D'Angelo J.P., West D.B. — Mathematical Thinking: Problem-Solving and Proofs :: Электронная библиотека попечительского совета мехмата МГУ

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D'Angelo J.P., West D.B. — Mathematical Thinking: Problem-Solving and Proofs

D'Angelo J.P., West D.B. — Mathematical Thinking: Problem-Solving and Proofs

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Название: Mathematical Thinking: Problem-Solving and Proofs

Авторы: D'Angelo J.P., West D.B.

Аннотация:

This survey of both discrete and continuous mathematics focuses on the logical thinking skills necessary to understand and communicate fundamental ideas and proofs in mathematics, rather than on rote symbolic manipulation. Coverage begins with the fundamentals of mathematical language and proof techniques (such as induction); then applies them to easily-understood questions in elementary number theory and counting; then develops additional techniques of proofs via fundamental topics in discrete and continuous mathematics. Topics are addressed in the context of familiar objects; easily-understood, engaging examples; and over 700 stimulating exercises and problems, ranging from simple applications to subtle problems requiring ingenuity.

ELEMENTARY CONCEPTS. Numbers, Sets and Functions. Language and Proofs. Properties of Functions. Induction. PROPERTIES OF NUMBERS. Counting and Cardinality. Divisibility. Modular Arithmetic. The Rational Numbers. DISCRETE MATHEMATICS. Combinatorial Reasoning. Two Principles of Counting. Graph Theory. Recurrence Relations. CONTINUOUS MATHEMATICS. The Real Numbers. Sequences and Series. Continuity. Differentiation. Integration. The Complex Numbers.

For anyone interested in learning how to understand and write mathematical proofs, or a reference for college professors and high school teachers of mathematics.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: Second Edition

