Why Read This Book? - Index of

More documents

Recommendations

Info

7.3 More on Limits 217 values of 0 and 1 are sprinkled up and down the domain like grains of salt and pepper. No one knows for sure which is the salt and which is the pepper. � EXERCISE 7.2.20 The signum function, sgn x, is defined in the following way: Show limx→0 sgn x does not exist. ⎧ ⎪⎨ −1, if x0 (7.10) EXERCISE 7.2.21 Use familiar values of sin x to show that limx→0 sin(1/x) does not exist. 7.3 More on Limits 7.3.1 One-Sided Limits In defining limx→a f(x) = L, we insisted f be defined on both sides of a and all points to the nearby left and right of a map into Nɛ(L).Iff is only defined on one side of a, or perhaps if values of f to the immediate left of a behave differently from those to the immediate right (as in sgn x), we can discuss the limit of f(x) as x approaches a from the left or from the right separately. Instead of using entire deleted neighborhoods of a, we use only the left or right half of them. Definition 7.3.1 Suppose f is defined on the interval (a, b). Then we say lim x→a + f(x) = L (read “as x approaches a from the right”) if for all ɛ>0, there exists δ>0 such that a
218 Chapter 7 Functions of a Real Variable ByTheorem 7.2.15, limx→2(x 2 − 1) = 3 and limx→2 1/(x − 1) = 1 (in the two-sided sense). Thus, by the ⇒ direction of Exercise 7.3.2, lim x→2 −(x 2 − 1) = 3 and lim x→2 + 1/(x − 1) = 1. But then lim x→2 − f(x) = 3 and lim x→2 + f(x) = 1, so that limx→2 f(x) fails to exist by the ⇐ direction of Exercise 7.3.2. � EXERCISE 7.3.4 Let f be defined in the following way: f(x) = � 2 sgn x, if x ≤ 1 x 3 + 1, if x>1 Evaluate the following, with reference to applicable results. (a) limx→0− f(x) (b) limx→0 + f(x) (c) limx→0 f(x) (d) limx→1− f(x) (e) limx→1 + f(x) (f) limx→1 f(x) √ EXERCISE 7.3.5 Show limx→0 + x = 0. 7.3.2 Sequential Limits (7.13) Suppose f is a function defined on a deleted neighborhood of a, where f(x) → L as x → a. Let 〈an〉 be any sequence of real numbers that converges to a, but where every an is different from a. Now consider the sequence 〈f(an)〉. (See Fig. 7.4.) It seems intuitively clear that f(an) → L as n →∞, and this is a fairly straightforward fact to demonstrate. f (a 3 ) L f (an ) f (a2 ) f (a1 ) a 1 a 2 an a Figure 7.4 Convergence of a function in terms of sequences of domain elements. a 3
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 10 and 11:
3.10 Irrational Numbers 107 3.11 Re
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 52 and 53:
Page 54 and 55:
Page 56 and 57:
1.4 Negations of Statements 33 2. F
Page 58 and 59:
1.4 Negations of Statements 35 want
Page 60 and 61:
Solution 1.4 Negations of Statement
Page 62 and 63:
Solution 1. Everyone in this class
Page 64 and 65:
1.5 How We Write Proofs 41 Our purp
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
Page 72 and 73:
2.1.2 Properties of Multiplication
Page 74 and 75:
2.2 Ordering Properties of the Real
Page 76 and 77:
(c) For all real numbers a, a2 ≥
Page 78 and 79:
2.3 Absolute Value 55 (⇐) Now sup
Page 80 and 81:
2.4 The Division Algorithm 57 mn =
Page 82 and 83:
2.5 Divisibility and Prime Numbers
Page 84 and 85:
2.5 Divisibility and Prime Numbers
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
Page 96 and 97:
Example 3.3.2 Consider the family o
Page 98 and 99:
3.3 Families of Sets 75 x ∈ � A
Page 100 and 101:
3.3 Families of Sets 77 and ... and
Page 102 and 103:
