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D'Angelo J.P., West D.B. Ч Mathematical thinking: problem-solving and proofs
D'Angelo J.P., West D.B. Ч Mathematical thinking: problem-solving and proofs

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Ќазвание: Mathematical thinking: problem-solving and proofs

јвторы: D'Angelo J.P., West D.B.

јннотаци€:

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.


язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

»здание: 2-nd edition

√од издани€: 1999

 оличество страниц: 433

ƒобавлена в каталог: 24.11.2013

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
-tuple      9 82Ч83 101Ч105 117 155 188 206 215 217 239
2-line form      113 114
Absolute value      4 11 19Ч21 88 93 259 279 305 312 326 329 351 362Ч363 368 370
Addition      3 11 16Ч17 51 74 86 98 131 133 143 149Ч150 258 296 361Ч362 369 371Ч383
Additive identity      16 102 144 195 380
Additive inverse      3 16 144 149Ч150 375 381
Adjacency relation      208Ч211 228
Adjacent vertices      208 219Ч220
AGM inequality      5 12 22Ч23 29 43 258 268 274 291 359 398
Algebraic closure      368
Algorithm      126
Alternating series      279 291
Analysis by cases      18Ч20 37 189 193
Andre, Antoine Desire      173
Antiderivative      345
Antipodal points      293 301
Antisymmetric property      17 141
Appel, Kenneth      223
Approximation      264Ч265 284Ч285 307Ч313 318Ч322 338 353 396
Archimedean property      258Ч259 263 265 269Ч270
Archimedes      258
Area      20Ч21 42 163 174 185 298 307 314 337Ч348 351Ч354 358 360 397
Arithmetic mean (average)      5 165 257
Arithmetic properties      x xi 3 16Ч17 51Ч52 76Ч77 139 142Ч144 149 164Ч165 257Ч258 273Ч274 361 371Ч379 382Ч383
Arrangement      101Ч104 107 111 154 182Ч183 185 187 201 253
Art Gallery Problem      203 227Ч228 231
Associative property      16 86 112 150 369 373Ч375 383
Average      xix 5 38 156 165 177Ч178 181 184 186Ч189 192 308 320 391
Average value      337 346
Axioms      x xiii 2 15Ч18 51Ч52 160 256Ч258 267 270 276 373 377Ч379 382Ч383
Babylonian problem      2Ч3 15 21 24
Ballot list      187
Ballot path      244
Ballot problem      xi 170 172Ч173 182 185 187 206 244 391
Barnett, A.      177
Base q representation      x 76Ч78 80 94 96 137 269
Baseball problems      xx 168 170 184 186Ч187 198 281 335
Basis step      52 54Ч64 70 124Ч125 234
Bayes' formula      176 186 391
Bell numbers      252
Bernoulli trials      170 175 179Ч180
Bernoulli, Jakob      170
Bernoulli, John      317
Bertrand's paradox      174 185
Bertrand, Joseph Louis      172 173
Biconditional      32Ч36 48
Bijection      80Ч98 102Ч105 111Ч112 115Ч117 121Ч122 144Ч145 147 155 173 187Ч188 196 208Ч211 240 251Ч254 266 347Ч349 353 358 373Ч374 382Ч383 386Ч393
Billiard problem      156 158 162
Binary encoding      82Ч83 103 105
Binary expansion      166 264Ч266 278 288 394
Binary list (n-tuple)      82 103 105 109 116 120 188 206 215 217 239Ч240
Binary operation      131 143 150 155 252 296 373
Binary representation      77 96 139 144 149
Binomial coefficients      102Ч110 113Ч122 135Ч136 171Ч173 175 178 182 187Ч188 195Ч197 201 210 222 227 230 237 244Ч245 247Ч252 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      215Ч219 225 231
Bipartition      215Ч219 231
Bisection method      278 300 306Ч307 319
Block-walking      106 109 188 240
Bolzano Ч Weierstrass theorem      xi 277Ч279 289 302 304 306 365
Boundary      15 188 223Ч227 231 393
Bounded function      12 13 24 31 269 303 305 310 312 326Ч327 332Ч345 357 366Ч367
Bounded interval      302Ч303 313 335
