Matemática discreta ( Tradução da 3ª edição edição norte-americana)

UMA INTRODUÇÃO TRADUÇÃO DA 3

a

EDIÇÃO NORTE-AMERICANA

Dirigida a interessados em matemática discreta de maneira geral, esta obra apresenta uma visão geral de toda a matemática a partir da perspectiva da matemática discreta. O livro cobre desde a natureza da matemática (definições, teoremas, provas e contraexemplos), lógica, conjuntos e relações, até técnicas avançadas de prova, probabilidades discretas, teoria dos números, teoria dos grafos e álgebra abstrata, incluindo aplicações em criptografia.

Esta terceira edição conta com muitos problemas e exercícios que exploram as interligações entre os diversos temas tratados.

Aplicações: Esta obra pode ser adotada como livro-texto em disciplinas de matemática discreta em cursos de engenharia, matemática e ciência da computação. Texto didático e agradável, pode ser lido por qualquer interessado na matéria.

ISBN 13 978-85-221-2534-0 ISBN 10 85-221-2534-1

MATEMÁTICA DISCRETA: UMA INTRODUÇÃO

MATEMÁTICA DISCRETA

EDWARD R. SCHEINERMAN

EDWARD R. SCHEINERMAN

OUTRAS OBRAS INTRODUÇÃO À LÓGICA MATEMÁTICA Carlos Alberto F. Bispo, Luiz B. Castanheira e Oswaldo Melo S. Filho LÓGICA PARA COMPUTAÇÃO Flávio Soares Corrêa da Silva, Marcelo Finger e Ana Cristina Vieira de Melo

EDWARD R. SCHEINERMAN

MATEMÁTICA DISCRETA

FUNDAMENTOS DA CIÊNCIA DA COMPUTAÇÃO Tradução da 2a edição Internacional Behrouz Forouzan e Firouz Mosharraf INTRODUÇÃO À TEORIA DA COMPUTAÇÃO Tradução da 2a edição Norte-Americana Michael Sipser

UMA INTRODUÇÃO TRADUÇÃO DA 3a EDIÇÃO NORTE-AMERICANA

Para suas soluções de curso e aprendizado, visite www.cengage.com.br

Matemática_18x26.indd 1

9 788522 125340

4/8/16 10:53 AM

MATEMÁTICA DISCRETA Uma introdução Tradução da 3a edição norte-americana

Book 1.indb 1

04/04/2016 16:30:22

Sumรกrio

Ao estudante ..................................................................................................................xvii &RPR OHU XP OLYUR GH PDWHPiWLFD.......................................................................................................[YLLL 3~EOLFR OHLWRU H SUp UHTXLVLWRV ...............................................................................................................xxi Ao professor ....................................................................................................................xxi 7ySLFRV FREHUWRV SHUFRUUHQGR DV VHo}HV .............................................................................................xxii 3ODQR GH FXUVRV WtSLFRV .........................................................................................................................xxii Caracterรญsticas especiais......................................................................................................................xxiii O que hรก de novo nesta terceira ediรงรฃo ......................................................................... xxv Agradecimentos ........................................................................................................... xxvii (VWD QRYD HGLomR ................................................................................................................................[[YLL 'D VHJXQGD HGLomR .............................................................................................................................[[YLL 'D SULPHLUD HGLomR ............................................................................................................................[[YLLL &$3ร 78/2 1

Book 1.indb 7

)XQGDPHQWRV .............................................................................................. 1 1

Alegria .................................................................................................. 1 3RU TXr"................................................................................................................... 1

2

Falando (e escrevendo) sobre matemรกtica ............................................ 2 Precisamente! .......................................................................................................... 2 8P SRXFR GH DMXGD ................................................................................................. 3 ([HUFtFLRV ................................................................................................................ 4

3

De๏ฌ niรงรฃo .............................................................................................. 5 5HFDSLWXODQGR.......................................................................................................... 8 ([HUFtFLRV ................................................................................................................ 8

