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Example 3.4.2 For any positive integer n, 1 2 + 2 2 + 3 2 +···+n 2 = 3.4 The Principle of Mathematical Induction 79 n� k 2 = k=1 n(n + 1)(2n + 1) 6 (3.35) Example 3.4.3 Let A be a given set, and {Bk} n k=1 a family of n ≥ 1 sets. Then A ∪ (B1 ∩ B2 ∩···∩Bn) = (A ∪ B1) ∩ (A ∪ B2) ∩···∩(A ∪ Bn) (3.36) Example 3.4.4 Suppose there are n ≥ 1 people in a room, and everyone shakes hands with everyone else exactly once. Then the number of handshakes that takes place is n(n − 1)/2. � Example 3.4.5 Let G be a connected, acyclic graph on n vertices, where n ≥ 1. Then G has n − 1 edges. � Notice that Examples 3.4.1–3.4.5 all make a statement about a finite but unspecified n number of things, and you want to prove that the claim is true for any n ≥ 1. You might have found Eq. (3.34) handy if you had been in grammar school with Carl Friedrich Gauss in the 1780s. Gauss, a very precocious child, showed amazing mathematical ability at a very early age. A somewhat embellished story goes that when Gauss was eight years old, it was raining one day during recess, Internet access was down, and his teacher needed to keep the children in the class busy for a while. So he told them to add up the first 100 natural numbers without their calculators. Gauss figured out how to get the result quickly in the following way. By writing the sum twice, once in reverse order, he added vertically, term by term: 1 + 2 + 3 + 4 +···+ 99 + 100 100 + 99 + 98 + 97 +···+ 2 + 1 101 + 101 + 101 + 101 +···+101 + 101 � �� 100 terms Gauss observed that 100 × 101 is twice the desired result, so he quickly reported the result of 5050. If you perform a similar trick replacing 100 with an arbitrary n, you get Eq. (3.34). Though Gauss’s technique might seem sufficient as a proof, there is something a little sloppy about making a claim that involves a “dot dot dot” in it. The PMI is a theorem derived from the WOP that eliminates this untidiness. � �
80 Chapter 3 Sets and Their Properties So what is the PMI? And what does the WOP have to do with it? Let’s conduct a thought experiment to set it up, first in its standard form. Then in Section 3.5 we will look at some variations. Suppose we consider a set S, which is assumed to be a subset of the positive integers, and suppose S is known to have the following properties: (I 1) 1 ∈ S (I 2) If n ≥ 1 and n ∈ S, then n + 1 ∈ S. Then what precisely is S? Now I 1 says 1 ∈ S, but then I 2 applies to guarantee that, since 1 ∈ S, then 1 + 1 = 2 ∈ S. But then I 2 applies again to guarantee that 2 + 1 = 3 ∈ S, and so on. Now S ⊆ N is assumed, and it appears that every positive integer is also in S, so that N ⊆ S. So surely S = N. Fair enough. But this argument has its own “dot dot dot” and is a bit less than rigorous. Another way to look at this same argument is still a little open ended, but comes closer to the actual proof as we will present it. If it were true that S �= N, then since S ⊆ N, it must be that N �⊆ S. Thus there exists k ∈ N such that k/∈ S.By I1,k �= 1, so that k − 1 is a positive integer. By the contrapositive of I 2, since k/∈ S, then k − 1 /∈ S either. Therefore k − 1 �= 1, so that k − 2 ∈ N, and since k − 1 /∈ S, I 2 implies k − 2 /∈ S. Here’s where the “dot dot dot” comes in, and we have that k, k − 1,k− 2,... /∈ S. But this crashes head-on into the fact that 1 ∈ S. If we look to the WOP, then we can clean up these arguments and find in the WOP enough strength to provide a rigorous proof that any subset of the positive integers that has properties I 1–I 2 must, in fact, be the entire set of positive integers. This is the Principle of Mathematical Induction. Theorem 3.4.6 (PMI). Suppose S is a subset of the positive integers with the following properties: (I 1) 1 ∈ S (I 2) If n ≥ 1 and n ∈ S, then n + 1 ∈ S. Then S = N. The proof of the PMI is a great example of proof by contradiction. If you would like to try to prove it yourself, then take a glance at the hints 7,8,9,10,11 below if you need to. Here is the proof. Proof. Suppose S is a subset of the positive integers with properties I 1–I 2. Suppose also that S �= N. Then N �⊆ S. Thus there exists n ∈ N such that n/∈ S. Ifwe 7 The theorem says [(S ⊆ N) ∧ I1∧ I2]→(S = N). Suppose this is false. 8 State N �⊆ S in ∃ form. 9 Let T = N − S. What does the WOP say about T ? 10 If T has a smallest element a, what can you say about a − 1? 11 But if a − 1 ∈ S, then what is true of a?
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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
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Preface to the First Edition xix ea
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Acknowledgments It takes an entire
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Notation and Assumptions 0 Suppose
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0.2 Assumptions about the Real Numb
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0.2 Assumptions about the Real Numb
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0.2.3 Other Assumptions 0.2 Assumpt
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P A R T I Foundations of Logic and
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Language and Mathematics 1 One main
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1.1 Introduction to Logic 13 Now im
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The compound statement we call “p
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1.1 Introduction to Logic 17 exampl
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1.2 If-Then Statements 19 Definitio
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1.2 If-Then Statements 21 (h) The o
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1.2 If-Then Statements 23 (g) If ei
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p : Meghan is at least 25 years old
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1.3 Universal and Existential Quant
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Page 60 and 61: Solution 1.4 Negations of Statement
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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 70 and 71: 2.1 Basic Algebraic Properties of R
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
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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 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
Page 120 and 121: 3.7 Equivalence Classes and Partiti
Page 126 and 127: 3.8 Building the Rational Numbers 1
Page 128 and 129: 3.8 Building the Rational Numbers 1
Page 130 and 131: 3.10 Irrational Numbers 107 EXERCIS
Page 132 and 133: 3.10 Irrational Numbers 109 To the
Page 134 and 135: EXERCISE 3.10.2 √ 2 is irrational
Page 136 and 137: (a) {(x, y) : x ≤ y} (b) {(x, y)
Page 138 and 139: 3.11 Relations in General 115 (O3)
Page 140 and 141: 3.11 Relations in General 117 at al
Page 142 and 143: Functions 4 Second only to sets, fu
Page 144 and 145: 4.1 Definition and Examples 121 pro
Page 146 and 147: Solution We show that T satisfies p
Page 148 and 149: 4.2 One-to-one and Onto Functions 4
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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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