Why Read This Book? - Index of

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Index 355 Tower of Hanoi, 84 Transcendental functions, 215 Transcendental numbers, 304 Transitive property of equality in real numbers, 3, 45, 46, 91 on sets, 91, 92 Transitive property (S3, T3, W3), 116–117 Transpositions, 271–272 Triangle, Pascal’s. See Pascal’s triangle Triangle-type inequalities, 55–56, 194n, 237 Trichotomy law, 6, 51 Trigonometric functions, 215, 215f Trivial morphisms, 281, 334 Trivial subgroups, 253 Trivial subrings, 294 Truth table, 13 Two-sided ideals, 306 U U1-U3 properties (generated ideals), 309–312 U1-U3 properties (generated subgroups), 254–255 UFDs. See Unique factorization domains Unbounded natural numbers, in reals, 42 Unbounded positive integers, from above, 169 Unbounded sequences, 187–188 Uncountable irrationals, 140, 143 Uncountable real numbers, 143 Uniform continuity, 235–240 Union disjoint, 135n, 175 over family of sets, 75 of sets, 64 Unique existence, 32–33, 47 multiplicative inverse, 49 theorem, 56 Unique factorization domains (UFDs), 319–321 Uniqueness additive identity, 47 additive inverse, 47 of factorization, 89 gcd(a,b), 60–61 of image, 120, 120f, 125 multiplicative identity, 49 quotient, 57 remainder, 57 theorems, 47 Unit, of ring, 297 Unit circle, 215, 216 Unity element, 288 Universal quantifier, 28–29, 36–37 Universal set, 3 Universal side divisor, 327 Upper bounds, 6–7 V Valuation, 326, 327, 328 Venn diagrams, 63, 63f, 64f, 65f, 66f, 67 Vertical asymptote, 220 W W1-W4 properties (well order relation), 117 Weaker statements, 24 Well defined addition, 4, 46 addition of rational numbers, 105 binary operation, 244 cardinality of finite set, 138 functions, 119–120 mapping, 122 multiplication, 4, 49 multiplication of rational numbers, 105 operations on sets, 70, 71 Well order relation, 117 Well-ordering principle (WOP), 6, 57–58. See also Principle of Mathematical Induction as axioms, 78 PMI from, 78, 79, 80 Wellington, 101 Whole numbers, 2, 102 construction of, 102 as countably infinite, 140 extension of, to integers, 108 as successors, 102 WOP. See Well-ordering principle Wrangell, 100 X X1-X3 properties (roots), 106–107. See also Roots, of real numbers Y Y1-Y4 properties (ideals), 307. See also Ideals
356 Index Z Z1-Z4 properties (integer assumptions), 103 Zermelo, Ernst, 118 Zero, 48 characteristic, 301 divisors of, 298 infinitesimals v., 169 minus, 48 one �= to, 5, 49, 52 Zero products, principle of, 51, 106, 298, 314, 315
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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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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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Page 338 and 339: 9.8 Integral Domains 315 EXERCISE 9
Page 340 and 341: 9.8 Integral Domains 317 Corollary
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Page 344 and 345: 9.10 Principal Ideal Domains 321 al
Page 346 and 347: 9.10 Principal Ideal Domains 323 So
Page 348 and 349: 9.11 Euclidean Domains 325 next exe
Page 350 and 351: 9.11 Euclidean Domains 327 Exercise
Page 352 and 353: Proof. Let f and g be nonzero polyn
Page 354 and 355: 9.12 Polynomials over a Field 331 W
Page 356 and 357: 9.13 Polynomials over the Integers
Page 358 and 359: 9.14 Ring Morphisms 335 Example 9.1
Page 360 and 361: 9.14 Ring Morphisms 337 EXERCISE 9.
Page 362 and 363: 9.15 Quotient Rings 339 this, howev
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Page 366 and 367: 9.15 Quotient Rings 343 However, th
Page 368 and 369: Index =,51 ɛ (epsilon), 167-168 ɛ
Page 370 and 371: Index 347 Cosets, 263, 264, 265 Lag
Page 372 and 373: Index 349 Gauchy sequences, 202-206
Page 374 and 375: Index 351 identity, 124 relations v
Page 376 and 377: Index 353 Quaternions, 248, 253, 25
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