Bounded region      49 224Ч228 231 338
Bounded sequence      258 261Ч262 267Ч270 274Ч279 284 287Ч292 300 302 305 360
Bounded set      12 258 268 270 302 352 365Ч368 378 381Ч382
breakpoints      339 343 357Ч358
Butterfly effect      293
Calculus      xЧxvi 68 69 87 94 118 271 287 307Ч360 397Ч400
Canonical expansion      263Ч266 269 278
Canonical representation      158 263
Cantor, Georg      266
Card problems      89 100 103Ч104 115 118Ч119 138 171 200 252 316 387
Cardinality      xii 76 87Ч92 98 270
Carmichael numbers      149
Cartesian coordinates      9
Cartesian Plane      361
Cartesian product      9 23 87Ч90 98 101 247Ч248
Casting out nines      144
Catalan numbers      173 244Ч245 249Ч254
Catalan recurrence      245 249 253 394
Catalan, Eugene Charles      244
Cauchy Convergence Crit.      276Ч279 363
Cauchy mean value theorem      316Ч317 321 332Ч333
Cauchy sequence      xi xii xvi 276Ч279 282Ч283 291 304 326Ч327 363 369 378Ч380 383
Cauchy, Augustin      276
Ceiling function      126 135 138 252 389
Census problem      22
Chain rule      312Ч313 319 323 330Ч331 349Ч351 356 358 396Ч397
Chairperson Identity      108 117Ч118
Change of variables      94 168 344 347Ч348 353
Characteristic, equation      241 242 250 370 393
Characteristic, polynomial      241 242Ч243 249
Checkerboard problems      xvii 48 50 58 192 251 386
Chess-player problem      192Ч193
Chessboard      75 167
Chinese remainder theorem      xi 139 145Ч146 154 390
Chord      174 185 228 306 391
Chromatic number      219Ч222 231
Chromatic polynomial      221Ч222 231 253
circle      xix 14 45 49 159 174 185 188 220Ч221 227 236 250Ч251 307 323Ч324 337 351Ч353 364 369Ч370
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 302Ч304 313
Closed, set      365Ч368 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      25Ч26 124 128 237Ч238
Coefficients (also binomial coefficients) in expansion      264 266 270
Coefficients (also binomial coefficients) in polynomials      11 41 46 59Ч60 69 87 122 131Ч132 155 160Ч162 166Ч167 231 234 361 367
Coefficients (also binomial coefficients) in power series      247Ч249
Coefficients (also binomial coefficients) in recurrence      234 241Ч243 252
Coefficients (also binomial coefficients), multinomial      170 183 188
Coin problems      xvii xviii 8 45 50 63Ч64 75 96 100 103 108 118 135 137 175 178 186 200 251 254
Coloring      154 219Ч223 227Ч228
Combinatorial proof      xii 104Ч108 115Ч122 188 230 248 332 388
Commutative property      16 86 150 369 373Ч375 381 383
Compact set      302 365Ч366 368 370
Comparison Test      282Ч283 286Ч287 291Ч292
Complement      9 185 193Ч194 210 224 229 365 370 398
Complete bipartite graph      225
Complete graph      216 220 230 392
Complete ordered field      15 258 372Ч382
Completeness axiom      17 54 160 256Ч258 267 270Ч271 276 284 299 302 340 378 381Ч382
complex numbers      168 324 361Ч370
Component of graph      212Ч213 216 222
composition      85Ч88 97Ч98 112 150 155 277 295 298 310 312Ч313 356 368 370Ч371 374
compound statements      31Ч33
Conclusion (of conditional)      32Ч40
Condensation test      292
Conditional probability      170 174Ч177 186
Conditional statement      32Ч49 56 58Ч59
Congruence      xii 142Ч155 190 199 390
Congruent modulo n      142
Conjugate (complex numbers)      362 369
Conjunction      32Ч35 37 48Ч49
Connected graph      212Ч216 224Ч225 228Ч230
Connection relation      212Ч213
Constant coefficients      236Ч238 241Ч243 249Ч252
Constant function      131Ч133 141 241 301 304Ч305 315Ч316 340 346 349 351 358Ч359