Год издания: 1999

Количество страниц: 412

Добавлена в каталог: 10.04.2008

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Предметный указатель

Greatest common divisor      123 30 133 8 145 154 164 168 193
Greatest lower bound      257 269
Group      150 155
Haken, Wolfgang      223
Hall’s condition      218 231 393
Handshake problem      50 58 60 1 205
Harmonic mean      5 6
Harmonic series      282
Homogeneous      234 5 241 3 250 252
Hypothesis (of conditional)      32 41
Icosahedron xx      203 226
Ideal      132 3 138 379
Identity element      16 144 149 50 155 362 383
Identity function      13 4 81 86 95 98 111 2 356 374
Identity permutation      111 2 121
Iiyective/iiyection      83 7 90 8 102 113 122 145 153 204 223 306 316 321 352 372 4 377 383 386—7
Image      10 4 21 24 81 1 87 92—4 102 166 223 268 9 302 306 333 348 370 373 382
Imaginary part      362 4 369
Improper integral      345 358 360
incident      204 211 218
Inclusion relation      141 193
Inclusion — Exclusion Principle      189 193 8 201 2 221 2 243 392
Increasing function      13 46 69 83 5 94 7 322 333 348 50 352 357—9
Increasing sequence      261 2 267 277—8
Indefinite integral      347
Independent events      170 175 179 85
Independent set      215 6 219 22
Index of summation      53 5 107 120 358 387
Indicator variable      179
Indirect proof      35 38 40
Induction      50 75 78 107 10 115—28 134 8 161 215 9 225 39 250—2 278 339 370 4
Induction, hypothesis      56 59 64 67 70
Induction, parameter      56 59 61
Induction, step      52 63
Infimum      257 260 2 268 9 284 302— 3 313 338 9 342 352
Infinite set      88 91 94 98
Infinitely differentiable      319
Inhomogeneous      234 242 244 249
Initial values      234 5 238 43 249 54
Integer combination      124 30 134
Integer part      264
Integer point      105 129 158 161 190 1
Integers      6 8 17 26 7 36 39 80 89 94 123 59 371—7
Integrable      340 8 354 357
integral      337 340 1 345 9 353 358— 60 397 8
Integral domain      133
integrand      341 345 353 355
Integration      337 60
Integration by parts      345 348 352 360
Interchange of limits      306 7 325 328 351 354 5
Interior      224 227
Intermediate Value Theorem      299 301 304 6 352 395—6
Intersection      9 10 20 23 4 34 5 49 51 172 175 6 194 8 226 231 250
Intersective identity      195
interval      9 12 23 57 90 259 65 294 8 301 6 313 47 352 7 365—6
Inverse      3 16 7 133 144 5 150—1 155 190 248 362 4 375 378 381
Inverse Composition Formula      85
Inverse function      81 2 94 8 112 121 158 173 306 313 333 348 50 356
Inverse image      14 84 6 92
Irrational number      156 160 2
Irreducible      133 4 138
Isolated vertex      212
Isomorphism      207 11 216
Isotherm      14
Iterate      98 112 122 235 246
Iteration      113 4 373
Jewel thieves problem      xx 293 301
Jordan curve theorem      224
Key problems      187 199
Kirchhoff, Gustav      214
Kleitman, Daniel J.      189
Knockout tournament      55 6
Konigsberg bridge problem      202 7
Kronecker, Leopold      258
Kuratowski’s Theorem      226
L-tiling      62 75
Lagrange, Joseph Louis      151
Lattice path      105 6 116 173 187 244
Leaf      214 6 231 393
Least common multiple      135
Least common refinement      339 42 357
Least upper bound      17 257 62 268—9 300 302 306 338 9 344 378 394
Leibniz, Gottfried Wilhelm      183
Lemma      39
Length (in graph)      205 210 6 223 226
Leonardo of Pisa      238
Level curve      14
Level set      14 5 141 178
lim sup      284 292 355 6 360 370
LIMIT      259 98 302 306 8 312 4 317— 9 324 30 333 7 341 2 345—6 350 353 60 363 8 377 9 382-3
Limit Comparison Test      287 291 2 395
Limit point      290
Limits of integration      94 340
Line      12 4 25 6 75 83 129 156 158 9 161 166 173 179 187 192 260 308 315 318 320 324
Linear, approximation      307 13 318 333 396
Linear, combination      234
Linear, equations      25 6 45 6 123 128 237 240 242
Linear, function      11 308 315
Linear, recurrence      234 5 241 4
Linearity of expectation      178 80 187
Linearity of Integration      342
Linearity of recurrence      234 240
LIST      9 52 77 9 94 100 8 111 3 121 2 147 154 172 175 183 188 191 199 205 239 251 4 281 359
Local maximum/minimum      314 5
Logarithm xi      94 136 142 291 337 349 50 356 358 60 364 397—8
Logical connective      32 4
Logical equivalence      33 5 37 42 48—9
Loop      112
Lower bound      31 199 200 257 275 306 322 339
Lower sum      339 45 358 9 397
Lowest terms      27 157 66 263
Lucas, Edouard      232
L’Hopital, Guillaume Francois de      317
L’Hopital’s Rule      317 8 333 5 350 359
Magnitude      361 70
Marriage problem      202 217 8
Matching      218 231
Mathematical statement      27 8
Maximal path      213 4 224 5 230
Maximal trail      205 6
Maximum      9 15 22 24 89 121 152 188 191 199 203 214 217 229 31 257 261 272 302 313 5 332 334 336 359 366 7
Maximum — Minimum Theorem      302 306 313 315
Mean value theorem      315 7 321 33
Measure zero      290
membership      6 8 34 5
Mersenne prime      136
Method of descent      64 5 71 161
Minimum      9 22 24 47 8 89 121 181 199 219 228 31 257 302 313—5 332 359 366 8 370
Modular arithmetic      139 55 183 190 199 229 379 390
Modulus      142
Monomial      11 108 183 187
Monotone Convergence Theorem      259 261 2 265 267 270 274 6 281 284 286 289 322 389 394
Monotone function      13 84 95 313 333 341 344 348 357 359
Monotone sequence      261 2 267 70 276— 7 288 9 300
Monotone sublist      191 199
multinomial      108 147 170 182 4 187
Multiplication      3 11 16 7 51 4 77 86 95 131 4 143 4 147 50 154— 5 258 270 296 361 2 369 83
Multiplicative identity      16 51 102 144 150 195 248 372 382
Multiplicative inverse      3 16 133 144 5 151 190 362 378
Multiplicity      125 211
Multiplier Effect      280 1
Mutually exclusive      175
Natural numbers      6 8 50 80 87 90 193 9 233 4 247 8 258 371—83
Necklace problems xx      154 293 301
Negation      27 8 30 1 38 40 44—6
Negative number      3 16 18 77 129 30 375 380 2
Neighborhood      294 9 308 18 332—3
Neighbors      208 218 9 230
Nested interval property      270
Newspaper Problem      139 146 7 154