4

Teorema .............................................................................................. 10 $ QDWXUH]D GD YHUGDGH ........................................................................................... 11 6H HQWmR ................................................................................................................. 12 6H H VRPHQWH VH ..................................................................................................... 14 ( RX H QmR............................................................................................................. 15 'HVLJQDo}HV SDUD XP WHRUHPD............................................................................... 16 $ยฟUPDomR YHUGDGHLUD SRU YDFXLGDGH .................................................................... 17

04/04/2016 16:30:24

viii

Matemática discreta

5HFDSLWXODQGR........................................................................................................ 18 ([HUFtFLRV .............................................................................................................. 18

&$3Ë78/2 2

5

Prova ................................................................................................... 20 8PD SURYD PDLV FRPSOH[D.................................................................................... 25 3URYD GH WHRUHPDV GR WLSR ³VH H VRPHQWH VH´....................................................... 27 3URYDQGR HTXDo}HV H GHVLJXDOGDGHV ..................................................................... 29 5HFDSLWXODQGR........................................................................................................ 30 ([HUFtFLRV .............................................................................................................. 30

6

Contraexemplo.................................................................................... 31 5HFDSLWXODQGR........................................................................................................ 33 ([HUFtFLRV .............................................................................................................. 33

7

Álgebra de Boole................................................................................. 34 0DLV RSHUDo}HV ...................................................................................................... 37 5HFDSLWXODQGR........................................................................................................ 38 ([HUFtFLRV .............................................................................................................. 38

&ROHo}HV................................................................................................... 45 8

9

Listas ................................................................................................... 45 Contagem de listas de dois elementos................................................................... 46 /LVWDV PDLV ORQJDV ................................................................................................. 48 5HFDSLWXODQGR........................................................................................................ 52 ([HUFtFLRV .............................................................................................................. 52 Fatorial ................................................................................................ 54 0XLWR EDUXOKR HP WRUQR GH ............................................................................... 54 1RWDomR GH SURGXWR ............................................................................................... 56 5HFDSLWXODQGR........................................................................................................ 57 ([HUFtFLRV .............................................................................................................. 57

10 Conjuntos I: introdução, subconjuntos ................................................ 59 ,JXDOGDGH GH FRQMXQWRV.......................................................................................... 60 6XEFRQMXQWR .......................................................................................................... 62 &RQWDJHP GH VXEFRQMXQWRV ................................................................................... 65 &RQMXQWR SRWrQFLD ................................................................................................. 66 5HFDSLWXODQGR........................................................................................................ 67 ([HUFtFLRV .............................................................................................................. 67 11 Quantificadores................................................................................... 68 ([LVWH..................................................................................................................... 68 Para todo ............................................................................................................... 70 1HJDomR GH D¿UPDo}HV TXDQWL¿FDGDV ................................................................... 71 &RPELQDomR GH TXDQWL¿FDGRUHV ............................................................................ 72 5HFDSLWXODQGR........................................................................................................ 73 ([HUFtFLRV .............................................................................................................. 73

Book 1.indb 8

04/04/2016 16:30:24

Sumário

ix

12 Conjuntos II: operações ....................................................................... 75 8QLmR H LQWHUVHFomR ............................................................................................... 75 7DPDQKR GH XPD XQLmR ......................................................................................... 77 'LIHUHQoD H GLIHUHQoD VLPpWULFD............................................................................. 80 3URGXWR FDUWHVLDQR ................................................................................................. 85 5HFDSLWXODQGR........................................................................................................ 85 ([HUFtFLRV .............................................................................................................. 86 13 Prova combinatória: dois exemplos ..................................................... 89 5HFDSLWXODQGR........................................................................................................ 92 ([HUFtFLRV .............................................................................................................. 92 &$3Ë78/2 3

