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7.7 Uniform Continuity 237 Next, let’s split up the fraction in the right-hand side of Eq. (7.36) and apply the triangle inequality. Also, notice that x, y > 0 implies that 1/(x + 1),1/(y + 1), x/(x + 1), and y/(y + 1) are all less than 1. So we have � � � � xy + x + y � � �(x + 1)(y + 1) � ≤ � � � � xy � � �(x + 1)(y + 1) � + � � � � x � � �(x + 1)(y + 1) � + � � � � y � � �(x + 1)(y + 1) � � � � � � = � x � � � � y � � �(x + 1) � �(y + 1) � + � � � � � � x � � � � 1 � � �(x + 1) � �(y + 1) � � � � � � + � y � � � � 1 � � �(y + 1) � �(x + 1) � ≤ 3 (7.37) Having arrived at |f(x) − f(y)| ≤ 3 |x − y|, we are ready to write a proof. Let ɛ>0 be given, and let δ = ɛ/3. Then for any x, y ≥ 0 such that |x − y| 0 such that |f(x) − f(y)| ≤ m |x − y| for all x, y ∈ A, then f is uniformly continuous on A. If y �= x, the hypothesis condition of Exercise 7.7.3 is equivalent to � � � � f(x) − f(y) � � � x − y � ≤ m (7.39) which means that f has a bound on the slopes of lines through any two points on its graph. Loosely speaking, if the steepness of f (as measured by slopes of secant �
238 Chapter 7 Functions of a Real Variable lines) is bounded, then f is uniformly continuous. The converse of Exercise 7.7.3 is not true. Shortly, we will point out a function f that is uniformly continuous on a set A, but for which inequality (7.39) does not hold across A for any m>0. Naturally, we want to include a demonstration that a continuous function need not be uniformly continuous on a set. EXERCISE 7.7.4 What does it mean for f not to be uniformly continuous on A? Example 7.7.5 Show that f(x) = 1/x is not uniformly continuous on (0, 1). Solution First, we do some scratchwork. The vertical asymptote at x = 0 will provide us with x and y values that are very close together but whose functional values can differ as much as a strategically chosen ɛ-value. We have to play with the inequality � � � � 1 1 � � − � x y � = |y − x| |x||y| ≥ ɛ (7.40) and find some ɛ>0 so that, regardless of δ>0, we can find two points x and y that are within δ of each other and satisfy inequality (7.40). No obvious ɛ-value jumps out at us, so let’s try ɛ = 1 and see if we can proceed. Inequality (7.40) itself suggests a way to find x and y. Whatever we decide to let x be, we can let y = x + δ/2 so that |y − x| = δ/2
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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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Page 212 and 213: EXERCISE 6.1.15 Consider the sequen
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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
Page 226 and 227: a1 a4 aN 1 2 aN aN 1 1 aN 1 3 a5 a3
Page 228 and 229: Example 6.4.7 The terms a0 = 1 a1 =
Page 230 and 231: Functions of a Real Variable 7 Func
Page 232 and 233: 7.1 Bounded and Monotone Functions
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Page 236 and 237: 7.2 Limits and Their Basic Properti
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Page 240 and 241: 7.3 More on Limits 217 values of 0
Page 242 and 243: 7.4 Limits Involving Infinity 219 W
Page 244 and 245: 7.4 Limits Involving Infinity 221 T
Page 246 and 247: (a) limx→a f(x) =−∞ (b) limx
Page 248 and 249: 7.5 Continuity 225 If f is not cont
Page 250 and 251: 1 2 3 4 5 6 Figure 7.6 Some example
Page 252 and 253: |a − x| |xa| = |x − a| |x||a| 7
Page 254 and 255: 7.6 Implications of Continuity 231
Page 256 and 257: 7.6 Implications of Continuity 233
Page 258 and 259: 7.7 Uniform Continuity 235 we decla
Page 262 and 263: 7.7.2 Uniform Continuity and Compac
Page 264 and 265: P A R T I I I Basic Principles of A
Page 266 and 267: Groups 8 In its simplest terms, alg
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
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Page 278 and 279: 8.2 Subgroups 255 Notice how proper
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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
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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
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Page 306 and 307: 8.6 Group Morphisms 283 Notice that
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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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