give a formal definition
or construction of the set of integers, the operations addition and multiplication, and
the order relation . We have not chosen this approach. Nowhere in this appendix (and
nowhere in this book) will we explicitly define the term integer, the and operations,
A-1 APPENDIX Axioms of Number Theory
or the relation . Instead, the integers are characterized implicitly as a set together with
operations and and a relation that satisfy our three axioms. These three axioms
together capture all of the essential properties of the integers. The situation is much the same
as in Euclidean geometry. Points, lines, and planes are normally taken as undefined terms
that gain a meaning only implicitly through the postulates that describe their properties.
A.1 What Is a Number System?
Rings and fields
To craft our first axiom of number theory, we wish to identify the fundamental algebraic
properties of the operations and on the set Z. Properties such as associativity,
commutativity, and distributivity come quickly to mind. We will include these properties
and others in our first axiom for number theory. It turns out that Q , R, and C also share
many of the same algebraic properties enjoyed by Z. In our journey through number theory,
we will meet other, more exotic number systems (such as Zn in Chapter 8 and Z[i ] in
Chapter 13) that also share these properties. Because of this commonality, mathematicians
decided that it would be useful to have a special name, ring, for any set with operations
that satisfy these properties. Our first axiom about the integers will be that Z is a ring.
To begin with, let’s be more specific about what we mean by an operation. The most
familiar operations, such as addition and multiplication, take two numbers as input and
combine them to produce a result that is a single number. Such operations are called
binary operations. We formally define this as follows.
DEFINITION A.1.1
Let S be a set. A binary operation * on S is a function that assigns to each pair of elements,
x and y in S, an element x * y that is also in S.
Notice that in order for * to be a valid binary operation on S, the set S must be closed
under the operation *. That is, for every pair of elements x and y in S, x * y must also be an
element of S. For example, is a valid binary operation on R because if we take any pair
of real numbers x and y, their sum x y is also a real number. We express this idea by
saying that the real numbers are closed under addition. Similarly, the real numbers are
closed under subtraction, so – is a binary operation on R.
EXAMPLE 1
Subtraction is not a binary operation on N because when we subtract one natural
number from another, we do not necessarily obtain a natural number. For instance,
3 5 2, which is not a natural number. Thus, N is not closed under subtraction.
A.1 What Is a Number System? A-2
EXAMPLE 2
Define the operation * on the real numbers by
x * y x 2 y 3 5.
For instance,
3 * 4 32 43 5
68.
Then * is a binary operation because to every pair of real numbers x and y, it assigns
the real number x2 y3 5.
Now that we have defined a binary operation, we are ready to define a ring.1
DEFINITION A.1.2
A ring is a set R together with two binary operations, and , that satisfy the following
properties.
Properties of Addition
A1. Associativity. For every x, y, and z in R, ( x y ) z x ( y z ).
A2. Commutativity. For every x and y in R, x y y x.
A3. Identity. R contains an additive identity, 0, such that for every x in R, x 0 x.
A4. Additive Inverses. For every x in R, there is an additive inverse, (x ), in R such
that x (x ) 0.
Properties of Multiplication
M1. Associativity. For every x, y, and z in R, ( x y ) z x ( y z ).
M2. Commutativity. For every x and y in R, x y y x.
M3. Identity. R contains a multiplicative identity, 1, such that for every x in R,
x 1 x.
Properties Relating Addition and Multiplication
R1. Distributivity. For every x, y, and z in R, x ( y z ) ( x y ) ( x z ).
1
Often the definition of ring does not include our rules M2 and M3.
We have actually defined what most mathematicians would call a
commutative ring with unity.
A-3 APPENDIX Axioms of Number Theory
The sets Z, Q , R, and C together with their usual operations of addition and
multiplication are all examples of rings. We will not prove that these sets are rings.
In fact, one of our basic axioms (Axiom 1) about the integers will be that Z is a ring.
However, the set of natural numbers, N, is not a ring. (Why not?)
Notice that the definition of a ring requires every element to have an additive
inverse but does not require every element to have a multiplicative inverse. Given
an element x in a ring R, there may or may not exist an element x1 such that
x x1 1. For example, in Z the only numbers that have multiplicative inverses are 1
and 1. In contrast, in Q every nonzero element has a multiplicative inverse. There is a
special name for those rings in which every nonzero element has an inverse.
DEFINITION A.1.3
Let R be a ring with at least one nonzero element. We say R is a field if the following
additional property holds:
M4. Multiplicative Inverses. For every nonzero element x in R, there is a
multiplicative inverse x1 in R such that x x1 1.
EXAMPLE 3
Q , R, and C are all fields, but Z is not a field.
