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7.6 Implications of Continuity 233 (c) Use your result from part (b) to prove the fundamental theorem of algebra for the case a2n+1 > 0. (d) Prove the fundamental theorem of algebra for the case a2n+1 < 0 by applying part (c) to −P(x). Theorem 7.1.7 and Corollary 7.1.9 demonstrated that strict monotonicity of a function on a set implies that it is one-to-one (hence invertible on the image of the set). The converse of Corollary 7.1.9 is clearly not true, as is illustrated by f(x) = 1/x on the nonzero real numbers, which is one-to-one but not monotone. However, if f is one-to-one and continuous on a set, then it must be strictly monotone. The IVT will come in handy in the proof of the next result. EXERCISE 7.6.5 If f is continuous and one-to-one on [a, b], then f is strictly monotone on [a, b]. 4,5 The IVT can also help us prove that a continuous, invertible function has a continuous inverse. Specifically, if f is continuous and invertible on [a, b], then it is continuous and one-to-one. By Exercise 7.6.5, f is strictly monotone on [a, b] also. If f is strictly increasing, then f(a)
234 Chapter 7 Functions of a Real Variable When you prove Exercise 7.6.7, you can write a very elegant proof if you will use a slightly different form of the definition of continuity than that in expression (7.17). The following statements are all equivalent: |x − a|
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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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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
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
Page 226 and 227: a1 a4 aN 1 2 aN aN 1 1 aN 1 3 a5 a3
Page 228 and 229: Example 6.4.7 The terms a0 = 1 a1 =
Page 230 and 231: Functions of a Real Variable 7 Func
Page 232 and 233: 7.1 Bounded and Monotone Functions
Page 234 and 235: 50 40 30 20 7 1 ε 7 7 2 ε 0 ( ( F
Page 236 and 237: 7.2 Limits and Their Basic Properti
Page 238 and 239: 7.2 Limits and Their Basic Properti
Page 240 and 241: 7.3 More on Limits 217 values of 0
Page 242 and 243: 7.4 Limits Involving Infinity 219 W
Page 244 and 245: 7.4 Limits Involving Infinity 221 T
Page 246 and 247: (a) limx→a f(x) =−∞ (b) limx
Page 248 and 249: 7.5 Continuity 225 If f is not cont
Page 250 and 251: 1 2 3 4 5 6 Figure 7.6 Some example
Page 252 and 253: |a − x| |xa| = |x − a| |x||a| 7
Page 254 and 255: 7.6 Implications of Continuity 231
Page 258 and 259: 7.7 Uniform Continuity 235 we decla
Page 260 and 261: 7.7 Uniform Continuity 237 Next, le
Page 262 and 263: 7.7.2 Uniform Continuity and Compac
Page 264 and 265: P A R T I I I Basic Principles of A
Page 266 and 267: Groups 8 In its simplest terms, alg
Page 268 and 269: 8.1 Introduction to Groups 245 are
Page 270 and 271: 8.1 Introduction to Groups 247 Defi
Page 272 and 273: ◦ 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 0
Page 274 and 275: Show that C × is an abelian group
Page 276 and 277: 8.2 Subgroups 253 G1 and G3 are aut
Page 278 and 279: 8.2 Subgroups 255 Notice how proper
Page 280 and 281: 8.2 Subgroups 257 Example 8.2.15 su
Page 282 and 283: EXERCISE 8.2.23 If G is a group and
Page 284 and 285: 8.3 Quotient Groups 261 [0] ={...,
Page 286 and 287: 8.3 Quotient Groups 263 is standard
Page 288 and 289: Step 2: Define an operation ∗H on
Page 290 and 291: (iii) (Z + 3.8) +Z (Z + 1.2) (iv)
Page 292 and 293: then for a given f ∈ S and any a
Page 294 and 295: Thus f 2 = f ◦ f = (132)(56) ◦
Page 296 and 297: 2 3 1 3 1 Figure 8.1 Rigid square u
Page 298 and 299: EXERCISE 8.4.12 Complete Table 8.64
Page 300 and 301: 8.5 Normal Subgroups 277 Theorem 8.
Page 302 and 303: 8.5 Normal Subgroups 279 point to b
Page 304 and 305: 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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