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2.1 Basic Algebraic Properties of Real Numbers 47 Proof 2. Suppose a, b, and c are real numbers such that a + c = b + c. By A7, there exists −c ∈ R such that c + (−c) = 0. Thus with several other properties from A1–A7 we have that a (A6) = a + 0 (A2) = a +[c + (−c)] (A4) = (a + c) + (−c) (A2) = (b + c) + (−c) (A4) = b +[c + (−c)] (A2) = b + 0 (A6) = b By A1 (the transitive property of equality), a = b. (2.1) Because addition is commutative, Theorem 2.1.1 also allows us to say that if c + a = c + b, then a = b. For if c + a = c + b, then a + c = c + a = c + b = b + c which by Theorem 2.1.1 implies a = b. Thus we have both left cancellation and right cancellation. We will eventually get a little lazy about referencing the real number properties that we need to use, especially the three properties of equality. At first, however, we need to make sure for our own sakes that we understand precisely which ones we use and when we use them. The next two results are uniqueness theorems. Recall what it means to say that a particular mathematical object is unique: If x1 and x2 are both assumed to have the desired property, then x1 = x2. Theorem 2.1.2 The additive identity is unique. Proof. Suppose 01 and 02 are both additive identities. Then 01 = 01 + 02 = 02 Assumption A7 guarantees that every real number a has an additive inverse −a for which a +−a = 0. This is an axiom. But nothing about the axioms A1–A22 precludes (at first glance) that a real number might have more than one additive inverse. The next theorem shows that every real number has a unique additive inverse. Once we prove this, we will then be able to talk about the additive inverse of a. Understand what the uniqueness component of the existence of additive inverses would say. For all b1,b2,ifb1 and b2 are both additive inverses of a, then b1 = b2. Theorem 2.1.3 The additive inverse of a real number is unique. Proof. Pick a ∈ R, and suppose b1,b2 ∈ R are both additive inverses of a. Then a + b1 = 0 and a + b2 = 0. Thus by the transitive property of equality, a + b1 = a + b2, so that b1 = b2 by Theorem 2.1.1. Therefore the additive inverse of a is unique.
48 Chapter 2 Properties of Real Numbers Theorem 2.1.4 For every a ∈ R, −(−a) = a. Proof. Pick a ∈ R. Then −a exists in R, where a + (−a) = 0. But then −(−a) also exists in R, where (−a) +[−(−a)] =0. Thus (−a) +[−(−a)] =a + (−a), which by cancellation yields −(−a) = a. Theorem 2.1.5 For all a, b ∈ R, −(a + b) = (−a) + (−b). Proof. Pick a, b ∈ R. Then −a and −b both exist in R. By A3, (−a) + (−b) is also a real number. Exploiting associativity and commutativity of addition, we have (a + b) +[(−a) + (−b)] =[a + (−a)]+[b + (−b)] =0 + 0 = 0 (2.2) Thus −(a + b) = (−a) + (−b). One of the oldest gags around says that, if it quacks like a duck, it must be a duck. That is, the defining characteristic of ducks is that they quack. Now suppose you know that there is only one duck in the world, and you have just found an animal that quacks. Then, by golly, you have found the duck. In Theorem 2.1.5, the duck we are looking for is −(a + b). That is, we are looking for some real number, which, when added to (a + b), yields zero. After all, that is the quacking feature of −(a + b). Theorem 2.1.5 claims that the number (−a) + (−b) quacks, and Eq. (2.2) is the demonstration of that claim. Finally, since the additive inverse of a real number is unique, we know that this quacking thing we have found is indeed −(a + b). Theorem 2.1.5 gives us the right to make a statement like the following: The additive inverse of a sum is the sum of the additive inverses. It is quite common in mathematics to investigate the truth of a statement of the form The X of the Y is equal to the Y of the X. (2.3) Statements of this form can be very powerful if true. However, sometimes such a statement is false. For example, let X = “square root” and let Y = “sum” and you have the statement √ a + b = √ a + √ b. EXERCISE 2.1.6 Calling on your experience from previous math courses, construct several statements with the same form as (2.3), some of which are true and some false. Theorem 2.1.7 −0 = 0. The behavior of zero yields the next theorem. Proof. Since 0 is the additive identity, 0 + (−0) =−0. Also, since −0 is the additive inverse of 0, by definition it satisfies 0 + (−0) = 0. Thus we have −0 = 0 + (−0) = 0.