&RQWDJHP H UHODo}HV ................................................................................ 97 14 Relações ............................................................................................. 97 3URSULHGDGHV GH UHODo}HV .................................................................................... 100 5HFDSLWXODQGR ..................................................................................................... 101 ([HUFtFLRV ............................................................................................................ 102 15 Relações de equivalência ................................................................. 104 5HFDSLWXODQGR...................................................................................................... 111 ([HUFtFLRV ............................................................................................................ 111 16 Partições ............................................................................................ 114 Contagem de classes/partes ................................................................................. 116 5HFDSLWXODQGR...................................................................................................... 119 ([HUFtFLRV ............................................................................................................ 119 17 Coeficientes binomiais ...................................................................... 121 &iOFXOR GH .................................................................................................... 125 2 WULkQJXOR GH 3DVFDO .......................................................................................... 126 8PD )yUPXOD SDUD ....................................................................................... 129 &RQWDQGR FDPLQKRV UHWLFXODGRV .......................................................................... 131 5HFDSLWXODQGR...................................................................................................... 132 ([HUFtFLRV ............................................................................................................ 132

18 Contagem de multiconjuntos............................................................. 136 0XOWLFRQMXQWRV .................................................................................................... 137 ............................................................................................ 139 )yUPXODV SDUD (VWHQGHQGR R 7HRUHPD %LQRPLDO SDUD SRWrQFLDV QHJDWLYDV ................................ 142 5HFDSLWXODQGR...................................................................................................... 145 ([HUFtFLRV ............................................................................................................ 146 19 Inclusão-exclusão.............................................................................. 148 &RPR XWLOL]DU D LQFOXVmR H[FOXVmR ...................................................................... 151 'HVRUGHQDo}HV .................................................................................................... 154 8PD IyUPXOD H[WHQVD .......................................................................................... 157 5HFDSLWXODQGR...................................................................................................... 157 ([HUFtFLRV ............................................................................................................ 158

Book 1.indb 9

04/04/2016 16:30:25

x

Matemรกtica discreta

&$3ร 78/2 4

0DLV SURYDV............................................................................................ 163 20 Contradiรงรฃo ...................................................................................... 163 3URYD SHOD FRQWUDSRVLWLYD .................................................................................... 163 Reductio ad absurdum......................................................................................... 165 3URYD SRU FRQWUDGLomR H sudoku .......................................................................... 169 8PD TXHVWmR GH HVWLOR ......................................................................................... 170 5HFDSLWXODQGR...................................................................................................... 170 ([HUFtFLRV ............................................................................................................ 170 21 Contraexemplo mรญnimo..................................................................... 172 %RD RUGHQDomR ..................................................................................................... 177 5HFDSLWXODQGR...................................................................................................... 183 ([HUFtFLRV ............................................................................................................ 183 ( SRU ยฟP .......................................................................................................... 184 22 Induรงรฃo ............................................................................................. 185 $ PiTXLQD GD LQGXomR ......................................................................................... 185 )XQGDPHQWRV WHyULFRV ......................................................................................... 187 3URYD SRU LQGXomR ............................................................................................... 188 3URYD GH HTXDo}HV H GHVLJXDOGDGHV ..................................................................... 190 2XWURV H[HPSORV ................................................................................................. 192 ,QGXomR IRUWH ...................................................................................................... 194 8P H[HPSOR PDLV FRPSOLFDGR ........................................................................... 196 8PD TXHVWmR GH HVWLOR ......................................................................................... 199 5HFDSLWXODQGR ..................................................................................................... 199 ([HUFtFLRV ............................................................................................................ 200 23 Relaรงรตes de recorrรชncia .................................................................... 205 5HODo}HV GH UHFRUUrQFLD GH SULPHLUD RUGHP ........................................................ 205 5HODo}HV GH UHFRUUrQFLD GH VHJXQGD RUGHP ........................................................ 209 2 FDVR GD UDL] UHSHWLGD ........................................................................................ 213 6HTXrQFLDV JHUDGDV SRU SROLQ{PLRV .................................................................... 215 5HFDSLWXODQGR...................................................................................................... 222 ([HUFtFLRV ............................................................................................................ 223

&$3ร 78/2

)XQo}HV .................................................................................................. 229 24 Funรงรตes............................................................................................. 229 Domรญnio e imagem .............................................................................................. 231 *UiยฟFRV GH IXQo}HV ............................................................................................. 233 &RQWDJHP GH IXQo}HV .......................................................................................... 235 )XQo}HV LQYHUVDV ................................................................................................. 236 1RYDPHQWH FRQWDJHP GH IXQo}HV ...................................................................... 240 5HFDSLWXODQGR...................................................................................................... 242 ([HUFtFLRV ............................................................................................................ 242

