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4.8 Cartesian Products and Cardinality 147 coordinate is f(a2), and so on. Since distinct m-tuples from B m represent distinct functions from A to B, we have proved the following. Theorem 4.8.8 If A and B are non-empty sets with |A| = m and |B| = n, then there are n m distinct functions from A to B. Here is an informal way to think of Theorem 4.8.8. Let A = Nm and imagine we have m empty slots numbered 1, 2,...,m. Into each slot we insert precisely one element of B, where any element of B may be used repeatedly if we like. Filling the slots in this way is equivalent to defining a function f : Nm → B. There are n choices of element to place in the first slot. For each of these ways to fill the first slot, there are n ways to fill the second slot, so that there are n 2 ways to fill the first two slots. Continuing, we have that there are n m ways to fill all m slots. Theorem 4.8.8 motivates the notation B A to represent the set of all functions from A to B. What if we must fill each of the n slots with one of the m elements of B, but we are not allowed to use any element of B more than once? How many ways can this be done? First note that if such a task can be done at all, it must be that m ≤ n. We have a choice of n objects for the first slot. Then, regardless of which element of B was placed in the first slot, there are n − 1 possible elements for the second slot, and so on. Thus the number of ways to fill the m slots with distinct elements of B is n(n − 1)(n − 2)...(n− m + 1). We write this as P(n, m) = n(n − 1)(n − 2)...(n− m + 1) = n! (n − m)! (4.23) which is called the number of permutations of n objects taken m at a time. Filling the slots with elements of B so that there is no repetition is the same as constructing a function from B to Nm that is one-to-one. With this, we have an informal proof of the following. A rigorous proof should be done by induction. EXERCISE 4.8.9 Let A and B be non-empty sets with |A| = m ≤ n = |B|. Then there are P(n, m) distinct one-to-one functions from A to B. 18 If Eq. (4.23) is to be taken as meaningful for m = 0, it implies that there is one way to arrange zero of n objects. It also suggests that there exists a unique (empty) function from the empty set to any finite set. If U is a finite univeral set and A is a subset of U, then Exercise 4.6.10 implies the complement rule, which says |A| = |U| − � �A C� � (4.24) Sometimes the task of determining the cardinality of a set is more easily done by exploiting Eq. (4.24). 18 Induct on m ≥ 1.
148 Chapter 4 Functions EXERCISE 4.8.10 Let A and B be non-empty sets with |A| = m ≤ n = |B|. How many distinct functions from A to B are not one-to-one? Suppose A is a set with n elements and (a1,...,an) is an element of A n , where all coordinates of the n-tuple are distinct. We may think of such an n-tuple as a way of ordering the elements of A. EXERCISE 4.8.11 If A is a set with n elements, in how many distinct ways can its elements be ordered? 4.8.3 Applications How many ways are there to put together a meal from all the cafeteria offerings? To arrange your CD collection on your dresser? To name a baby boy from a list of family preferences? Results from this section can help us answer some questions of the sort “How many ways are there to ...?” Example 4.8.12 Your university cafeteria has the following menu for today’s lunch. Meats : Meatloaf, Chicken, Fish sticks Starchy vegetables : Potatoes, Rice, Corn, Pasta Green vegetables : Beans, Broccoli, Salad, Spinach Breads : Rolls, Corn bread Desserts : Chocolate cake, Pudding If you choose one item from each category of the menu, how many different meals could you put together? Solution If we let A1 be the set of meat offerings, A2 the set of starchy vegetables, and so on, then each potential complete meal is an element of � 5 k=1 Ak. By Exercise 4.8.7, there are 3 × 4 × 4 × 2 × 2 = 192 possible meals. � In Example 4.8.12, counting the number of possible meals can be visualized in the following way. We have five empty slots to fill on our plate, the first to be filled with a choice of meat, the second with a choice of starchy vegetable, and so on. Furthermore, the number of choices available to fill a particular slot is unaffected by the way any of the previous slots have been filled or the way the remaining slots will be filled. Multiplying the number of ways to fill each slot illustrates the multiplication rule.Ifann-step process is such that the kth step can be done in ak ways, and the number of ways each step can be done is unaffected by the choice made for any other step, then the total number of ways to perform all n steps is � nk=1 ak.
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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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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 172 and 173: 4.8 Cartesian Products and Cardinal
Page 174 and 175: 4.9 Combinations and Partitions 151
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
Page 190 and 191: 5.1 The Least Upper Bound Axiom 167
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
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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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