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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: en

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

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Absolute value      4 11 19 21 88 93 259 279 305 312 326 329 351 362 3 368 370
Addition      3 11 16 7 51 74 86 98 131 133 143 149 50 258 296 361—2 369 371 83
Additive identity      16 102 144 195 380
Additive inverse      3 16 144 149 50 375 381
Adjacency relation      208 11 228
Adjacent vertices      208 219 20
AGM inequality      5 12 22 3 29 43 258 268 274 291 359 398
Algebraic closure      368
Algorithm      126
Alternating series      279 291
Analysis by cases      18 20 37 189 193
Andre, Antoine Desire      173
Antiderivative      345
Antipodal points      293 301
Antisymmetric property      17 141
Appel, Kenneth      223
Approximation      264 5 284 5 307—13 318 22 338 353 396
Archimedean property      258 9 263 265 269 70
Archimedes      258
Area      20 1 42 163 174 185 298 307 314 337 48 351 1 358 360 397
Arguments      272 6 279 290
Arithmetic mean (average)      5 165 257
Arithmetic properties x, xi      3 16 7 51 2 76 7 139 142 1 149 164— 5 257 8 273 4 361 371 9 382-3
Arrangement      101 4 107 111 154 182 3 185 187 201 253
Art Gallery Problem      203 227 8 231
Associative property      16 86 112 150 369 373 5 383
Average value      337 346
Average xix      5 38 156 165 177 8 181 184 186 9 192 308 320 391
Axioms x      xiii 2 15 8 51 2 160 256— 8 267 270 276 373 377 9 382-3
Babylonian problem      2 3 15 21 24
Ballot list      187
Ballot path      244
Ballot problem      xi 170 172 3 182 185 187 206 244 391
Barnett, A.      177
Base q representation      x 76 8 80 94 96 137 269
Baseball problems      xx 168 170 184 186 7 198 281 335
Basis step      52 54 64 70 124 5 234
Bayes’ formula      176 186 391
Bell numbers      252
Bernoulli trials      170 175 179 80
Bernoulli, Jakob      170
Bernoulli, John      317
Bertrand, Joseph Louis      172 173
Bertrand’s paradox      174 185
Biconditional      32 6 48
Bijection      80 98 102 5 111 2 115—7 121 2 144 5 147 155 173 187—8 196 208 11 240 251 4 266 347— 9 353 358 373 4 382 3 386-93
Billiard problem      156 158 162
Binary, encoding      82 3 103 105
Binary, expansion      166 264 6 278 288 394
Binary, list (n-tuple)      82 103 105 109 116 120 188 206 215 217 239 10
Binary, operation      131 143 150 155 252 296 373
Binary, representation      77 96 139 144 149
Binomial coefficients      102 10 113 22 135 6 171 3 175 178 182 187—8 195 7 201 210 222 227 230 237 244 5 247 52 254 281 309 333
Binomial distribution      178 179 182
Binomial theorem      96 104 106 109 120 136 178 183 188 196 201 292 309 316 369
Bipartite graph      215 9 225 231
Bipartition      215 9 231
Bisection method      278 300 306 7 319
Block-walking      106 109 188 240
Bolzano — Weierstrass theorem      xi 277 9 289 302 304 306 365
Boundary      15 188 223 7 231 393
Bounded, function      12 13 24 31 269 303 305 310 312 326 7 332 45 357 366—7
Bounded, interval      302 3 313 335
Bounded, region      49 224 8 231 338
Bounded, sequence      258 261 2 267 70 274 9 284 287 92 300 302 305 360
Bounded, set      12 258 268 270 302 352 365 8 378 381 2
breakpoints      339 343 357 8
Butterfly effect      293
Calculus      x-xvi 68 69 87 94 118 271 287 307 60 397 100
Canonical expansion      263 6 269 278
Canonical representation      158 263
Cantor, Georg      266
Card problems      89 100 103 4 115 118 9 138 171 200 252 316 387
Cardinality      xii 76 87 92 98 270
Carmichael numbers      149
Cartesian coordinates      9
Cartesian Plane      361
Cartesian product      9 23 87 90 98 101 247 8
Casting out nines      144
Catalan numbers      173 244 5 249 54
Catalan recurrence      245 249 253 394
Catalan, Eugene Charles      244
Cauchy Convergence Crit      276 9 363
Cauchy mean value theorem      316 7 321 332 3
Cauchy sequence      xi xii xvi 276 9 282 3 291 304 326 7 363 369 378 80 383
Cauchy, Augustin      276
Ceiling function      126 135 138 252 389
Census problem      22
Chain rule      312 3 319 323 330 4 349 51 356 358 396—7
Chairperson Identity      108 117 8
Change of variables      94 168 344 347 8 353
Checkerboard problems      xvii 48 50 58 192 251 386
Chess-player problem      192 3
Chessboard      75 167
Chinese remainder theorem      xi 139 145 6 154 390
Chord      174 185 228 306 391
Chromatic number      219 22 231
Chromatic polynomial      221 2 231 253
