Why Read This Book? - Index of

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4.9 Combinations and Partitions 153 Theorem 4.9.3 Suppose |A| = n, and let n1,...,np be positive integers such that �p k=1 np = n. Then the number of ways to partition A into distinguishable subsets of sizes n1,...,np is given by � � n n! = n1,...,np n1!n2! ...np! (4.28) Theorem 4.9.3 says that the subsets of A are distinguishable. If all of the subset sizes n1,...,np are distinct, then all of the p subsets of A are clearly distinguishable. However, if ni = nj for some distinct i and j, Eq. (4.28) distinguishes between two partitions that differ only in that the elements of the ith and jth subsets are interchanged. For example, if A ={1, 2, 3, 4, 5}, then the partition is considered to be different from A1 ={2} A2 ={1, 4} A3 ={3, 5} (4.29) A1 ={2} A2 ={3, 5} A3 ={1, 4} (4.30) As an example, consider that we are going to split a group of 12 children into four groups of three. Equation (4.28) assumes that each group has a unique feature associated with it, so that swapping two entire groups would be considered a different outcome. 4.9.3 Applications Example 4.9.4 The State Lottery Commission draws six balls at random from a bin of 51 numbered balls, where order does not matter in determining if someone wins. There are �51� 6 ways to draw these balls. Thus if you buy one lottery ticket, your chances of winning are one in about 18 million. � Example 4.9.5 Your club of 50 members is going to choose a president, vicepresident, secretary, a banquet committee of four, and a service project committee of five. No one may serve in more than one capacity. This process can be thought of as partitioning a 50-element set into subsets of size 1, 1, 1, 4, 5, and 38. The number of ways this can be done is � 50 1 �� 49 1 �� 48 1 �� 47 4 �� 43 5 � 50! = = a whole bunch (4.31) 1!1!1!4!5!38! � Let’s put together all the counting techniques we have developed to answer some more complex questions.
154 Chapter 4 Functions Example 4.9.6 Suppose you toss a coin 10 times and observe the sequence of outcomes. From Example 4.8.14, we know that there are 2 10 possible outcomes. We want to count the number of these outcomes that have exactly 3 heads. Imagine we have the numbers in N10 ={1,...,10} written on slips of paper. An outcome of the 10 coin tosses with exactly 3 heads can be uniquely specified by taking a 3-element subset of N10 to represent which of the 10 slots are heads. There are C(10, 3) = (10 · 9 · 8)/(3 · 2 · 1) = 120 such choices. � Example 4.9.7 A box of light bulbs contains 100 bulbs, 3 of which are defective. If we choose 10 bulbs from the box, we want to calculate how many ways there are to get precisely one of the defective bulbs. Such a choice of 10 bulbs consists of 9 of the 97 good bulbs chosen without regard to order, and 1 of the defective bulbs chosen from the 3 bulbs. Multiplying these, we have C(97, 9) × C(3, 1) ways of choosing the bulbs in which precisely one is defective. � Exercise 4.6.7 says that if A and B are disjoint, then |A ∪ B| = |A| + |B|. Applied to counting the number of ways of performing a task, we call this theorem the sum rule. If counting the number of ways of performing a task must be broken up into disjoint cases, the sum rule says simply that the calculations for the different cases are added to produce the total number of ways of performing the task. Example 4.9.8 A license plate consists of up to seven characters, taken from the 26 letters and the digits 0–9. If there is room for spaces, they are not counted in the arrangement; for example, there is no distinction between TANGENT and TAN GENT . We count the total number of plates that can be issued. There are different cases based on the number of characters used. The number of onecharacter license plates is 36; the number of two-character license plates is 362 , and so on. Thus the total number of license plates is 36 + 36 2 + 36 3 +···+36 7 = 36(1 +···+36 6 ) = 36 � 367 � −1 36 36−1 = 35 (367 − 1) � Example 4.9.9 Your club of 30 women and 20 men is going to choose a committee of five. We calculate the number of ways to choose the committee if it must include at least one man. The number of ways to choose the committee with no restrictions is C(50, 5), and the number of ways to choose the committee with no men is C(30, 5). By the complement rule, the number of ways to choose the committee where there is at least one man is C(50, 5) − C(30, 5). � EXERCISE 4.9.10 Pizza Peddler offers 12 different toppings and is having a special on their two-topping pizzas. How many distinct ways are there to order one of their special pizzas? EXERCISE 4.9.11 You overheard that a TRUE/FALSE test of 10 questions had 7 TRUE and 3 FALSE answers. If you take the test with this knowledge, how many ways are there to fill out the answer sheet?
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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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Page 126 and 127: 3.8 Building the Rational Numbers 1
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Page 130 and 131: 3.10 Irrational Numbers 107 EXERCIS
Page 132 and 133: 3.10 Irrational Numbers 109 To the
Page 134 and 135: EXERCISE 3.10.2 √ 2 is irrational
Page 136 and 137: (a) {(x, y) : x ≤ y} (b) {(x, y)
Page 138 and 139: 3.11 Relations in General 115 (O3)
Page 140 and 141: 3.11 Relations in General 117 at al
Page 142 and 143: Functions 4 Second only to sets, fu
Page 144 and 145: 4.1 Definition and Examples 121 pro
Page 146 and 147: Solution We show that T satisfies p
Page 148 and 149: 4.2 One-to-one and Onto Functions 4
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Page 152 and 153: A 1 x A B f (A 1 ) Figure 4.4 The i
Page 154 and 155: 4.4 Composition and Inverse Functio
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Page 158 and 159: 4.5 Three Helpful Theorems 135 The
Page 160 and 161: 4.6 Finite Sets 137 EXERCISE 4.5.3
Page 162 and 163: 4.7 Infinite Sets 139 The next exer
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Page 166 and 167: 4.7 Infinite Sets 143 EXERCISE 4.7.
Page 168 and 169: EXERCISE 4.8.3 If A1,A2, and B are
Page 170 and 171: 4.8 Cartesian Products and Cardinal
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Page 174 and 175: 4.9 Combinations and Partitions 151
Page 178 and 179: 4.9 Combinations and Partitions 155
Page 180 and 181: 4.10 The Binomial Theorem 157 EXERC
Page 182 and 183: EXERCISE 4.10.3 Determine the follo
Page 184 and 185: (c) The coefficient of ab 3 c 2 in
Page 186 and 187: P A R T I I Basic Principles of Ana
Page 188 and 189: The Real Numbers 5 Let’s look at
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Page 192 and 193: 5.2 The Archimedean Property 169 EX
Page 194 and 195: 5.2 The Archimedean Property 171 No
Page 196 and 197: 5.3 Open and Closed Sets 173 Thus A
Page 198 and 199: 5.4 Interior, Exterior, Boundary, a
Page 200 and 201: 5.4 Interior, Exterior, Boundary, a
Page 202 and 203: 5.5 Closure of Sets 179 The constru
Page 204 and 205: 5.6 Compactness 181 EXERCISE 5.6.3
Page 206 and 207: 5.6 Compactness 183 EXERCISE 5.6.13
Page 208 and 209: Sequences of Real Numbers 6.1 Seque
Page 210 and 211: 6.1 Sequences Defined 187 Example 6
Page 212 and 213: EXERCISE 6.1.15 Consider the sequen
Page 214 and 215: L 1 e L L 2 e . . . . . . 1 2 3 4 N
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
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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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