Mathematical Reasoning- Writing and Proof, Version 2.1, 2014a

More documents

Recommendations

Info

184 Chapter 4. Mathematical Induction (b) Why is it not possible to use mathematical induction to prove a proposition of the form For each real number x with x 1, P.x/, where P.x/ is some predicate? 18. Evaluation of proofs See the instructions for Exercise (19) onpage100 from Section 3.1. n.3n 1/ (a) For each natural number n, 1 C 4 C 7 CC.3n 2/ D . 2 Proof. We will prove this proposition using mathematical induction. So we let P.n/ be the open sentence 1 C 4 C 7 CC.3n 2/: Using n D 1, we see that 3n 2 D 1 and hence, P.1/is true. We now assume that P.k/ is true. That is, k.3k 1/ 1 C 4 C 7 CC.3k 2/ D : 2 We then see that .k C 1/ .3k C 2/ 1 C 4 C 7 CC.3k 2/ C .3.k C 1/ 2/ D 2 k.3k 1/ .k C 1/ .3k C 2/ C .3k C 1/ D 2 2 3k 2 k C .6k C 2/ D 3k2 C 5k C 2 2 2 3k 2 C 5k C 2 2 D 3k2 C 5k C 2 : 2 We have thus proved that P.k C 1/ is true, and hence, we have proved the proposition. n.3n 1/ (b) For each natural number n, 1 C 4 C 7 CC.3n 2/ D . 2 Proof. We will prove this proposition using mathematical induction. So we let P.n/ D 1 C 4 C 7 CC.3n 2/:
4.1. The Principle of Mathematical Induction 185 Using n D 1, we see that P.1/ D 1 and hence, P.1/ is true. We now assume that P.k/ is true. That is, We then see that 1 C 4 C 7 CC.3k 2/ D k.3k 1/ : 2 P.k C 1/ D 1 C 4 C 7 CC.3k 2/ C .3.k C 1/ 2/ k.3k 1/ D C 3.kC 1/ 2 2 D 3k2 k C 6k C 6 4 2 D 3k2 C 5k C 2 2 .k C 1/ .3k C 2/ D : 2 We have thus proved that P.k C 1/ is true, and hence, we have proved the proposition. (c) All dogs are the same breed. Proof. We will prove this proposition using mathematical induction. For each natural number n, weletP.n/ be Any set of n dogs consists entirely of dogs of the same breed. We will prove that for each natural number n, P.n/ is true, which will prove that all dogs are the same breed. A set with only one dog consists entirely of dogs of the same breed and, hence, P.1/ is true. So we let k be a natural number and assume that P.k/ is true, that is,thateverysetofk dogs consists of dogs of the same breed. Now consider a set D of k C 1 dogs, where D Dfd 1 ;d 2 ;:::;d k ;d kC1 g: If we remove the dog d 1 from the set D, wethenhaveasetD 1 of k dogs, and using the assumption that P.k/ is true, these dogs must all be of the same breed. Similarly, if we remove d kC1 from the set D, we again have a set D 2 of k dogs, and these dogs must all be of the same
Page 3:
Mathematical Reasoning Writing and
Page 7 and 8:
Contents Note to Students Preface v
Page 9 and 10:
Contents v 7.4 Modular Arithmetic .
Page 11 and 12:
Note to Students vii Progress Chec
Page 13 and 14:
Preface ix the emphasis is on the f
Page 15 and 16:
Preface xi With the exception of S
Page 17 and 18:
Preface xiii students with a thorou
Page 19 and 20:
Chapter 1 Introduction to Writing P
Page 21 and 22:
1.1. Statements and Conditional Sta
Page 23 and 24:
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
1.2. Constructing Direct Proofs 15
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
Page 41 and 42:
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
1.3. Chapter 1 Summary 31 1.3 Chapt