Constant sequence      259 261 283 381
Constant term      3 97 180 238 243 387
containment      6 7 23 49 212
continuous      see "Uniform"
Continuous at a point      296Ч298 302Ч305 311Ч312 317Ч319 327 332 346 359
Continuous complex function      366Ч367 370
Continuous compounding      337 350 357
Continuous function      165 223 296Ч306 311Ч313 315 320 327Ч336 344Ч360
Continuous on an interval      296Ч299 302Ч306 315Ч316 327Ч329 336 344Ч347 357
Continuously differentiable      318Ч319 333 346Ч348 355
Contradiction, proof by      26Ч27 35Ч40 43 48Ч49 130 161 190Ч193 262 267 286Ч287 366Ч368
Contrapositive      35Ч38 40 43 64 83 92 130 136 148 190 205 215 299 304 386 389 392Ч393 395
Convergence      see "Uniform"
Convergence at a point      294Ч298 324Ч326
Convergence of sequences      256 259Ч265 268 274Ч292 279 295Ч298 302 304 318Ч322 343 350 363Ч370 378Ч380
Convergence of series      279Ч284 288 290Ч292 324 351 353 363Ч364
Convergence, absolute      282 291 363Ч364 370
Convergence, conditional      291
CONVERSE      32 34Ч37 42 44 66 141 155 159 271 276 279 282 290 366
Convex function      xi 233 253 320Ч322 334Ч335 397
Convex polygon      245
Corollary      39
Cosine      xi 69 94 96 319 324 350Ч353 364 369 398
Countable set      89 92Ч94 98 161 167 256 266Ч268 270 290 387
Countably infinite      89
Counterexample      24 37Ч38 65 155 161 166 177 184
Counting      23 48 79Ч80 101Ч110 116Ч122 136 139 154 172 194Ч197 201 221Ч222 228Ч231 232Ч254 386Ч389 393
Counting two ways      53 103Ч105 108 115 117 120Ч121 178 202
Coupon Collector Problem      170 180
Crossing edges      223
Cube (graph)      206 209 212 215 229
Cube (object)      73Ч74 119Ч120 203 226
Cube root      23 27 83 93 162 331 369
Cubic equation      168
Cubic polynomial      11 74 97 332
Curvature      307 322Ч324 336
Curve      14 159 167 203Ч204 223Ч221 228 307 323Ч324 342 353
CYCLE      27 91 170 184 187
Cycle description      114
Cycle in graph      211Ч217 220Ч221 224Ч231 392
Cycle in permutation      113Ч115 121Ч122 147
Cyclic shift      155
Dart Board Problem      xviii 123 129Ч131 137 202
de l'Hopital, Guillaume Francois      317
De Morgan's laws      33 35 49
de Morgan, Augustus      33
December 31 game      75
Decimal expansion      162 166 256 263Ч271 280 286 289Ч290 330 353 395
Decimal representation      xx 76Ч77
deck of cards      100 103 118Ч119 171 200
Decreasing function      12 95
Decreasing sequence      167 274 292
Defined on      10
Degree (polynomial)      11 21 59Ч60 74 82 87 108Ч110 122 132 134 155 168 187 237Ч238 241Ч243 249 253 288 292 306 334 367Ч368 392Ч393
Degree (vertex)      204Ч215 222 226Ч231
Degree-sum Formula      206Ч207 226Ч227
Deleted neighborhood      294Ч295 317
Denominator      157 161 165Ч166 249 266
Derangement      193Ч197 245Ч246 252
Derivative      308Ч316 319 322 330Ч335 346Ч349 355 359 396Ч397
Derivative, kth      319 334
Descartes, Rene      9
Diagonalization argument      xvi 266Ч267
Dice problems      103 118Ч119 171Ч172 182 185 187 193 197 254
Difference of functions      60 72 300Ч301 315
Difference of sets      9 21 C2
Difference, kth      253 333
Difference, quotient      308Ч315 325 330Ч333 336
Differentiable      307Ч324 329Ч336 346Ч351 355Ч356 359 397
Differential equation      350 358 397
Differentiation      xi 87 307Ч336 345Ч356
Diophantine equation      xii 128Ч129 134 137 146 389
Diophantus      128
Direct proof      36Ч40 43 53
Dirichlet drawer principle      189
Dirichlet, Peter G. Lejeune      153 189
Disconnected graph      212 216 227Ч230
Discontinuous function      298 305 333
