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2.5 Divisibility and Prime Numbers 59 EXERCISE 2.4.9 Appeal to the division algorithm to explain precisely why every integer is either even or odd, but not both. 2.5 Divisibility and Prime Numbers Suppose a and b are nonzero integers and we apply the division algorithm to write b = aq + r. The situation where r = 0 motivates a new term. Definition 2.5.1 Suppose a and b are nonzero integers. We say that a divides b, written a | b, provided there exists an integer k such that b = ak. We also say that a is a divisor of b. Ifa | b where a/∈{±1, ±b}, we call a a proper divisor of b. Ifa does not divide b, we write a ∤ b. Notice that if a does not divide b, then writing b = aq + r according to the division algorithm will imply 0
60 Chapter 2 Properties of Real Numbers (D1) g | a and g | b (D2) If h is any positive integer such that h | a and h | b, then h | g also. Then g is called a greatest common divisor of a and b, and is denoted gcd(a, b). Some remarks about Definition 2.5.6 are in order. First, nothing about the definition of gcd(a, b) (or of any term for that matter) guarantees that any such thing exists. It merely lays out criteria by which some positive integer earns the right to be called gcd(a, b). Second, property D1 states that any integer that might qualify as gcd(a, b) will in fact be a common divisor of a and b. Third, we need to explain our choice of property D2 as the other criterion, for it might seem a bit unnatural. If property D2 is supposed to describe what it means for g to be greatest of all the common divisors of a and b, you might think a more natural way to say it would be (D2 ′ ) Ifh is any positive integer such that h | a and h | b, then h ≤ g. Well, we could do it that way, but we don’t. Here’s why. It all centers around how you decide to measure greatness. In the integers, we can measure relative greatness by ≤. But as you will see in your later work in algebra, there are other more abstract settings where we discuss divisibility, then ask about a gcd, but we do not necessarily have a notion of ≤ to use as our criterion for greatness. The point is that it is better to develop a criterion for greatness of a divisor that stays within the context of divisibility rather than relying on some external measure like ≤, which might or might not already exist. All this is to say that property D2 is our measure of greatness among all common divisors of a and b. If g is going to qualify as gcd(a, b), it has the property that any positive integer h that comes down the pike also having property D1 cannot, in some sense, be greater than g. Our way of saying h is no greater than g is that h | g. In the past, you have probably found gcd(a, b) by breaking a and b down into their prime factorizations to see how many 2s, 3s, 5s, and so on, that you can factor out of each. Then you built the gcd to contain just the right number of each prime factor represented. This might work well practically, but (1) we have no logical basis for this in the work we have done so far because we have not discussed prime numbers yet, and (2) it is not particularly useful as leverage in later theorems. So, given nonzero integers a and b, how do we even know that there exists some positive integer having properties D1–D2? And if such an integer does exist, how many can there be? The following theorem claims that gcd(a, b) exists uniquely. Furthermore, hidden inside its proof is some additional information about gcd(a, b) that comes in handy later. Most of the verification of the details is left to you as an exercise. Theorem 2.5.7 Suppose a and b are nonzero integers. Then there exists a unique positive integer g with the following properties:
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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
Page 32 and 33: P A R T I Foundations of Logic and
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Page 42 and 43: 1.2 If-Then Statements 19 Definitio
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Page 60 and 61: Solution 1.4 Negations of Statement
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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
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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
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Page 102 and 103: Example 3.4.2 For any positive inte
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Page 108 and 109: 3.5 Variations of the PMI 85 EXERCI
Page 110 and 111: (g) (a −m ) −n = a mn (h) (ab)
Page 112 and 113: Strong Induction 3.5 Variations of
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Page 130 and 131: 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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