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A Transition to Abstract Mathematic
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A Transition to Abstract Mathematic
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For Topo my little mouse
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Contents Why Read This Book? xiii P
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3.10 Irrational Numbers 107 3.11 Re
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8.2 Subgroups 252 8.2.1 Subgroups D
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Why Read This Book? One of Euclid
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Preface A Transition to Abstract Ma
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Preface to the First Edition This t
Page 20 and 21: Preface to the First Edition xix ea
Page 22 and 23: Acknowledgments It takes an entire
Page 24 and 25: Notation and Assumptions 0 Suppose
Page 26 and 27: 0.2 Assumptions about the Real Numb
Page 28 and 29: 0.2 Assumptions about the Real Numb
Page 30 and 31: 0.2.3 Other Assumptions 0.2 Assumpt
Page 32 and 33: P A R T I Foundations of Logic and
Page 34 and 35: Language and Mathematics 1 One main
Page 36 and 37: 1.1 Introduction to Logic 13 Now im
Page 38 and 39: The compound statement we call “p
Page 40 and 41: 1.1 Introduction to Logic 17 exampl
Page 42 and 43: 1.2 If-Then Statements 19 Definitio
Page 44 and 45: 1.2 If-Then Statements 21 (h) The o
Page 46 and 47: 1.2 If-Then Statements 23 (g) If ei
Page 48 and 49: p : Meghan is at least 25 years old
Page 50 and 51: 1.3 Universal and Existential Quant
Page 56 and 57: 1.4 Negations of Statements 33 2. F
Page 58 and 59: 1.4 Negations of Statements 35 want
Page 60 and 61: Solution 1.4 Negations of Statement
Page 62 and 63: Solution 1. Everyone in this class
Page 64 and 65: 1.5 How We Write Proofs 41 Our purp
Page 66 and 67: 1.5 How We Write Proofs 43 the nega
Page 68 and 69: Properties of Real Numbers 2 It’s
Page 72 and 73: 2.1.2 Properties of Multiplication
Page 74 and 75: 2.2 Ordering Properties of the Real
Page 76 and 77: (c) For all real numbers a, a2 ≥
Page 78 and 79: 2.3 Absolute Value 55 (⇐) Now sup
Page 80 and 81: 2.4 The Division Algorithm 57 mn =
Page 82 and 83: 2.5 Divisibility and Prime Numbers
Page 84 and 85: 2.5 Divisibility and Prime Numbers
Page 86 and 87: Sets and Their Properties 3 All of
Page 88 and 89: U A B Figure 3.5 Venn diagram illus
Page 90 and 91: (f) For all sets A, B, and C,ifA
Page 92 and 93: 3.2 Proving Basic Set Properties 69
Page 94 and 95: 3.3 Families of Sets 71 Notice that
Page 96 and 97: Example 3.3.2 Consider the family o
Page 98 and 99: 3.3 Families of Sets 75 x ∈ � A
Page 100 and 101: 3.3 Families of Sets 77 and ... and
Page 102 and 103: Example 3.4.2 For any positive inte
Page 104 and 105: 3.4 The Principle of Mathematical I
Page 106 and 107: Proof. To clean up the notation, we
Page 108 and 109: 3.5 Variations of the PMI 85 EXERCI
Page 110 and 111: (g) (a −m ) −n = a mn (h) (ab)
Page 112 and 113: Strong Induction 3.5 Variations of
Page 114 and 115: 3.6 Equivalence Relations 91 EXERCI
Page 116 and 117: 3.6 Equivalence Relations 93 beginn
Page 118 and 119: 3.6 Equivalence Relations 95 constr
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3.7 Equivalence Classes and Partiti
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3.8 Building the Rational Numbers 1
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3.8 Building the Rational Numbers 1
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3.10 Irrational Numbers 107 EXERCIS
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3.10 Irrational Numbers 109 To the
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EXERCISE 3.10.2 √ 2 is irrational
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(a) {(x, y) : x ≤ y} (b) {(x, y)
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3.11 Relations in General 115 (O3)
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3.11 Relations in General 117 at al
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Functions 4 Second only to sets, fu
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4.1 Definition and Examples 121 pro
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Solution We show that T satisfies p
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4.2 One-to-one and Onto Functions 4
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4.2 One-to-one and Onto Functions 1
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A 1 x A B f (A 1 ) Figure 4.4 The i
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4.4 Composition and Inverse Functio
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4.4 Composition and Inverse Functio
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4.5 Three Helpful Theorems 135 The
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4.6 Finite Sets 137 EXERCISE 4.5.3
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4.7 Infinite Sets 139 The next exer
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4.7 Infinite Sets 141 of A. This or
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4.7 Infinite Sets 143 EXERCISE 4.7.