What is a number system? It is easy to take a set S and concoct a couple of strange
binary operations, such as the operation in Example 2. But should S together with
these operations be called a number system? In most cases, S will not have much interesting structure. If we can show that S satisfies all (or even most) of the ring properties,
however, then we are certainly justified in calling it a number system because then we
can do arithmetic as usual in S.
Our first axiom
Our first axiom of number theory, the assumption that Z is a ring, means that we can
do arithmetic as usual in Z.
AXIOM 1 ALGEBRAIC PROPERTIES OF THE INTEGERS
Z is a ring.
EXERCISES A.1
Reasoning and Proofs
1. Define the operation * on the real numbers by x * y | x y |. Is * a binary
operation? Is * associative? Is * commutative?
A.1 What Is a Number System? A-4
2. a. Consider the set of even integers with the usual addition and multiplication
operations, and . Do they form a ring? If not, list those ring properties
that are not satisfied.
b. Consider the set of odd integers with the usual and . Do they form a ring?
If not, list those ring properties that are not satisfied.
3. Let i denote the imaginary number i √1, and let S {x i | x 僆 R} with
the usual addition and multiplication operations, and . Is S a ring? If not, list
those ring properties that are not satisfied.
4. Let S denote the set of 2 2 invertible matrices with real entries. Let and denote matrix addition and multiplication.
a. Is S a ring? If not, list those ring properties that are not satisfied.
b. Is S a field?
5. Let S denote the set of functions from the real numbers to the real numbers.
Let and denote addition and multiplication of functions.
a. Is S a ring? If not, list those ring properties that are not satisfied.
b. Is S a field?
6. Let R be any ring. We are assured of the existence of the multiplicative
identity, 1, in the ring and that the ring will be closed under addition. Therefore,
the following elements will all be in R:
1, 1 1, 1 1 1, 1 1 1 1, . . . .
Can we conclude that R must contain infinitely many elements? Why or
why not?
7. Let U {apple, banana, cranberry} be a set, and define two operations on the set
U by the following tables:
a. Explain why
a
b
c
a
c
a
b
b
a
b
c
b
c
and
a
b
c
a
a
b
c
c
b
b
b
b
a
c
c
b
a
are binary operations.
b. Is U a ring? Explain why or why not.
c. Is U a field? Explain why or why not.
A-5 APPENDIX Axioms of Number Theory
8. Let R {T, F }. If you wish, you may think of these as “True” and “False.” The
logical operations XOR (exclusive or) and AND are defined by the following tables:
XOR
T
F
AND
T
F
T
F
T
T
T
F
F
T
F
F
F
F
Show that R with the operations XOR and AND is a ring. Is R a field?
A.2 Order Properties of the Integers
Axiom 1 assures us that we can add and multiply integers. But we also know intuitively
that the integers are ordered: We can compare any two integers using , the “less than”
relation. We list our assumptions about in the following axiom.
AXIOM 2 ORDER PROPERTIES OF THE INTEGERS
(i )
Transitive Property.
(ii ) Trichotomy Property.
For every a, b, and c in Z, if a b and b c, then a c.
For every a and b in Z, exactly one of the following holds:
a b, a b, or b a.
(iii ) Additive Property.
For every a, b, and c in Z, if a b, then a c b c.
(iv) Multiplicative Property.
For every a, b, and c in Z, if a b and 0 c, then ac bc.
(v) Order of Identities.
0 1.
We will use a b as a shorthand to mean that either a b or a b. Notice that
Axiom 2 gives us the relation but doesn’t mention the relation . This is easy to fix:
We just define a b to mean b a.
A particularly important subset of the integers is the positive integers, a notion that
we can define using our new relation, .
DEFINITION A.2.1
Let n 僆 Z. Then n is said to be positive if 0 n. In this case, n is called a natural number
or a positive integer. The set of all natural numbers is denoted by N.
There are a couple of related sets of integers that we will also define.
DEFINITION A.2.2
Let n 僆 Z. If n 0, then n is said to be negative. If 0 n, then n is said to be nonnegative.
A.2 Order Properties of the Integers A-6
The Well-Ordering Principle
So far, we have assumed that Z is a ring (Axiom 1) that satisfies the order properties
of Axiom 2. However, Z is not the only such ring. The rings Q and R also satisfy these
order properties. One property that distinguishes Z from these other rings Q and R
is that for any integer n, there is a next integer n 1 that comes immediately after
it, with no other integers in between. For example, if we are given the number 37, we
know that 38 comes just after it. However, there is no “next” rational or real number
that immediately follows 37. This property of the integers does not follow from the
ring properties and the order properties alone. To prove this property of the integers
(and many others), we will also need a third axiom, the Well-Ordering Principle, which
we now state. Amazingly, the Well-Ordering Principle together with Axioms 1 and 2
fully characterize the structure of the integers.
