Why Read This Book? - Index of

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3.10 Irrational Numbers 107 3.11 Relations in General 111 4 Functions 119 4.1 Definition and Examples 119 4.2 One-to-one and Onto Functions 125 4.3 Image and Pre-Image Sets 128 4.4 Composition and Inverse Functions 131 4.4.1 Composition of Functions 132 4.4.2 Inverse Functions 133 4.5 Three Helpful Theorems 135 4.6 Finite Sets 137 4.7 Infinite Sets 139 4.8 Cartesian Products and Cardinality 144 4.8.1 Cartesian Products 144 4.8.2 Functions Between Finite Sets 146 4.8.3 Applications 148 4.9 Combinations and Partitions 151 4.9.1 Combinations 151 4.9.2 Partitioning a Set 152 4.9.3 Applications 153 4.10 The Binomial Theorem 157 II Basic Principles of Analysis 163 5 The Real Numbers 165 5.1 The Least Upper Bound Axiom 165 5.1.1 Least Upper Bounds 166 5.1.2 Greatest Lower Bounds 168 5.2 The Archimedean Property 169 5.2.1 Maximum and Minimum of Finite Sets 170 5.3 Open and Closed Sets 172 5.4 Interior, Exterior, Boundary, and Cluster Points 175 5.4.1 Interior, Exterior, and Boundary 175 5.4.2 Cluster Points 176 5.5 Closure of Sets 178 5.6 Compactness 180 6 Sequences of Real Numbers 185 6.1 Sequences Defined 185 6.1.1 Monotone Sequences 186 6.1.2 Bounded Sequences 187 Contents ix
x Contents 6.2 Convergence of Sequences 190 6.2.1 Convergence to a Real Number 190 6.2.2 Convergence to Infinity 196 6.3 The Nested Interval Property 197 6.3.1 From LUB Axiom to NIP 198 6.3.2 The NIP Applied to Subsequences 199 6.3.3 From NIP to LUB Axiom 201 6.4 Cauchy Sequences 202 6.4.1 Convergence of Cauchy Sequences 203 6.4.2 From Completeness to the NIP 205 7 Functions of a Real Variable 207 7.1 Bounded and Monotone Functions 207 7.1.1 Bounded Functions 207 7.1.2 Monotone Functions 208 7.2 Limits and Their Basic Properties 210 7.2.1 Definition of Limit 210 7.2.2 Basic Theorems of Limits 213 7.3 More on Limits 217 7.3.1 One-Sided Limits 217 7.3.2 Sequential Limits 218 7.4 Limits Involving Infinity 219 7.4.1 Limits at Infinity 220 7.4.2 Limits of Infinity 222 7.5 Continuity 224 7.5.1 Continuity at a Point 224 7.5.2 Continuity on a Set 228 7.5.3 One-Sided Continuity 230 7.6 Implications of Continuity 231 7.6.1 The Intermediate Value Theorem 231 7.6.2 Continuity and Open Sets 233 7.7 Uniform Continuity 235 7.7.1 Definition and Examples 236 7.7.2 Uniform Continuity and Compact Sets 239 III Basic Principles of Algebra 241 8 Groups 243 8.1 Introduction to Groups 243 8.1.1 Basic Characteristics of Algebraic Structures 243 8.1.2 Groups Defined 246
Page 2 and 3: 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 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
Page 56 and 57: 1.4 Negations of Statements 33 2. F
Page 58 and 59: 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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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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