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EXERCISE 4.8.3 If A1,A2, and B are
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4.8 Cartesian Products and Cardinal
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4.8 Cartesian Products and Cardinal
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4.9 Combinations and Partitions 151
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4.10 The Binomial Theorem 157 EXERC
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EXERCISE 4.10.3 Determine the follo
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(c) The coefficient of ab 3 c 2 in
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P A R T I I Basic Principles of Ana
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The Real Numbers 5 Let’s look at
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5.1 The Least Upper Bound Axiom 167
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5.2 The Archimedean Property 169 EX
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5.2 The Archimedean Property 171 No
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5.3 Open and Closed Sets 173 Thus A
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5.4 Interior, Exterior, Boundary, a
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5.4 Interior, Exterior, Boundary, a
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5.5 Closure of Sets 179 The constru
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5.6 Compactness 181 EXERCISE 5.6.3
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5.6 Compactness 183 EXERCISE 5.6.13
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Sequences of Real Numbers 6.1 Seque
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6.1 Sequences Defined 187 Example 6
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EXERCISE 6.1.15 Consider the sequen
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L 1 e L L 2 e . . . . . . 1 2 3 4 N
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6.2 Convergence of Sequences 193 EX
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6.2 Convergence of Sequences 195 To
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6.3 The Nested Interval Property 19
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a1 a4 aN 1 2 aN aN 1 1 aN 1 3 a5 a3
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Example 6.4.7 The terms a0 = 1 a1 =
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Functions of a Real Variable 7 Func
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7.1 Bounded and Monotone Functions
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50 40 30 20 7 1 ε 7 7 2 ε 0 ( ( F
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7.2 Limits and Their Basic Properti
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7.2 Limits and Their Basic Properti
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7.3 More on Limits 217 values of 0
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7.4 Limits Involving Infinity 219 W
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7.4 Limits Involving Infinity 221 T
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(a) limx→a f(x) =−∞ (b) limx
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7.5 Continuity 225 If f is not cont
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1 2 3 4 5 6 Figure 7.6 Some example
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|a − x| |xa| = |x − a| |x||a| 7
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7.6 Implications of Continuity 231
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7.6 Implications of Continuity 233
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7.7 Uniform Continuity 235 we decla
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7.7 Uniform Continuity 237 Next, le
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7.7.2 Uniform Continuity and Compac
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P A R T I I I Basic Principles of A
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Groups 8 In its simplest terms, alg
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8.1 Introduction to Groups 245 are
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8.1 Introduction to Groups 247 Defi
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◦ 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 0
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Show that C × is an abelian group
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8.2 Subgroups 253 G1 and G3 are aut
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8.2 Subgroups 255 Notice how proper
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8.2 Subgroups 257 Example 8.2.15 su
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EXERCISE 8.2.23 If G is a group and
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8.3 Quotient Groups 261 [0] ={...,
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8.3 Quotient Groups 263 is standard
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Step 2: Define an operation ∗H on
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(iii) (Z + 3.8) +Z (Z + 1.2) (iv)
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then for a given f ∈ S and any a
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Thus f 2 = f ◦ f = (132)(56) ◦
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2 3 1 3 1 Figure 8.1 Rigid square u
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EXERCISE 8.4.12 Complete Table 8.64
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8.5 Normal Subgroups 277 Theorem 8.
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8.5 Normal Subgroups 279 point to b
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8.6 Group Morphisms 281 EXERCISE 8.
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8.6 Group Morphisms 283 Notice that
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Ker � e x 2 x maps to x Ker � G
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Rings 9 We can create algebraic str
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9.1 Rings and Fields 289 (R9) Multi
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9.1 Rings and Fields 291 we define
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9.2 Subrings 293 In addition to pro
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9.2 Subrings 295 p1 = q1 = 3. Verif
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9.3 Ring Properties 297 Theorem 9.3
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9.3 Ring Properties 299 EXERCISE 9.
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9.4 Ring Extensions 301 in the inte
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9.4 Ring Extensions 303 Example 9.4
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9.4 Ring Extensions 305 We insist t
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9.5 Ideals 307 An ideal, say a left
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9.6 Generated Ideals 309 Proof. Sup
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EXERCISE 9.6.6 In the integers, wha
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9.7 Prime and Maximal Ideals 313 Ev
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9.8 Integral Domains 315 EXERCISE 9
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9.8 Integral Domains 317 Corollary
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9.9 Unique Factorization Domains 31
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9.10 Principal Ideal Domains 321 al
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9.10 Principal Ideal Domains 323 So
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9.11 Euclidean Domains 325 next exe
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9.11 Euclidean Domains 327 Exercise
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Proof. Let f and g be nonzero polyn
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9.12 Polynomials over a Field 331 W
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9.13 Polynomials over the Integers
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9.14 Ring Morphisms 335 Example 9.1
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9.14 Ring Morphisms 337 EXERCISE 9.
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9.15 Quotient Rings 339 this, howev
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9.15 Quotient Rings 341 Because the
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9.15 Quotient Rings 343 However, th
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Index =,51 ɛ (epsilon), 167-168 ɛ
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Index 347 Cosets, 263, 264, 265 Lag
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Index 349 Gauchy sequences, 202-206
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Index 351 identity, 124 relations v
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Index 353 Quaternions, 248, 253, 25
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Index 355 Tower of Hanoi, 84 Transc
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