AXIOM 3 THE WELL-ORDERING PRINCIPLE
Every nonempty set of positive integers has a smallest element.
If we replace “integers” in the statement of the Well-Ordering Principle by “rational
numbers” or “real numbers,” then the Well-Ordering Principle would not be true. For
example, the set of positive rational numbers does not have a smallest element, as we
proved in Section 1.3, Example 2.
The grand assumptions
We summarize all of the assumptions that we make about the integers in the following.
A.2.3 AXIOMS FOR Z
Axiom 1. The integers are a ring.
Axiom 2. The integers satisfy the properties (i )–(v) of ⬍.
Axiom 3. The Well-Ordering Principle is true.
All of our results in number theory will be based on these three axioms.
Observe that since Z is a ring, the numbers 0 and 1 are defined to be the additive
identity and the multiplicative identity, respectively. But nowhere in our axioms do we
mention 2 or 3. This is easy to fix. We can simply define 2 to be the number 1 1 and
define 3 as 1 1 1. Similarly, we will allow ourselves to write the symbols 4, 5, 6, . . . ,
and it is clear what we mean.
EXERCISES A.2
Reasoning and Proofs
1. Using the list of properties of for the integers given in Axiom 2, write a list of
properties of for the integers.
A-7 APPENDIX Axioms of Number Theory
2. Using the list of properties of for the integers given in Axiom 2, write a list of
properties of for the integers.
3. In the statement of the Well-Ordering Principle (Axiom 3), if we were to replace
positive integers by negative integers, would the new statement be true?
4. In the statement of the Well-Ordering Principle (Axiom 3), we use the term
smallest element. Give a formal definition of what it means for x to be the smallest
element of a set S with an order relation .
5. Let S be a nonempty subset of the natural numbers. Prove that S has a unique
smallest element.
6. Define an order relation on the complex numbers as follows. If a bi 僆 C
and c di 僆 C, we define a bi c di if
(i ) a c,
or (ii ) a c and b d.
Does this relation on C satisfy the properties listed in Axiom 2? If not,
demonstrate which of the properties are not satisfied.
7. Is it possible to define an order relation on the set C of complex numbers that
satisfies the properties of Axiom 2? Prove or disprove.
A.3 Building Results from our Axioms
Now that we have explicitly listed all of the assumptions that we will make about the
integers, we may begin discovering and proving theorems using our axioms. Once
we have proved theorems, we may use these theorems in turn to prove other theorems. Our hope is that eventually the theorems we prove will exhibit one or more
of the intangible qualities of beauty, elegance, depth, or even practical usefulness.
To professional mathematicians, this process of building and discovery is known
as doing mathematics. But before we get carried away, let’s remind ourselves that at
this point in our nascent journey, we have only the axioms at our disposal; we must
not be tempted to use other assumptions, even ones that seem patently obvious, in
our proofs.
In this section, we will use our axioms to prove some basic facts about the integers.
In the past, you have probably taken many of these facts for granted, such as the fact
that when any integer is multiplied by 0, the result is 0. You may not have ever realized that it’s possible to prove this, but it is! This section culminates with a proof of
another basic fact about the integers: there is no integer that is both even and odd.
One of the great things about the axiomatic approach is that once we have stated our
axioms, we no longer have to take anything else on faith—we can prove everything
from the axioms.
A.3 Building Results from our Axioms A-8
Proving things about rings
We begin be proving a few facts that are valid in an arbitrary ring. The general study of
rings and fields belongs to the domain of abstract algebra rather than number theory.
However, because we assumed that the integers are a ring (Axiom 1), every theorem we
prove about rings will also hold for the integers.
LEMMA A.3.1
ADDITIVE CANCELLATION LEMMA
Let R be a ring. Then for every r, s, and z in R, if r z s z, then r s.
PROOF We prove this directly. Let r, s, z 僆 R such that
r z s z.
By the definition of a ring, z has an additive inverse, z, in R. Adding z to both sides
of this equation, we get
( r z ) (z ) ( s z ) (z ).
Using the associativity of , we may regroup the terms in our equation:
r [ z (z )] s [ z (z )].
Since z (z ) 0, we have
r 0 s 0.
Since 0 is the additive identity, it follows that
r s.
A key step in this proof involved adding the same quantity to both sides of an equation. Is that step permitted by our axioms? Yes, it turns out that this follows from the
fact that , as a binary operation, is a function. One familiar property of functions that
you’re used to is
x y ⇒ f (x) f ( y).