Newton, Sir Isaac      318
Newton’s method      318 22 334—5
NIM      69 70 94 96
Nondecreasing      8 13 95 122 187 256 261 5 268 321 2 333
Nonincreasing      13 119 122 254 261 3 284
Nowhere differentiable function xi      324 329 30 336
Numerator      157 165 6 290 317—8
Octahedron xx      203 226
Odd cycle      216 219 20
odd number      8 26 7 29 36 49 63— 6 90 111 143 152 165 191 205 215 7 228 254 386
Odd permutation      111
Oh notation      22 110 154
One      16 51 372 381 2
One-to-one correspondence      76 80 1 91 3 105 107 391
One-to-one/onto functions      93
Open ball/set      365 6 370
Open interval      9 76 90 98 294 7 302— 3 316 319 322 346
Operator      86 7 333
Order axioms      16 7 52 377
Order of element      147 8 155
Order of growth      46 110
Order of recurrence      233 43
Order relation      141 382
Ordered field      15 7 160 258 270 372 377 382 3
Ordered pair      9 16 98 159 361
Origin      13 91 2 105 6 156 61 361—4
Outerplanar graph      227 8 231
Palindrome xviii      153
PARAMETER      56 61 67 9 159 180 247
Parametric equations      159 60 166—7
Parenthesization      253
Parity      8 48 9 111 121 139 10 143 167 190 202 215 252 390
Partial sum      279 2 286 290 2 353
Particular solution      242 3 249
Partite set      216 7 225
Partition of integer      122 254
Partition of interval      261 1 339 45 353 357—9
Partition of region      62 236 253
Partition of set      101 109 141 148 171 2 176 190 216 7 221 230 1 334 379
Pascal, Blaise      106
Pascal’s formula      106 8 120 387
Pascal’s triangle      106
Path      105 6 158 173 187 208 211—6 222 30 244 253 391-2
Path-connected      224
Penny Problem x      8 13 18 113 122
Perfect matching      218 9 231
Perfect number      136
Periodic      264 290
Permutation      102 111 6 121 2 141 148 153 193 196 246 388
Petersen graph      229 231 392
PI      53 69 94 96 266 290 332 337 351 3 359 60 364 369—70
Pigeonhole Principle      189 93 198—9 202 217 228 231 270 277
Planar/plane graph      223 8
Poker problems      100 103 119 171
Polya, George      136
Polygon      119 203 228 233 245
Polynomial      11 59 60 87 109 10 131 4 182 234 237 8 367—70
Positive number      3 5 8 157 166 257 259 274 283 319 363 8
Positive set      16 375 82
Postage Stamp Problem      137
Power series      324 350 354 6 360 363 4 369 70
Power set      6 7 82 152
Prime      123 39 144 62 148 166—8 183 190 193 200 388 91
Prime factor      149 193 4 196
Prime factorization      xi 125 135 161
Principal ideal      132 3 138
probability      xi xx 100 103 118 9 167 8 170 84 193 4 197 200 246 254 271 281 291 335 7 353
Probability space      173 80 184 7 391
Process of elimination      41
PRODUCT      2 3 15
Product of formal power series      247
Product of functions      11 87
Product of numbers      361 373 376 380
Proper coloring      220 1
Proper subset      6 80 89 98 218
Proposition      39
Pythagoras      353
Pythagorean Theorem      162 3
Pythagorean triples      164 167 8
Quadratic equation      2 3 7 21 159 162 168 384