Page 51 and 52:
Chapter 2 Logical Reasoning 2.1 Sta
Page 53 and 54:
2.1. Statements and Logical Operato
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
Page 61 and 62:
2.2. Logically Equivalent Statement
Page 63 and 64:
Page 65 and 66:
Page 67 and 68:
Page 69 and 70:
Page 71 and 72:
2.3. Open Sentences and Sets 53 The
Page 73 and 74:
2.3. Open Sentences and Sets 55 5.
Page 75 and 76:
2.3. Open Sentences and Sets 57 in
Page 77 and 78:
2.3. Open Sentences and Sets 59 Exa
Page 79 and 80:
2.3. Open Sentences and Sets 61 Whe
Page 81 and 82:
2.4. Quantifiers and Negations 63
Page 83 and 84:
2.4. Quantifiers and Negations 65 6
Page 85 and 86:
2.4. Quantifiers and Negations 67 N
Page 87 and 88:
Page 89 and 90:
2.4. Quantifiers and Negations 71 :
Page 91 and 92:
2.4. Quantifiers and Negations 73 C
Page 93 and 94:
2.4. Quantifiers and Negations 75
Page 95 and 96:
2.4. Quantifiers and Negations 77 ?
Page 97 and 98:
Page 99 and 100:
2.5. Chapter 2 Summary 81 De Morgan
Page 101 and 102:
3.1. Direct Proofs 83 mathematics i
Page 103 and 104:
3.1. Direct Proofs 85 3. This is a
Page 105 and 106:
3.1. Direct Proofs 87 Please rememb
Page 107 and 108:
3.1. Direct Proofs 89 Proof. We ass
Page 109 and 110:
3.1. Direct Proofs 91 The problem i
Page 111 and 112:
3.1. Direct Proofs 93 This means th
Page 113 and 114:
3.1. Direct Proofs 95 4. Write a fi
Page 115 and 116:
3.1. Direct Proofs 97 Let a be an i
Page 117 and 118:
3.1. Direct Proofs 99 The point .a
Page 119 and 120:
3.1. Direct Proofs 101 (a) Proposit
Page 121 and 122:
3.2. More Methods of Proof 103 2. T
Page 123 and 124:
3.2. More Methods of Proof 105 mean
Page 125 and 126:
3.2. More Methods of Proof 107 Prop
Page 127 and 128:
3.2. More Methods of Proof 109 belo
Page 129 and 130:
3.2. More Methods of Proof 111 Nonc
Page 131 and 132:
3.2. More Methods of Proof 113 (b)
Page 133 and 134:
3.2. More Methods of Proof 115 19.
Page 135 and 136:
3.3. Proof by Contradiction 117 A l
Page 137 and 138:
3.3. Proof by Contradiction 119 Wri
Page 139 and 140:
3.3. Proof by Contradiction 121 1.
Page 141 and 142:
3.3. Proof by Contradiction 123 whi
Page 143 and 144:
3.3. Proof by Contradiction 125 r i
Page 145 and 146:
3.3. Proof by Contradiction 127 ? (
Page 147 and 148:
3.3. Proof by Contradiction 129 18.
Page 149 and 150:
3.4. Using Cases in Proofs 131 Proo
Page 151 and 152: 3.4. Using Cases in Proofs 133 The
Page 153 and 154: 3.4. Using Cases in Proofs 135 by 1
Page 155 and 156: 3.4. Using Cases in Proofs 137 Theo
Page 157 and 158: 3.4. Using Cases in Proofs 139 (b)
Page 159 and 160: 3.5. The Division Algorithm and Con
Page 177 and 178: 3.6. Review of Proof Methods 159 Pr
Page 179 and 180: 3.6. Review of Proof Methods 161 Co
Page 181 and 182: 3.6. Review of Proof Methods 163 8.
Page 183 and 184: 3.6. Review of Proof Methods 165 ca
Page 185 and 186: 3.7. Chapter 3 Summary 167 Exercis
Page 187 and 188: Chapter 4 Mathematical Induction 4.
Page 189 and 190: 4.1. The Principle of Mathematical
Page 201: 4.1. The Principle of Mathematical
Page 207 and 208: 4.2. Other Forms of Mathematical In
Page 219 and 220: 4.3. Induction and Recursion 201 Us
Page 221 and 222: 4.3. Induction and Recursion 203 Th