(1)
That is, the value returned by a function depends only on the input. In other words,
equal inputs produce equal outputs. (Don’t confuse statement (1) with its converse,
f (x) f ( y) ⇒ x y, which holds only for one-to-one functions.)
Since we have assumed that is a function, statement (1) holds for . Because is a function with two inputs, and the function name is written in between its inputs
rather than preceding them, statement (1) for the function takes a slightly different
form:
a b, c d ⇒ a c b d.
A-9 APPENDIX Axioms of Number Theory
But this says exactly that any two equations can be added to produce another equation!
A special case of this is that we may add the same quantity to both sides an equation.
A familiar basic fact about the integers is that any number times zero is zero. This is
not one of our axioms, but we can prove it!
LEMMA A.3.2
Let R be a ring. Then for every x in R, x 0 0 x 0.
PROOF Let R be a ring, and let x 僆 R.
Since 0 is the additive identity,
0 0 0.
Multiply both sides of this equation by x to get
x (0 0) x 0.
Now applying distributivity yields
( x 0 ) ( x 0 ) x 0.
Add the additive identity, 0, to the right side of this equation to get
( x 0 ) ( x 0 ) ( x 0 ) 0.
Then use the commutativity of to obtain
( x 0 ) ( x 0 ) 0 ( x 0 ).
Now the Additive Cancellation Lemma (A.3.1) allows us to cancel ( x 0 ) from both
sides of our equation. This gives
x 0 0.
By the commutativity of , we also have
0 x 0.
The next lemma asserts the uniqueness of the additive identity in a ring.
LEMMA A.3.3
Let R be a ring. Then R has only one additive identity element.
PROOF Let R be a ring, and suppose that both y and z are additive identity elements of R.
[To show: y ⴝ z.]
A.3 Building Results from our Axioms A-10
Since y is an additive identity,
z y z.
(2)
y z y.
(3)
Since z is an additive identity,
By commutativity,
y z z y.
It now follows from equations (2) and (3) that y z. Thus, there is only one additive
identity.
The next lemma asserts that every element of a ring has a unique additive inverse.
LEMMA A.3.4
Let R be a ring, and let x be in R. Then x has only one additive inverse.
PROOF Let x be an element of a ring R, and suppose that the elements a and b in R
are both additive inverses of x. [To show: a ⴝ b.]
Since each of a and b is an additive inverse of x,
x a 0 and x b 0.
Thus,
x a x b.
It now follows from the commutativity of and the Additive Cancellation Lemma
(A.3.1) that a b.
First consequences of the order properties
Having derived a few properties that hold in any ring, we now turn our attention to
order properties—that is, properties of the relation .
Axiom 2 guarantees that certain familiar order properties hold, but there are many
other familiar order properties that we would like have. For example, Axiom 2 part (iv),
the Multiplicative Property of , tells us that multiplying an inequality by a positive
number preserves the inequality. But no axiom tells us what happens when we multiply
an inequality by a negative number. You learned in algebra class that when you multiply
an inequality by a negative number, you flip the direction of the inequality. We no longer
have to take this statement on faith—we can actually prove it from the axioms!
A-11 APPENDIX Axioms of Number Theory
LEMMA A.3.5
Suppose that x, y, and z are integers such that x y and z 0. Then yz xz.
PROOF We begin with our hypothesis that x y, then add x to both sides of this
inequality to get
x (x ) y (x ).
It follows that
0 y (x ).
We now use Axiom 2 part (iv), the Multiplicative Property of , to multiply the
inequality z 0 by the positive integer y (x). This gives us the inequality
z [ y (x )] 0 [ y (x )].
Now by the distributive property and Lemma A.3.2, this becomes
zy z(x ) 0
(4)
Adding zx to both sides of the equation and simplifying (this uses several ring properties and Lemma A.3.2), we get
yz xz.
(5)
This completes the proof.
As we continue to prove results about the integers, we will often use the ring
properties freely, without explicitly appealing to every ring property that we use. In
Exercise 26, you are asked to fill in the steps that allowed us to go from inequality (4)
to inequality (5) in the preceding proof.
LEMMA A.3.6
Let n be an integer. Then n 2 0.
PROOF Let n be an integer. By the Trichotomy Property, n 0, n 0, or n 0.
We consider these three cases separately.
Case 1 n 0.
We can multiply both sides of this inequality by n (using the Multiplicative Property
of ) to get
n n 0 n.
It now follows from Lemma A.3.2 that n2 0. Hence, n2 0.