Example 3.4.2 For any positive inte
Page 104 and 105:
3.4 The Principle of Mathematical I
Page 106 and 107:
Proof. To clean up the notation, we
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
Page 114 and 115:
3.6 Equivalence Relations 91 EXERCI
Page 116 and 117:
3.6 Equivalence Relations 93 beginn
Page 118 and 119:
3.6 Equivalence Relations 95 constr
Page 120 and 121:
3.7 Equivalence Classes and Partiti
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
3.8 Building the Rational Numbers 1
Page 128 and 129:
3.8 Building the Rational Numbers 1
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
Page 150 and 151:
4.2 One-to-one and Onto Functions 1
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
Page 156 and 157:
4.4 Composition and Inverse Functio
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
Page 164 and 165:
4.7 Infinite Sets 141 of A. This or
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
Page 172 and 173:
4.8 Cartesian Products and Cardinal
Page 174 and 175:
4.9 Combinations and Partitions 151
Page 176 and 177:
Page 178 and 179:
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
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 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 256 and 257: 7.6 Implications of Continuity 233
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.
Page 306 and 307:
8.6 Group Morphisms 283 Notice that
Page 308 and 309:
Ker � e x 2 x maps to x Ker � G
Page 310 and 311:
Rings 9 We can create algebraic str
Page 312 and 313:
9.1 Rings and Fields 289 (R9) Multi
Page 314 and 315:
9.1 Rings and Fields 291 we define
Page 316 and 317:
9.2 Subrings 293 In addition to pro
Page 318 and 319:
9.2 Subrings 295 p1 = q1 = 3. Verif
Page 320 and 321:
9.3 Ring Properties 297 Theorem 9.3
Page 322 and 323:
9.3 Ring Properties 299 EXERCISE 9.
Page 324 and 325:
9.4 Ring Extensions 301 in the inte
Page 326 and 327:
9.4 Ring Extensions 303 Example 9.4
Page 328 and 329:
9.4 Ring Extensions 305 We insist t
Page 330 and 331:
9.5 Ideals 307 An ideal, say a left
Page 332 and 333:
9.6 Generated Ideals 309 Proof. Sup
Page 334 and 335:
EXERCISE 9.6.6 In the integers, wha
Page 336 and 337:
9.7 Prime and Maximal Ideals 313 Ev
Page 338 and 339:
9.8 Integral Domains 315 EXERCISE 9
Page 340 and 341:
9.8 Integral Domains 317 Corollary
Page 342 and 343:
9.9 Unique Factorization Domains 31
Page 344 and 345:
9.10 Principal Ideal Domains 321 al
Page 346 and 347:
9.10 Principal Ideal Domains 323 So
Page 348 and 349:
9.11 Euclidean Domains 325 next exe
Page 350 and 351:
9.11 Euclidean Domains 327 Exercise
Page 352 and 353:
Proof. Let f and g be nonzero polyn
Page 354 and 355:
9.12 Polynomials over a Field 331 W
Page 356 and 357:
9.13 Polynomials over the Integers
Page 358 and 359:
9.14 Ring Morphisms 335 Example 9.1
Page 360 and 361:
9.14 Ring Morphisms 337 EXERCISE 9.
Page 362 and 363:
9.15 Quotient Rings 339 this, howev
Page 364 and 365:
9.15 Quotient Rings 341 Because the
Page 366 and 367:
9.15 Quotient Rings 343 However, th
Page 368 and 369:
Index =,51 ɛ (epsilon), 167-168 ɛ
Page 370 and 371:
Index 347 Cosets, 263, 264, 265 Lag
Page 372 and 373:
Index 349 Gauchy sequences, 202-206
Page 374 and 375:
Index 351 identity, 124 relations v
Page 376 and 377:
Index 353 Quaternions, 248, 253, 25
Page 378 and 379:
Index 355 Tower of Hanoi, 84 Transc
show all

Why Read This Book? - Index of

Create successful ePaper yourself

Delete template?

Save as template?