Quadratic formula x      2 3 12 23 30 46 96 162 167 249
Quadratic polynomial      3 11 97 243
Quadrisection      365
Quantifier      28 45
Quotient      308 15 325 330 3 336
Quotient rule      310 313 323 331
Raabe’s test      283
Rabbits and Cadillacs      233 239
Radius of convergence      351 355 6 360
Random      103 167 170 4 178 80 185 7 193 4 198
Random variable      170 177 81 334 336 391
Ratio Test      283 291 2 351 363—4
Rational numbers      6 17 26 7 89 156 68 265 71 290 368 71 376—83
Rational Zeros Theorem      161 2 167
Real numbers      3 7 11 25 37 9 44—55 59 60 83 90 93 8 131 5 156—61 165 8 256 360 377-83
Real part      362
Real-valued fen      11 84 306 366 370
Reciprocal      xviii 16 21 162 165 269 292 306 7 324 381 395
rectangle      xix 9 20 48 62 75 116 119 135 163 200 298 314 338 9 341 347 8 365 6 370 386—9
Recurrence relation      232 54
Reducible polynomial      133 4
Refinement (partition)      283 339 42 357
Reflexive      17 140 1 148 152 208 376
Region xviii-xx      10 49 123 61 2 163 174 185 6 198 223 7 235 6 242 249 250 1 337 8 344 354 392—3
Relation      140 8 152 5 157 190 193 207 13 371 83
Relatively prime      124 35 144 6 150—5 162 166 8 190 193 196 200 221
Remainder      12 32 137 9 142 145 51
Repeating decimal      280
Riemann integral      345
Riemann, G. F. B.      345
Ring      xii 133 232 236 379
Rolle’s Theorem      315 317
Roman numerals      76
Root      23 27 83 93 240 3 319 321 364 367 369 70
Root test      284 292 355 370
Rule of product      101 2 115 6 193
Rule of sum      101 115 6 193
Russell, Bertrand      39
Scaling operator      87
Schroeder — Bemstein Thm.      x xiv 91
Second-order recurrence      238 41 246
Selection      101 7 117 22 186 7 218 240 248 251 388 393
Selection with repetition      107 8 121 187 200 1 248 388 391
SEQUENCE      52 3 63 70 73 5 89 94 98 135 233 92 295 8 302 5 318—9 340 2 358 60 363 70 377—83
Sequence/series of functions      289 324 30 335 6 354 8 397
Sequential (continuity, convergence, limit)      295 8 300 302 304 350
Series      180 271 279 92 324 30 335 350 6 360 363 4 369 70 395—8
Set      6 10
Shifting the index      55 238 387
Simple curve      223 4
Simple graph      204 225
Simpson’s paradox      xi 177 186
Sine      xi 94 286 295 298 309 319 324 336 350 3 364 369 398
Size of finite set      88 91 101 5 116—23
Smooth      307 319 334
Sorting      100 112 4 121
Spanning tree      214 216
Speed      126 289 308 316
Square root      3 4 30 42 160 2 167 250 256 8 263 274 309 324 362
Squeeze theorem      273 289 291 297 310 312
Stirling number      334
Stirling’s formula      360
Strictly increasing      13 85 96 289 349
Strong induction      63 70 75 96 124—8 132 134 137 251 386 389 393
Subgraph      204 6 209 22 226 31 392
Subsequence      277 9 287 90 302 304 365 7 379 80 383 395
Subset      6 10 193—6
Substitution      18 9 22 68 70 127—8 164 238 245 6 252 359 384 391
Subtraction      3 16 7 77 135 375—6
Successor function      372 3

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