A.3 Building Results from our Axioms A-12
Case 2 n 0.
In this case,
n2 n n 0 0 0.
Since n2 0, we conclude that n2 0.
Case 3 n 0.
Using Lemma A.3.5, we can multiply both sides of this inequality by n to get
n n 0 n,
then apply Lemma A.3.2 to get
n2 0.
Hence, n2 0.
In all three cases, we have shown that n2 0.
Consequences of the Well-Ordering Principle
So far we have derived a few consequences of Axiom 1 (the ring properties)
and Axiom 2 (the order properties). But up to this point, we have not used the
Well-Ordering Principle (Axiom 3).
We would like to use our axioms to prove that after each integer n, there is a next
integer, n 1, that immediately follows it with no other integers in between. Recall
that this was part of our motivation for introducing the Well-Ordering Principle as an
axiom. Before we can prove this property of the integers, though, we first need to know
that there are no integers between 0 and 1.
LEMMA A.3.7
There is no integer x such that 0 x 1.
PROOF (By contradiction.)
Assumption: Suppose there is an integer x such that 0 x 1.
Consider the set S consisting of all positive integers less than 1:
S { x 僆 N | x 1 }.
It follows from our Assumption that S is nonempty. Thus, by the Well-Ordering
Principle (Axiom 3), S has a smallest element, m. Since m is in S, we know that
0 m 1.
(6)
Now we can multiply this inequality by m (using the Multiplicative Property of ) to get
0 m2 m.
A-13 APPENDIX Axioms of Number Theory
(7)
Since m is an integer and Z is a ring, m2 m m is also an integer. It then follows
from inequality (7) that m2 is a positive integer that is smaller than m. By the Transitive
Property of and inequality (6), m2 1. Thus, m2 僆 S. Hence, m2 is an element of S
that is smaller than m. This contradicts that m is the smallest element of S. ⇒⇐
Since we have reached a contradiction, our Assumption must be false. We conclude
that there is no integer between 0 and 1.
COROLLARY A.3.8
Let n 僆 N. Then n 1.
PROOF You will prove this in Exercise 18.
We now use Lemma A.3.7 to prove that if n is an integer, there is no integer
between n and n 1.
LEMMA A.3.9
Let n be an integer. Then there is no integer x such that n x n 1.
PROOF Let n be an integer.
Assumption: Suppose there is some integer x such that
n x n 1.
Since Z is a ring, n has an additive inverse, n 僆 Z. We can add n to the inequality above and simplify to get
0 x (n ) 1.
Since x and n are both integers, x ( n ) is also an integer. This contradicts
Lemma A.3.7. ⇒⇐
Hence, our Assumption must be false. We conclude that there is no integer x such
that n x n 1.
LEMMA A.3.10
Let a and b be natural numbers. If ab 1, then a 1 and b 1.
PROOF Let a, b 僆 N such that ab 1.
Assumption: Suppose that a 1 or b 1.
Without loss of generality, we may suppose that a 1. But a 1 by
Corollary A.3.8, so it must be that
a 1.
A.3 Building Results from our Axioms A-14
Multiplying both sides of this inequality by b (using the Multiplicative Property of ),
we get
ab b.
However, ab 1 and hence 1 b. This contradicts Corollary A.3.8. ⇒⇐
Hence, a 1 and b 1.
We define even and odd integers as follows.
DEFINITION A.3.11
Let a 僆 Z. We say that a is even if there exists n 僆 Z such that a 2n. We say a is odd
if there exists n 僆 Z such that a 2n 1.
It is intuitively clear that every integer is either even or odd, and no integer is
both even and odd. However, proving these assertions is another matter! The following
lemma states that the number 1 is not even.
LEMMA A.3.12
There does not exist x in Z such that 2x 1.
PROOF (By contradiction.)
Assumption: Suppose there exists x 僆 Z such that 2x 1.
By the Trichotomy Property, we know that either x 0, x 0, or x 0.
Case 1 x 0.
We can multiply this inequality by 2 to get 2x 0. By our Assumption, 2x 1,
and hence 1 0. This contradicts the Order of Identities (Axiom 2 part (v)). ⇒⇐
Case 2 x 0.
It follows that 2x 0. But by our Assumption, 2x 1. Hence, 0 1. Since 0 1,
we know by the Trichotomy Property that 0 1. This is a contradiction. ⇒⇐
Case 3 x 0.
By our Assumption, 2x 1. By Lemma A.3.10, this implies that 2 1. But it is
not hard to prove that 2 1: Start with the Order of Identities, 1 0, and add 1
to both sides to get 2 1. This is a contradiction. ⇒⇐
Since we have reached a contradiction in all three cases, we conclude that our
Assumption must be false. Hence, there is no x in Z such that 2x 1.