circle      xix 14 45 49 159 174 185 188 220 1 227 236 250 1 307 323 1 337 351 3 364 369—70
Circuit      205 212
clock      22 143 198
Closed      17 131 375 377 379 383
Closed, curve      203 223 224 228
Closed, interval      9 73 98 268 270 294 297 302 1 313
Closed, set      365 8 370 375 377 379
Closed, trail      205 206 210 212
Closure property      16 150 164 368 377
Coconuts problem      137 389
Coefficients (also binomial coefficients) in equation      25 6 124 128 237—8
Coefficients (also binomial coefficients) in expansion      264 266 270
Coefficients (also binomial coefficients) in polynomials      11 41 46 59 60 69 87 122 131 2 155 160 2 166—7 231 234 361 367
Coefficients (also binomial coefficients) in power series      247 9
Coefficients (also binomial coefficients) in recurrence      234 241 3 252
Coefficients (also binomial coefficients), multinomial      170 183 188
Coin problems      xvii xviii 8 45 50 63 4 75 96 100 103 108 118 135 137 175 178 186 200 251 254
Coloring      154 219 23 227—8
Combinatorial proof      xii 104 8 115 22 188 230 248 332 388
Commutative property      16 86 150 369 373 5 381 383
Compact set      302 365 6 368 370
Comparison Test      282 3 286 7 291—2
Complement      9 185 193 4 210 224 229 365 370 398
Complete bipartite graph      225
Complete graph      216 220 230 392
Complete ordered field      15 258 372 82
Completeness axiom      17 54 160 256 8 267 270 1 276 284 299 302 340 378 381 2
Complex function      366 7 370
complex numbers      168 324 361 70
Component of graph      212 3 216 222
composition      85 8 97 8 112 150 155 277 295 298 310 312 3 356 368 370 1 374
compound statements      31 3
Compounding      337 350 357
Compounding on an interval      296 9 302 6 315—6 327 9 336 344 7 357
Compounding, function      165 223 296 306 311—3 315 320 327 36 344 60
Conclusion (of conditional)      32 40
Condensation test      292
Conditional      291
Conditional of sequences      256 259 65 268 274 92 279 295 8 302 304 318 22 343 3 350 363 70 378—80
Conditional of series      279 84 288 290 2 324 351 353 363 4
Conditional probability      170 174 7 186
Conditional statement      32 49 56 58—9
Congruence      xii 142 55 190 199 390
Congruent modulo n      142
Conjugate (complex numbers)      362 369
Conjunction      32 5 37 48—9
Connected graph      212 6 224 5 228—30
Connection relation      212 3
Constant, coefficients      236 8 241 3 249—52
Constant, function      131 3 141 241 301 304—5 315 6 340 346 349 351 358-9
Constant, sequence      259 261 283 381
Constant, term      3 97 180 238 243 387
containment      6 7 23 49 212
Continuous (see uniform.) at a point      296 8 302 5 311 2 317— 9 327 332 346 359
Continuously differentiable      318 9 333 346 8 355
Contradiction, proof by      26 7 35 40 43 48 9 130 161 190 3 262 267 286 7 366—8
Contrapositive      35 8 40 43 64 83 92 130 136 148 190 205 215 299 304 386 389 392 3 395
Convergence at a point      294 8 324—6
Convergence, absolute      282 291 363 4 370
CONVERSE      32 34 7 42 44 66 141 155 159 271 276 279 282 290 366
Convex function      xi 233 253 320 2 334 5 397
Convex polygon      245
Corollary      39
Cosine      xi 69 94 96 319 324 350 3 364 369 398
Countable set      89 92 4 98 161 167 256 266 8 270 290 387
Countably infinite      89
Counterexample      24 37 8 65 155 161 166 177 184
Counting      23 48 79 80 101 10 116— 22 136 139 154 172 194 7 201 221 2 228 31 232 54 386 9 393
Counting two ways      53 103 5 108 115 117 120 1 178 202
Coupon Collector Problem      170 180
Crossing edges      223
Cube (graph)      206 209 212 215 229
Cube (object)      73 4 119 20 203 226
Cube root      23 27 83 93 162 331 369
Cubic equation      168
Cubic polynomial      11 74 97 332
Curvature      307 322 4 336
Curve      14 159 167 203 1 223 1 228 307 323 4 342 353
CYCLE      27 91 170 184 187
Cycle description      114
Cycle in graph      211 7 220 1 224 31 392
Cycle in permutation      113 5 121 2 147
Cyclic shift      155
Dart Board Problem      xviii 123 129 31 137 202
de Morgan, Augustus      33
de Morgan’s laws      33 35 49
December      31
Decimal expansion      162 166 256 263 71 280 286 289 90 330 353 395
Decimal representation      xx 76 7
deck of cards      100 103 118 9 171 200
Decreasing function      12 95
Decreasing sequence      167 274 292
Defined on      10
Degree (polynomial)      11 21 59 60 74 82 87 108 10 122 132 134 155 168 187 237 8 241 3 249 253 288 292 306 334 367 8 392—3
Degree (vertex)      204 15 222 226 31
Degree-sum Formula      206 7 226—7
