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The Real Numbers 5 Let’s look at some defining characteristics of the area of mathematics we call analysis. Given a set S, its elements might be endowed with a measure of size. The size, or norm, of an element x is typically denoted |x|, like absolute value on the real numbers, or perhaps �x�. The norm of an element will always be a nonnegative real number, so that a norm is really just a function from S into the real numbers that has three defining characteristics. You have already run into these in Section 2.3. They are properties N1–N2 on page 54 and property N3 on page 55. Measuring the sizes of elements is only one type of structure that can be placed on a set that puts it squarely in the field of analysis, but some notion of measure with nonnegative real numbers is characteristic of structures in analysis. For example, some structures do not have a norm but have a way of measuring some idea of distance between elements. Such a measure of distance is called a metric. Whether S is endowed with a norm or a metric, the measure it represents is inextricably tied to the real numbers. So the set of real numbers is at the heart of analysis, and one could argue that no analysis into any structure except the real numbers should be undertaken until one understands the axioms and fundamental results of the theory of real numbers. Be that as it may, a norm or metric on S, if one exists, lends itself to much fruitful study: Sequences and their convergence, continuity, and calculus are but a few. In this chapter, we address more advanced properties of the real numbers that arise out of the assumptions from Chapter 0, in particular A20. All these properties are important not only because they apply to the real numbers, but also because they are typical of properties of many other structures in analysis. In this chapter and the one to follow, all sets are assumed to be subsets of the real numbers unless specified otherwise. 5.1 The Least Upper Bound Axiom The least upper bound (LUB) axiom is a standard axiom of the real numbers, endowing it with some of its familiar features. For example, the way we visualize 165
166 Chapter 5 The Real Numbers the real numbers as a smooth number line with no holes is due to the LUB axiom. Two other characteristics of the real numbers are logically equivalent to the LUB axiom. One is called the Nested Interval Property (NIP) and has to do with what you get when you intersect a whole bunch of real number intervals that have certain properties. The other is called completeness and has to do with the way certain sequences behave when the terms get close to each other. It is possible to assume any one of the LUB axiom, the NIP, or completeness and derive the other two as theorems. Assuming the LUB property as an axiom is probably most common, so we choose that route. In this chapter, we will prove the NIP from the LUB axiom and completeness from the NIP, and look into the converses of these theorems to show that all three are logically equivalent. In this section we explore the LUB property in some depth to get a feel for what it means, and we derive some immediately important implications. First, let’s present a definition. Definition 5.1.1 A set A is said to be bounded from above if there exists M1 such that a ≤ M1 for all a ∈ A. A is said to be be bounded from below if there exists M2 such that a ≥ M2 for all a ∈ A. A is said to be bounded if there exists M>0 such that |a| ≤ M for all a ∈ A. The first exercise appears to be a very obvious result, but it must be demonstrated in terms of Definition 5.1.1. Proving the ⇒ direction is easier. For if A is bounded, then the guaranteed M>0 such that |a| ≤ M for all a ∈ A should clearly suggest values of M1 and M2. However, in proving the ⇐ direction, you must use the existence of M1 and M2 to create a single, positive value of M such that −M ≤ a ≤ M for all a ∈ A. EXERCISE 5.1.2 A set of real numbers is bounded if and only if it is bounded from above and below. 1 5.1.1 Least Upper Bounds Suppose A is non-empty and bounded from above. Among all possible upper bounds for A, we would call L a least upper bound for A if L is an upper bound for A with the additional property that L ≤ N for every upper bound N. That is, among all upper bounds for A, none is any smaller than L. The LUB axiom says that every non-empty set of real numbers that is bounded from above has an LUB that is a real number. We state it again here for the sake of reference. (A20) Least upper bound property: If A is a non-empty subset of real numbers that is bounded from above, then there exists a real number L with the following properties: (L1) For every a ∈ A, we have that a ≤ L, and (L2) If N is any upper bound for A, it must be that N ≥ L. 1 To prove ⇐, let M = |M1| + |M 2| + 1. Use Exercise 2.3.7.