We are finally ready to prove that an integer cannot be both even and odd.
A-15 APPENDIX Axioms of Number Theory
LEMMA A.3.13
No integer can be both even and odd.
PROOF (By contradiction.)
Assumption: Suppose there exists an integer n that is both even and odd.
Since n is even, there is an integer r such that n 2r. Since n is odd, there is an
integer q such that n 2q 1. It follows that
2r 2q 1,
and hence
2[ r (q )] 1.
Since r (q) is an integer, this contradicts Lemma A.3.12. ⇒⇐
Thus, no integer can be both even and odd.
You will prove that every integer is either even or odd in Exercise 1 of Section A.4.
EXERCISES A.3
Reasoning and Proofs
In Exercises 1–5, be sure to justify each step of your argument. Every time
you use one of the ring properties, refer to it by name.
1. Let R be a ring with operations and , and let x be an element of R. Prove
that (x ) x. That is, prove that the inverse of the inverse of x is equal to x.
2. Let R be a ring with operations and , and let x and y be elements of R. Prove
that ( x y ) (x ) y and ( x y ) x (y ).
3. Let R be a ring with operations and , and let x and y be elements of R. Prove
that (x ) (y ) x y. (If necessary, you may use the results of Exercises 1
and 2 in your proof.)
4. Suppose that x is an element of a ring R. Prove that x (1) x.
5. Let R be a ring, and let x 僆 R. Suppose that x has a multiplicative inverse in R.
Prove that x has only one multiplicative inverse in R.
In the remainder of the exercises in this section, make sure that you justify
each step using the axioms for the integers or the results that we have proved
in this appendix.
6. Suppose that x and y are integers such that x y. Prove that y x.
A.3 Building Results from our Axioms A-16
7. Let x, y, z, w 僆 Z such that 0 x y and 0 z w. Prove that xz yw.
8. Let x, y 僆 Z such that 0 x y. Prove that 0 x 2 y 2.
9. Let a and b be natural numbers such that a 2 b 2. Prove that a b. (In your
proof, you may use the result of Exercise 8.)
10. a. Prove that the sum of any two even integers is an even integer.
b. Prove that the sum of any two odd integers is an even integer.
c. Prove that the sum of any even integer and any odd integer is an odd integer.
11. In Exercise 10, you proved everything there is to know about the parity of the
sums of even and odd integers. Write the corresponding three statements for
multiplication, and prove them.
12. Copy the Transitive Property of , the Additive Property of , and the
Multiplicative Property of from Axiom 2, but in each case replace every
occurrence of the symbol by the symbol. Prove the three new statements.
EXPLORATION Subtraction (Exercises 1317)
We can define a binary operation of subtraction on the integers, as follows:
x y x (y).
In this exploration, you will prove some results about this operation.
13. Explain how we know that – is a binary operation on Z.
14. Prove that for every x 僆 Z, x x 0.
15. Let x and y be integers. Prove that x y if and only if x y 0.
16. Prove that if x, y, and z are integers, then x ( y z ) ( x y ) ( x z ).
17. Prove that the operation – on Z is neither associative nor commutative.
18. Prove Corollary A.3.8.
19. Let a and b be positive integers. Prove that there exists an integer n such that
an b.
20. Let S be a nonempty set of integers. Suppose there is some integer m such that no
element of S is less than m. Prove that S has a smallest element.
21. Suppose that instead of the order property that 0 1, we only assumed that
0 1. Using this weaker assumption together with the other order properties
and the assumption that Z is a ring, prove that 0 1.
A-17 APPENDIX Axioms of Number Theory
22. Suppose that a and b are integers and ab 1. Prove that either a b 1
or a b 1.
23. Let x and y be integers. Prove that xy 0 if and only if both x and y are positive
or both x and y are negative.
24. Let x and y be integers. Prove that xy 0 if and only if x 0 or y 0.
25. Lemma A.3.1 concerns additive cancellation. In this exercise, you will prove a
corresponding result for multiplication. Let x, y, and z be integers with x 0 such
that x y x z. Prove that y z.
26. Fill in the missing steps in the proof of Lemma A.3.5 by proving that
inequality (4) implies inequality (5).
27. To make the proofs in the text shorter and clearer, we often leave out steps and
assume that the reader can fill them in. Rewrite the proof of Lemma A.3.12, filling
in every step and justifying each step using the axioms or one of our earlier results.
28. Let a, b, and c be integers with a bc. Prove that if a and c are natural numbers,
then b is also a natural number.
