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9.2 Subrings 293 In addition to properties R1–R10, a field K must have the following features. (K11) There exists a nonzero element e such that e · k = k for all k ∈ K (G4). (K12) For all nonzero k ∈ K, there exists k−1 such that k · k−1 = e (G5). (K13) Multiplication is commutative (abelian). Notice that properties K11–K13 complete the requirements for K × to be an abelian group under multiplication. Thus a shorthand way of defining a field K is to say that K is an abelian group under addition and that K × is an abelian group under multiplication. Example 9.1.10 The rational numbers and the real numbers are fields. Also, the complex numbers are a field. In Section 8.1, we showed that C with addition is an abelian group, and in Exercise 8.1.20, you showed that C × with multiplication is an abelian group. � Example 9.1.11 R2×2 is not a field because multiplication is not commutative. � EXERCISE 9.1.12 Not only is multiplication in R2×2 not commutative, � �but also 1 0 many elements do not have a multiplicative inverse. Show that has no 0 0 inverse by showing that the equation � �� 1 0 a b 1 0 = (9.11) 0 0 c d 0 1 has no solution 9.2 Subrings � � a b . c d Suppose R is a ring and S is a subset of R. IfS is also a ring under the same operations, we say that S is a subring of R. Demonstrating S is a subring, some of the properties R1–R10 are inherited from R, while some (the closure properties) must be shown for S. (S1) S is closed under addition (R2, H1). (S2) S contains the additive identity (R4, H2). (S3) S is closed under additive inverses (R5, H3). (S4) S is closed under multiplication (R8, H1).
294 Chapter 9 Rings A subring S of a ring R is merely a subgroup of the additive group that is also closed under multiplication. We call {0} the trivial subring, and all subrings other than {0} and R itself are called proper subrings. Example 9.2.1 The integers are a subring of the rationals, and the rationals are a subring of the real numbers. � If R has a unity, it is not necessary that the unity be in S in order for S to be a subring of R. Example 9.2.2 Let m be a positive integer, and write mZ ={km : k ∈ Z}, the set of integer multiples of m. This is another common notation for the set in Eq. (8.31), where we denoted by (m) the subgroup of the additive group of integers generated by m. Then mZ is a subring of the integers, because it is a subgroup of the integers under addition, and it is closed under multiplication. If m ≥ 2, then mZ does not contain 1. � EXERCISE 9.2.3 Find all subrings of Z6 and Z7. Example 9.2.4 Z2×2 is a subring of R2×2. � EXERCISE 9.2.5 Call a square matrix diagonal if its only nonzero entries lie on the main diagonal. Let D2×2 be the subset of Z2×2 consisting of the diagonal matrices. Then D2×2 is a subring of Z2×2. Example 9.2.6 In this example, we create a subring of the rational numbers that we will return to in Section 9.9. Let QOD be the subset of the rational numbers whose denominators are odd. There is more than one way to denote an element of QOD. The form � � m : m, n ∈ Z (9.12) 2n + 1 is an obvious way. But another useful way to denote the set is to exploit the prime factorization of the numerator and denominator, isolating 2 to keep it separate from all the other primes involved. This works for all elements except zero, which we throw in separately. QOD ={0}∪ � ± 2n p1p2 ···pr q1q2 ···qs : n ∈ W and pi,qi are odd primes � for all 1 ≤ i ≤ r and 1 ≤ i ≤ s (9.13) The amount of repetition among the pi and qi does not matter. And notice that the form of an element in Eq. (9.13) includes 1 by letting n = 0, r = s = 1 and
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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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Page 268 and 269: 8.1 Introduction to Groups 245 are
Page 270 and 271: 8.1 Introduction to Groups 247 Defi
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Page 274 and 275: Show that C × is an abelian group
Page 276 and 277: 8.2 Subgroups 253 G1 and G3 are aut
Page 278 and 279: 8.2 Subgroups 255 Notice how proper
Page 280 and 281: 8.2 Subgroups 257 Example 8.2.15 su
Page 282 and 283: EXERCISE 8.2.23 If G is a group and
Page 284 and 285: 8.3 Quotient Groups 261 [0] ={...,
Page 286 and 287: 8.3 Quotient Groups 263 is standard
Page 288 and 289: Step 2: Define an operation ∗H on
Page 290 and 291: (iii) (Z + 3.8) +Z (Z + 1.2) (iv)
Page 292 and 293: then for a given f ∈ S and any a
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Page 296 and 297: 2 3 1 3 1 Figure 8.1 Rigid square u
Page 298 and 299: EXERCISE 8.4.12 Complete Table 8.64
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Page 302 and 303: 8.5 Normal Subgroups 279 point to b
Page 304 and 305: 8.6 Group Morphisms 281 EXERCISE 8.
Page 306 and 307: 8.6 Group Morphisms 283 Notice that
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Page 312 and 313: 9.1 Rings and Fields 289 (R9) Multi
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Page 318 and 319: 9.2 Subrings 295 p1 = q1 = 3. Verif
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Page 322 and 323: 9.3 Ring Properties 299 EXERCISE 9.
Page 324 and 325: 9.4 Ring Extensions 301 in the inte
Page 326 and 327: 9.4 Ring Extensions 303 Example 9.4
Page 328 and 329: 9.4 Ring Extensions 305 We insist t
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Page 334 and 335: EXERCISE 9.6.6 In the integers, wha
Page 336 and 337: 9.7 Prime and Maximal Ideals 313 Ev
Page 338 and 339: 9.8 Integral Domains 315 EXERCISE 9
Page 340 and 341: 9.8 Integral Domains 317 Corollary
Page 342 and 343: 9.9 Unique Factorization Domains 31
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.
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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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