Page 223 and 224: 4.3. Induction and Recursion 205 if
Page 225 and 226: 4.3. Induction and Recursion 207 Ex
Page 227 and 228: 4.3. Induction and Recursion 209 11
Page 229 and 230: 4.3. Induction and Recursion 211 Pr
Page 231 and 232: 4.4. Chapter 4 Summary 213 4.4 Chap
Page 233 and 234: Chapter 5 Set Theory 5.1 Sets and O
Page 235 and 236: 5.1. Sets and Operations on Sets 21
Page 249 and 250: 5.2. Proving Set Relationships 231
Page 251 and 252: 5.2. Proving Set Relationships 233
Page 253 and 254:
5.2. Proving Set Relationships 235
Page 255 and 256:
Page 257 and 258:
Page 259 and 260:
Page 261 and 262:
Page 263 and 264:
5.3. Properties of Set Operations 2
Page 265 and 266:
Page 267 and 268:
Page 269 and 270:
Page 271 and 272:
Page 273 and 274:
5.4. Cartesian Products 255 is an o
Page 275 and 276:
5.4. Cartesian Products 257 6. For
Page 277 and 278:
5.4. Cartesian Products 259 boundar
Page 279 and 280:
5.4. Cartesian Products 261 Since u
Page 281 and 282:
5.4. Cartesian Products 263 9. Is t
Page 283 and 284:
5.5. Indexed Families of Sets 265 D
Page 285 and 286:
5.5. Indexed Families of Sets 267 W
Page 287 and 288:
5.5. Indexed Families of Sets 269 E
Page 289 and 290:
5.5. Indexed Families of Sets 271 t
Page 291 and 292:
5.5. Indexed Families of Sets 273 E
Page 293 and 294:
5.5. Indexed Families of Sets 275 9
Page 295 and 296:
5.6. Chapter 5 Summary 277 5.6 Chap
Page 297:
5.6. Chapter 5 Summary 279 Theorem
Page 300 and 301:
282 Chapter 6. Functions For this f
Page 302 and 303:
284 Chapter 6. Functions 2. Let s b
Page 304 and 305:
286 Chapter 6. Functions Nowhere do
Page 306 and 307:
288 Chapter 6. Functions For each m
Page 308 and 309:
290 Chapter 6. Functions Arrow Diag
Page 310 and 311:
292 Chapter 6. Functions 4. Let f W
Page 312 and 313:
294 Chapter 6. Functions Exploratio
Page 314 and 315:
296 Chapter 6. Functions (a) f.x/ D
Page 316 and 317:
298 Chapter 6. Functions verify tha
Page 318 and 319:
300 Chapter 6. Functions Progress C
Page 320 and 321:
302 Chapter 6. Functions and there
Page 322 and 323:
304 Chapter 6. Functions 5. Let A a
Page 324 and 325:
306 Chapter 6. Functions (a) Calcul
Page 326 and 327:
308 Chapter 6. Functions A f B A g
Page 328 and 329:
310 Chapter 6. Functions Definition
Page 330 and 331:
312 Chapter 6. Functions Let f W A
Page 332 and 333:
314 Chapter 6. Functions An Importa
Page 334 and 335:
316 Chapter 6. Functions Notice tha
Page 336 and 337:
318 Chapter 6. Functions (c) Draw a
Page 338 and 339:
320 Chapter 6. Functions 13. Let A
Page 340 and 341:
322 Chapter 6. Functions Hence, x a
Page 342 and 343:
324 Chapter 6. Functions a A f B g
Page 344 and 345:
326 Chapter 6. Functions Compositio
Page 346 and 347:
Page 348 and 349:
330 Chapter 6. Functions Proof of T
Page 350 and 351:
332 Chapter 6. Functions ? 6. Prove
Page 352 and 353:
334 Chapter 6. Functions 6.5 Invers
Page 354 and 355:
336 Chapter 6. Functions 3. If the
Page 356 and 357:
338 Chapter 6. Functions Definition
Page 358 and 359:
340 Chapter 6. Functions that f is
Page 360 and 361:
342 Chapter 6. Functions Theorems a
Page 362 and 363:
344 Chapter 6. Functions Now these
Page 364 and 365:
346 Chapter 6. Functions ? (a) If g