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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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1.4 Negations of Statements 33 2. F
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1.4 Negations of Statements 35 want
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Solution 1.4 Negations of Statement
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Solution 1. Everyone in this class
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1.5 How We Write Proofs 41 Our purp
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1.5 How We Write Proofs 43 the nega
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Properties of Real Numbers 2 It’s
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2.1 Basic Algebraic Properties of R
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2.1.2 Properties of Multiplication
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2.2 Ordering Properties of the Real
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(c) For all real numbers a, a2 ≥
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2.3 Absolute Value 55 (⇐) Now sup
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2.4 The Division Algorithm 57 mn =
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2.5 Divisibility and Prime Numbers
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2.5 Divisibility and Prime Numbers
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Sets and Their Properties 3 All of
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U A B Figure 3.5 Venn diagram illus
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(f) For all sets A, B, and C,ifA
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3.2 Proving Basic Set Properties 69
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3.3 Families of Sets 71 Notice that
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Example 3.3.2 Consider the family o
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3.3 Families of Sets 75 x ∈ � A
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3.3 Families of Sets 77 and ... and
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Example 3.4.2 For any positive inte
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3.4 The Principle of Mathematical I
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Proof. To clean up the notation, we
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3.5 Variations of the PMI 85 EXERCI
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(g) (a −m ) −n = a mn (h) (ab)
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Strong Induction 3.5 Variations of
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3.6 Equivalence Relations 91 EXERCI
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3.6 Equivalence Relations 93 beginn
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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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Page 142 and 143: Functions 4 Second only to sets, fu
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Page 146 and 147: Solution We show that T satisfies p
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Page 152 and 153: A 1 x A B f (A 1 ) Figure 4.4 The i
Page 154 and 155: 4.4 Composition and Inverse Functio
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Page 158 and 159: 4.5 Three Helpful Theorems 135 The
Page 160 and 161: 4.6 Finite Sets 137 EXERCISE 4.5.3
Page 162 and 163: 4.7 Infinite Sets 139 The next exer
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Page 166 and 167: 4.7 Infinite Sets 143 EXERCISE 4.7.
Page 168 and 169: EXERCISE 4.8.3 If A1,A2, and B are
Page 170 and 171: 4.8 Cartesian Products and Cardinal
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Page 174 and 175: 4.9 Combinations and Partitions 151
Page 180 and 181: 4.10 The Binomial Theorem 157 EXERC
Page 182 and 183: EXERCISE 4.10.3 Determine the follo
Page 184 and 185: (c) The coefficient of ab 3 c 2 in
Page 186 and 187: P A R T I I Basic Principles of Ana
Page 190 and 191: 5.1 The Least Upper Bound Axiom 167
Page 192 and 193: 5.2 The Archimedean Property 169 EX
Page 194 and 195: 5.2 The Archimedean Property 171 No
Page 196 and 197: 5.3 Open and Closed Sets 173 Thus A
Page 198 and 199: 5.4 Interior, Exterior, Boundary, a
Page 200 and 201: 5.4 Interior, Exterior, Boundary, a
Page 202 and 203: 5.5 Closure of Sets 179 The constru
Page 204 and 205: 5.6 Compactness 181 EXERCISE 5.6.3
Page 206 and 207: 5.6 Compactness 183 EXERCISE 5.6.13
Page 208 and 209: Sequences of Real Numbers 6.1 Seque
Page 210 and 211: 6.1 Sequences Defined 187 Example 6
Page 212 and 213: EXERCISE 6.1.15 Consider the sequen
Page 214 and 215: L 1 e L L 2 e . . . . . . 1 2 3 4 N
Page 216 and 217: 6.2 Convergence of Sequences 193 EX
Page 218 and 219: 6.2 Convergence of Sequences 195 To
Page 220 and 221: 6.3 The Nested Interval Property 19
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Page 228 and 229: Example 6.4.7 The terms a0 = 1 a1 =
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Page 232 and 233: 7.1 Bounded and Monotone Functions
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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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