A.4 The Principle of Mathematical Induction
The Principle of Mathematical Induction is a staple proof technique used throughout
this book and throughout mathematics. Chapter 2 contains several intuitive explanations (one involving dominoes) of why induction makes sense. But thinking more
rigorously, what justifies our use of the Principle of Mathematical Induction in a
proof ? We did not assume this principle as an axiom, so if we want to use it, we must
prove it. Yes, that’s right—our favorite tried-and-true proof technique, the Principle of
Mathematical Induction, itself needs a proof ! And here it is:
A.4.1 (A.K.A. 2.1.1) THE PRINCIPLE OF MATHEMATICAL INDUCTION
Suppose that P (n) is a statement about the natural number n.
If it is established that both
Base Case P (1) is true.
Inductive Step For every natural number k, if P (k) is true, then P (k 1) is also true.
Then P (n) is true for all natural numbers n.
PROOF Let P (n) be a statement about the natural number n. Suppose that both of the
following are true:
(i ) P (1) is true.
(ii ) For every natural number k, if P (k) is true, then P (k 1) is also true.
[To show: For all n 僆 N, P (n) is true. We will do this by contradiction.]
A.4 The Principle of Mathematical Induction A-18
Assumption: Suppose there exists n 僆 N such that P (n) is false.
Consider the set of all natural numbers for which the statement P (n) is false:
S {n 僆 N | P (n) is false}.
By our Assumption, S is nonempty. Thus by the Well-Ordering Principle (Axiom 3),
the set S has a smallest element, m.
Since m is in S, we know that
P (m) is false.
(1)
But condition (i ) of our hypothesis tells us that P (1) is true. Hence, m 1.
Since m 僆 N, it follows that m 1 (using Corollary A.3.8). Hence, m 1 0,
which implies that m 1 僆 N. Note that since m is the smallest element of S,
we have m 1 僆 S. Thus,
P (m 1) is true.
Now we may apply condition (ii ), with k m 1:
Since P (m 1) is true, P (m) is also true.
But this contradicts statement (1). ⇒⇐
Since we have reached a contradiction, our original Assumption must be false.
Hence, there does not exist n 僆 N such that P (n) is false. It follows that for
all n 僆 N, P (n) is true.
Note that the Well-Ordering Principle (Axiom 3) plays a key role in this proof. This
is no accident. It turns out that the Principle of Mathematical Induction and the WellOrdering Principle are actually equivalent—each of these Principles implies the other.
Indeed, the proof of Theorem A.4.1 shows that the Well-Ordering Principle implies
the Principle of Mathematical Induction. One can also show that the Principle of
Mathematical Induction implies the Well-Ordering Principle (see Exercise 7).
Because of this equivalence, we had a choice in our axiomatic development of number
theory. In this appendix, we chose to assume the Well-Ordering Principle as an axiom,
and we derived the Principle of Mathematical Induction as a theorem. Another valid
route would have been to take the Principle of Mathematical Induction as an axiom
(instead of the Well-Ordering Principle). In this event, we would have proceeded to
prove the Well-Ordering Principle as a theorem. Taking either route, we arrive quickly
at the same juncture. Which route we choose to take is simply a matter of taste.
A show of strength
The Principle of Strong Induction can also be proved using the Well-Ordering
Principle.
A-19 APPENDIX Axioms of Number Theory
A.4.2 (A.K.A. 2.2.1) THE PRINCIPLE OF STRONG INDUCTION
Suppose that P (n) is a statement about the natural number n.
If for every natural number n,
P (k) is true for all natural numbers k ⬍ n ⇒ P (n) is true,
(2)
then P (n) is true for all natural numbers n.
PROOF You will prove this in Exercise 6.
In fact, as you will show in Exercise 8, the Principle of Strong Induction also implies
the Well-Ordering Principle. Thus, we have three equivalent principles: the Principle
of Mathematical Induction, the Principle of Strong Induction, and the Well-Ordering
Principle. Any one of them implies the other two.
These axioms and your future in number theory
Our three axioms give you all the tools you’ll need for your further study of number
theory. Just as the Pythagorean theorem, an interesting relationship between the sides
of a right triangle, can be derived from Euclid’s very basic postulates of geometry, in
the same way many subtle and deep truths about the natural numbers can be derived
using only the three basic axioms stated in this appendix. By Chapter 6, we will have
used these axioms to prove that each natural number has a unique factorization into
prime numbers. This is followed by even spicier theorems, such as Fermat’s Little
Theorem in Chapter 9 and the Primitive Root Theorem in Chapter 10.