Page 366 and 367:
348 Chapter 6. Functions input and
Page 368 and 369:
350 Chapter 6. Functions (a) fx 2 R
Page 370 and 371:
Page 372 and 373:
354 Chapter 6. Functions 3. For eac
Page 374 and 375:
356 Chapter 6. Functions Since x 2
Page 376 and 377:
358 Chapter 6. Functions 3. Let gW
Page 378 and 379:
360 Chapter 6. Functions Onto func
Page 380 and 381:
Chapter 7 Equivalence Relations 7.1
Page 382 and 383:
364 Chapter 7. Equivalence Relation
Page 384 and 385:
Page 386 and 387:
Page 388 and 389:
Page 390 and 391:
Page 392 and 393:
Page 394 and 395:
Page 396 and 397:
Page 398 and 399:
Page 400 and 401:
Page 402 and 403:
Page 404 and 405:
Page 406 and 407:
Page 408 and 409:
Page 410 and 411:
Page 412 and 413:
Page 414 and 415:
Page 416 and 417:
Page 418 and 419:
Page 420 and 421:
Page 422 and 423:
Page 424 and 425:
Page 426 and 427:
Page 428 and 429:
Page 430 and 431:
Page 432 and 433:
Chapter 8 Topics in Number Theory 8
Page 434 and 435:
416 Chapter 8. Topics in Number The
Page 436 and 437:
Page 438 and 439:
Page 440 and 441:
Page 442 and 443:
Page 444 and 445:
Page 446 and 447:
Page 448 and 449:
Page 450 and 451:
Page 452 and 453:
Page 454 and 455:
Page 456 and 457:
Page 458 and 459:
Page 460 and 461:
Page 462 and 463:
Page 464 and 465:
Page 466 and 467:
Page 468 and 469:
Page 470 and 471:
Chapter 9 Finite and Infinite Sets
Page 472 and 473:
454 Chapter 9. Finite and Infinite
Page 474 and 475:
Page 476 and 477:
Page 478 and 479:
Page 480 and 481:
Page 482 and 483:
Page 484 and 485:
Page 486 and 487:
Page 488 and 489:
Page 490 and 491:
Page 492 and 493:
Page 494 and 495:
Page 496 and 497:
Page 498 and 499:
Page 500 and 501:
Page 502 and 503:
Page 504 and 505:
Page 506 and 507:
Page 508 and 509:
Page 510 and 511:
Appendix A Guidelines for Writing M
Page 512 and 513:
494 Appendix A. Writing Guidelines
Page 514 and 515:
496 Appendix A. Writing Guidelines
Page 516 and 517:
498 Appendix B. Answers for Progres
Page 518 and 519:
Page 520 and 521:
Page 522 and 523:
Page 524 and 525:
Page 526 and 527:
Page 528 and 529:
Page 530 and 531:
Page 532 and 533:
Page 534 and 535:
Page 536 and 537:
Page 538 and 539:
Page 540 and 541:
Page 542 and 543:
Page 544 and 545:
Page 546 and 547:
Page 548 and 549:
Page 550 and 551:
Page 552 and 553:
Page 554 and 555:
Appendix C Answers and Hints for Se
Page 556 and 557:
538 Appendix C. Answers for Exercis
Page 558 and 559:
Page 560 and 561:
Page 562 and 563:
Page 564 and 565:
Page 566 and 567:
Page 568 and 569:
Page 570 and 571:
Page 572 and 573:
Page 574 and 575:
Page 576 and 577:
Page 578 and 579:
Page 580 and 581:
Page 582 and 583:
Page 584 and 585:
Page 586 and 587:
Page 588 and 589:
Page 590 and 591:
Page 592 and 593:
Page 594 and 595:
Page 596 and 597:
Page 598 and 599:
Page 600 and 601:
Appendix D List of Symbols Symbol M
Page 602 and 603:
584 Appendix D. List of Symbols Sym
Page 604 and 605:
586 Index conditional, 33 condition
Page 606 and 607:
588 Index idempotent laws for sets,
Page 608 and 609:
590 Index open, 228 real function,
show all

Mathematical Reasoning- Writing and Proof, Version 2.1, 2014a

Create successful ePaper yourself

Delete template?

Save as template?