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1.3 Universal and Existential Quantifiers 31 (a) There is a function in F whose graph intersects the x-axis. (b) No function in F has a graph that intersects the x-axis. (c) There is a function in F whose graph does not intersect the x-axis. Sometimes a universal statement might be expressed informally by placing the ∀ phrase after the property that is universally true. This is merely a way to make a sentence sound more natural. We would not write the statement that way symbolically. The next example illustrates this with a range of expressions from informal to formal. Example 1.3.10 The following statements are equivalent. There is an element of set A that is less than every element of set B. There exists x ∈ A such that x
32 Chapter 1 Language and Mathematics 1.3.3 Unique Existence Sometimes it is important to know not only that something exists, but also that exactly one such thing exists. If there exists exactly one thing with a certain property, we say that it exists uniquely. The mathematical statement (∃!x)(P (x)) (1.17) is read, “There exists a unique x such that P(x).” To define unique existence, we must amend the statement (∃x)(P (x)) to include the additional stipulation that there is no more than one such x. The standard way of defining unique existence is the following. Definition 1.3.12 We say that there exists a unique x with property P provided two things are true. First, there exists x with property P, and second, for all x1 and x2,ifx1 and x2 both have property P, then x1 and x2 are the same. In other words, the logical statement (∃!x)(P (x)) is defined by [(∃x)(P (x))]∧[(∀x1)(∀x2)(P (x1) ∧ P(x2)) → (x1 = x2)] (1.18) Unique existence is really a compound AND statement. The first component says that at least one thing with the property exists, and the second component says that at most one thing with the property exists. Example 1.3.13 Reword the following statements of unique existence in a form like that in Definition 1.3.12. 1. There exists a unique real number x such that x3 = 8. 2. For all a ∈ R, there exists a unique b ∈ R such that a + b = 0. Solution 1. There exists a real number x such that x3 = 8, and for all x1,x2 ∈ R, ifx3 1 = 8 and x3 2 = 8, then x1 = x2. 2. For all a ∈ R, there exists b ∈ R such that a + b = 0, and for all b1,b2 ∈ R, if a + b1 = 0 and a + b2 = 0, then b1 = b2. When we write these unique existence statements in prose, we can give ourselves the freedom to omit the universal quantifier phrase, knowing it is understood. So we could reword these slightly as 1. There exists a real number x such that x3 = 8, and if x1,x2 ∈ R satisfy x3 1 = 8 and x3 2 = 8, then x1 = x2.
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A Transition to Abstract Mathematic
Page 4 and 5: A Transition to Abstract Mathematic
Page 6 and 7: For Topo my little mouse
Page 8 and 9: Contents Why Read This Book? xiii P
Page 10 and 11: 3.10 Irrational Numbers 107 3.11 Re
Page 12 and 13: 8.2 Subgroups 252 8.2.1 Subgroups D
Page 14 and 15: Why Read This Book? One of Euclid
Page 16 and 17: Preface A Transition to Abstract Ma
Page 18 and 19: Preface to the First Edition This t
Page 20 and 21: Preface to the First Edition xix ea
Page 22 and 23: Acknowledgments It takes an entire
Page 24 and 25: Notation and Assumptions 0 Suppose
Page 26 and 27: 0.2 Assumptions about the Real Numb
Page 28 and 29: 0.2 Assumptions about the Real Numb
Page 30 and 31: 0.2.3 Other Assumptions 0.2 Assumpt
Page 32 and 33: P A R T I Foundations of Logic and
Page 34 and 35: Language and Mathematics 1 One main
Page 36 and 37: 1.1 Introduction to Logic 13 Now im
Page 38 and 39: The compound statement we call “p
Page 40 and 41: 1.1 Introduction to Logic 17 exampl
Page 42 and 43: 1.2 If-Then Statements 19 Definitio
Page 44 and 45: 1.2 If-Then Statements 21 (h) The o
Page 46 and 47: 1.2 If-Then Statements 23 (g) If ei
Page 48 and 49: p : Meghan is at least 25 years old
Page 50 and 51: 1.3 Universal and Existential Quant
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Page 56 and 57: 1.4 Negations of Statements 33 2. F
Page 58 and 59: 1.4 Negations of Statements 35 want
Page 60 and 61: Solution 1.4 Negations of Statement
Page 62 and 63: Solution 1. Everyone in this class
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
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Page 78 and 79: 2.3 Absolute Value 55 (⇐) Now sup
Page 80 and 81: 2.4 The Division Algorithm 57 mn =
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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
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Page 102 and 103: 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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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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