A good understanding of the axioms, lemmas, and theorems of this appendix is a
powerful tool you can use as you begin your study of number theory. As you encounter
new proofs in this book, most steps will be clearly justified by specific definitions or
previous theorems. However, sometimes a seemingly obvious step will be taken without
any apparent justification. In these cases, a good knowledge of the material from this
appendix should enable you to provide a precise justification for these steps.
For example, when we begin our development of divisibility and primes in Chapter
3, the very first step in the proof of the very first lemma (Lemma 3.1.2) reads
Since d is a natural number, we know that 1 d.
Without the axioms, this might seem like a very reasonable step, but you would be
hard-pressed to give a rigorous reason why this is true! With the axioms, however, you
can easily justify the statement rigorously—just use Corollary A.3.8.
Following the axiomatic road requires constant vigilance. As you learn new
theorems, keep your eyes peeled for steps that lack precise justification. Then
challenge yourself to find those justifications. Your reward will be an understanding
of how great theorems can follow logically from a few simple axioms. Euclid would
approve!
A.4 The Principle of Mathematical Induction A-20
EXERCISES A.4
Reasoning and Proofs
1. a. Let n 僆 N. Prove that n is even or n is odd.
b. Use your result from part a to prove that every integer is even or odd.
2. In the proof of the Principle of Mathematical Induction (A.4.1), we used the
expression m 1, but we have not officially defined subtraction. (See Exercises
13–17 of Section A.3 for more on the subtraction operation.) Not to worry, we
simply define m 1 to mean m (1), where 1 is the additive inverse of 1.
In this exercise, you will use this definition to prove a couple of steps that we
used but did not formally show in the proof of the Principle of Mathematical
Induction (A.4.1).
a. Let m 僆 N such that m 1. Show that m 1 0.
b. Let m 僆 Z. Show that m 1 m. (This is necessary in the proof of
Theorem A.4.1 in order to know that m 1 僆 S.)
3. Read the proof of Lemma 3.1.2, and write a more rigorous version of the proof.
For each step of the proof, give a careful justification using the axioms, the results
from this appendix, or previous results in Chapter 3.
4. Read the proof of Lemma 3.1.3, and write a more rigorous version of the proof.
For each step of the proof, give a careful justification using the axioms, the results
from this appendix, or previous results in Chapter 3.
5. Find at least five places in proofs in Chapters 3–5 where otherwise unjustified
steps can be given a rigorous justification using the material in this appendix.
For each of these steps, give a careful justification.
Advanced Reasoning and Proofs
6. Prove the Principle of Strong Induction (A.4.2). [Hint: Begin as in the proof of
the Principle of Mathematical Induction (A.4.1).]
7. In this exercise, you will show that the Principle of Mathematical Induction (A.4.1)
implies the Well-Ordering Principle (Axiom 3). Assume that Z satisfies Axioms 1
and 2 (but do not assume Axiom 3), and let N be defined as in A.2.1. In addition,
assume the Principle of Mathematical Induction. Note that the proofs of all of our
results before Lemma A.3.7 rely only on Axioms 1 and 2, so you may use any of
these results.
a. Give another proof of Lemma A.3.7 using induction, but without using the
Well-Ordering Principle.
A-21 APPENDIX Axioms of Number Theory
b. The proof you wrote in part a allows you to use Lemma A.3.7 in the rest of
this exercise. Examine the proof of Lemma A.3.9, and explain why you may
also use that lemma.
c. Prove the Well-Ordering Principle. [Hint: Let S be a nonempty subset of N,
and assume that S does not have a smallest element. Consider the statement
P (n): The set S contains no elements that are less than or equal to n.]
8. Show that the Principle of Strong Induction (A.4.2) implies the Well-Ordering
Principle (Axiom 3).
9. In Exercise 8, you gave a proof of the Principle of Strong Induction (A.4.2) using
the Well-Ordering Principle. It is also possible to prove the Principle of Strong
Induction directly from the Principle of Mathematical Induction (A.4.1), without
appealing to the Well-Ordering Principle. Do so by using the following outline as
a guide.
I. Suppose you have a statement P (n) that satisfies the hypothesis of the
Principle of Strong Induction. What is this hypothesis, and what must you
show about P (n)?
II. For any n 僆 N, let Q (n) be the statement: P (m) is true for all m 僆 N that
satisfy m n.
III. Explain why Q (1) is true.
IV. Let k 僆 N. Prove that Q (k) implies Q (k 1). [Hint: Lemmas A.3.8
and A.3.9 may be useful for steps III and IV.]
V. Conclude that Q (n) is true for all n 僆 N.
VI. Use step V to conclude that P (n) is true for all n 僆 N.
A.4 The Principle of Mathematical Induction A-22