Deleted neighborhood      294 5 317
Denominator      157 161 165 6 249 266
Derangement      193 7 245 6 252
Derivative      308 16 319 322 330—5 346 9 355 359 396-7
Descartes, Rene      9
Diagonalization argument      xvi 266 7
Dice problems      103 118 9 171 2 182 185 187 193 197 254
Dice problems of functions      60 72 300 1 315
Dice problems of sets      9 21
Differentiable      307 24 329 36 346— 51 355 6 359 397
Differential equation      350 358 397
Differentiation      xi 87 307 36 345 56
Diophantine equation      xii 128 9 134 137 146 389
Diophantus      128
Direct proof      36 40 43 53
Dirichlet drawer principle      189
Dirichlet, Peter G. Lejeune      153 189
Disconnected graph      212 216 227 30
Discontinuous function      298 305 333
Disjoint      9 10 88 91 2 101 109 119 151 2 171 2 215 218 9 229 231 247 387 392
Disjunction      32 5 48—9
Distance      4 5 20 88 198 259 276 284 5 299 308 329 361 2 375
Distributive law      3 16 54 5 116 125— 6 165 179 80 281 342 361 369 371 374 5 378 381 383
Divergence      279 87 291 2 355 359 364 370
Divisibility      123 39 141 148 9 151—5 162 164 168 183 190 1 387-9
Division      3 16 7 26 42 126 133 156— 8 270 296 371 388 391
Division algorithm      126 131 2 142
Divisor      123 133
dodecahedron      xx 203 226
Dollars      xviii xix 22 45 134 5 146 186 8 254 280 332 337
Domain      10 4 21 24 52 3 81 84—8 92 4 133 294 299 301 333 348
Drawing of graph      204 209 223 7
Drummer Problem      100 111
Dyadic rational      165 6
Dyadic rational, edge (of graph)      204 202 31 250 253
Egyptian fraction      292
Element      6 8 14 28 9 34 9
Empty set      6 9 38 9 88 125 193—7
Endpoints (edges, paths, trails)      204 8 211 214 5 222 6 231
Endpoints (intervals)      264 5 297 314
EPSILON      2
Equality of sets      6 8 23—4
Equality relation      140 213
equation      241 242 250 370 393
Equivalence class/relation      xii 140 2 146 148 152 5 157 207 8 213 347 371 374 82 390
Erdos, Paul      191 199
Error function      308 13 332 346
Euclidean algorithm      126 9 131 134
Euler totient function      xi 193 6 200
Euler, Leonhard      147 155 244 193 203 205 244
Eulerian graph      205 6 228
Euler’s formula      224 7 231 250
Even number      8 9 26 7 64 65 96 103 153 192 204 207 215 6 228 293
Even permutation      111
Event      171 9 185 197 200 391
Eventually periodic      290
Exclusive OR      32
Existential quantifier      28 40 44 9
Expectation      170 177 81 187 8 234 334 337 391
Expected value      178
Exponential fen      283 292 324 328 335 345 349 50 356 359 364 369
exponentiation      85 96 316 358 383
Extremality      161 202 206 213
Face      187 198 224 8 231 393
Factor      104 123 6 134 9 148 9 151 154 157 163 8 183 193 1 200 241 247 9 368
Factorial      53 102 5 111 119 21 182 193 196 7 246 283 345 360 364
Factorization      xi-xii 55 59 123 125 131 133 6 149 157 161 240 331
Fermat, Pierre de      71 147 164
Fermat’s Last Theorem      164
Fermat’s Little Theorem      xi 147 8 152 155 183 206 390
Fibonacci numbers      xi 233 238 41 248 9 251 2 291
Field      xiii 15 7 24 97 150 1 160 258 270 361 2 368 372 377 83 392
Finger Game      181 188
Finite sequence      281
Finite set      88 9 94 98 102 105 141 171 183 193 374
First difference      253
First-order recurrence      235 8 252
Fixed point      13 113 122 196 200 333
Floor fen      126 135 198 230 1 252 264
Formal power series      247 8
Forward difference      333
Four-color problem      223 227
Fourier series      324 364
Fourier, Joseph      324
FRACTION      65 74 103 157 8 161 165—6 175 6 186 7 270 280 335 353
Fractional part      198 264 330
Friday the      13
Function      10 4 59 80 98 294—360
Functional digraph      112 5 122 147 154 204 370 388
Fundamental Thm. of Algebra      367 8
Fundamental Thm. of Arithmetic      125
Fundamental Thm. of Calculus      337 345 50 355 9 397
Gambler problems      186 7 254
Game      75
Games      xx 56 69 70 75 94 96 119 170 181 184 8 192 271 281 291
Gamma function      360
Gauss, Karl Friedrich      53 4 142 153
General solution      240 3 252
Generating function      247 9 254 291 336 394
Generator      132 3
Geometric mean      5
Geometric random variable      180
Geometric series      248 9 280 1 283 290 2 330 335 353 356 395
Geometric sum      55 6 73 78 9 265 335 358 9 386 397
Graph      113 154 202 31 253
Graph of function      12 5 23 4 83 97—8 260 299 300 306 8 315 318 24 329 334 6 339 41 348 354 392-7
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