Disjoint      9Ч10 88 91Ч92 101 109 119 151Ч152 171Ч172 215 218Ч219 229 231 247 387 392
Disjunction      32Ч35 48Ч49
Distance      4Ч5 20 88 198 259 276 284Ч285 299 308 329 361Ч362 375
Distributive law      3 16 54Ч55 116 125Ч126 165 179Ч180 281 342 361 369 371 374Ч375 378 381 383
Divergence      279Ч287 291Ч292 355 359 364 370
Divisibility      123Ч139 141 148Ч149 151Ч155 162 164 168 183 190Ч191 387Ч389
Division      3 16Ч17 26 42 126 133 156Ч158 270 296 371 388 391
Division algorithm      126 131Ч132 142
Divisor      123 133
dodecahedron      xx 203 226
Dollars      xviii xix 22 45 134Ч135 146 186Ч188 254 280 332 337
Domain      10Ч14 21 24 52Ч53 81 84Ч88 92Ч94 133 294 299 301 333 348
Drawing of graph      204 209 223Ч227
Drummer Problem      100 111
Dyadic rational      165Ч166
Edge (of graph)      204 202Ч231 250 253
Egyptian fraction      292
Element      6 8Ч14 28Ч29 34Ч39
Empty set      6 9 38Ч39 88 125 193Ч197
Endpoints (edges, paths, trails)      204Ч208 211 214Ч215 222Ч226 231
Endpoints (intervals)      264Ч265 297 314
Epsilon/2 arguments      272Ч276 279 290
Equality of sets      6Ч8 23Ч24
Equality relation      140 213
Equivalence class/relation      xii 140Ч142 146 148 152Ч155 157 207Ч208 213 347 371 374Ч382 390
Erdos, Paul      191 199
Error function      308Ч313 332 346
Euclidean algorithm      126Ч129 131 134
Euler totient function      xi 193Ч196 200
Euler's formula      224Ч227 231 250
Euler, Leonhard      147 155 244 193 203 205 244
Eulerian graph      205Ч206 228
Even number      8Ч9 26Ч27 64 65 96 103 153 192 204 207 215Ч216 228 293
Even permutation      111
Event      171Ч179 185 197 200 391
Eventually periodic      290
Exclusive OR      32
Existential quantifier      28Ч40 44Ч49
Expectation      170 177Ч181 187Ч188 234 334 337 391
Expected value      178
Exponential fcn.      283 292 324 328 335 345 349Ч350 356 359 364 369
exponentiation      85 96 316 358 383
Extremality      161 202 206 213
Face      187 198 224Ч228 231 393
Factor      104 123Ч126 134Ч139 148Ч149 151 154 157 163Ч168 183 193Ч194 200 241 247Ч249 368
Factorial      53 102Ч105 111 119Ч121 182 193 196Ч197 246 283 345 360 364
Factorization      xiЧxii 55 59 123 125 131 133Ч136 149 157 161 240 331
Fermat's last theorem      164
Fermat's little theorem      xi 147Ч148 152 155 183 206 390
Fermat, Pierre de      71 147 164
Fibonacci numbers      xi 233 238Ч244 248Ч249 251Ч252 291
Field      xiii 15Ч17 24 97 150Ч151 160 258 270 361Ч362 368 372 377Ч383 392
Finger Game      181 188
Finite sequence      281
Finite set      88Ч89 94 98 102 105 141 171 183 193 374
First difference      253
First-order recurrence      235Ч238 252
Fixed point      13 113 122 196 200 333
Floor fcn.      126 135 198 230Ч231 252 264
Formal power series      247Ч248
Forward difference      333
Four-color problem      223 227
Fourier series      324 364
Fourier, Joseph      324
FRACTION      65 74 103 157Ч158 161 165Ч166 175Ч176 186Ч187 270 280 335 353
Fractional part      198 264 330
Friday the 13th problem      xviii 152
Function      10Ч14 59 80Ч98 294Ч360
Functional digraph      112Ч115 122 147 154 204 370 388
Fundamental Thm. of Algebra      367Ч368
Fundamental Thm. of Arithmetic      125
Fundamental Thm. of Calculus      337 345Ч350 355Ч359 397
Gambler problems      186Ч187 254
Games      xx 56 69Ч70 75 94 96 119 170 181 184Ч188 192 271 281 291
Gamma function      360
Gauss, Karl Friedrich      53Ч54 142 153
General solution      240Ч243 252
Generating function      247Ч249 254 291 336 394
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