Book 1.indb 10

04/04/2016 16:30:25

Sumรกrio

xi

24 O princรญpio da casa do pombo .......................................................... 245 Teorema de Cantor .............................................................................................. 249 5HFDSLWXODQGR...................................................................................................... 251 ([HUFtFLRV ............................................................................................................ 251 26 Composiรงรฃo ...................................................................................... 253 $ IXQomR LGHQWLGDGH ............................................................................................ 257 5HFDSLWXODQGR...................................................................................................... 258 ([HUFtFLRV ............................................................................................................ 258 27 Permutaรงรตes ...................................................................................... 260 1RWDomR HP FLFORV ............................................................................................... 261 &iOFXORV FRP SHUPXWDo}HV.................................................................................. 264 7UDQVSRVLo}HV ...................................................................................................... 266 8PD DERUGDJHP JUiยฟFD ...................................................................................... 272 5HFDSLWXODQGR...................................................................................................... 274 ([HUFtFLRV ............................................................................................................ 274 28 Simetria ............................................................................................. 277 6LPHWULDV GH XP TXDGUDGR ................................................................................... 278 6LPHWULDV FRPR SHUPXWDo}HV .............................................................................. 279 &RPELQDomR GH VLPHWULDV .................................................................................... 280 'HยฟQLomR IRUPDO GH VLPHWULD ............................................................................... 282 5HFDSLWXODQGR...................................................................................................... 283 ([HUFtFLRV ............................................................................................................ 283 29 Tipos de notaรงรฃo ............................................................................... 284 : e 4 .................................................................................................................. 287 ยณ2ยด 3HTXHQR ....................................................................................................... 288 6ROR H WHWR............................................................................................................ 288 f f x H f ฤ ........................................................................................................ 289 5HFDSLWXODQGR...................................................................................................... 290 ([HUFtFLRV ............................................................................................................ 290 &$3ร 78/2 6

3UREDELOLGDGH ............................................................... 295 30 Espaรงo amostral ................................................................................ 296 5HFDSLWXODQGR...................................................................................................... 299 ([HUFtFLRV ............................................................................................................ 299 31 Eventos.............................................................................................. 301 &RPELQDomR GH HYHQWRV ...................................................................................... 303 2 SUREOHPD GRV DQLYHUViULRV ............................................................................... 306 5HFDSLWXODQGR...................................................................................................... 307 ([HUFtFLRV ............................................................................................................ 307 32 Probabilidade condicional e independรชncia...................................... 309 Independรชncia ..................................................................................................... 312

Book 1.indb 11

04/04/2016 16:30:25

xii

Matemática discreta

3URYDV UHSHWLGDV LQGHSHQGHQWHV .......................................................................... 314 2 SUREOHPD GH 0RQW\ +DOO ................................................................................. 315 5HFDSLWXODQGR...................................................................................................... 317 ([HUFtFLRV ............................................................................................................ 317

33 Variáveis aleatórias ........................................................................... 321 9DULiYHLV DOHDWyULDV FRPR HYHQWRV ...................................................................... 322 9DULiYHLV DOHDWyULDV LQGHSHQGHQWHV...................................................................... 323 5HFDSLWXODQGR...................................................................................................... 324 ([HUFtFLRV ............................................................................................................ 325 34 Valor esperado .................................................................................. 327 /LQHDULGDGH GR YDORU HVSHUDGR ............................................................................ 331 3URGXWR GH YDULiYHLV DOHDWyULDV ........................................................................... 335 Valor esperado como medida de centralidade ..................................................... 338 9DULkQFLD.............................................................................................................. 339 5HFDSLWXODQGR...................................................................................................... 343 ([HUFtFLRV ............................................................................................................ 343 &$3Ë78/2 7

Teoria dos números ................................................................................ 349 35 Divisão.............................................................................................. 349 'LY H 0RG .......................................................................................................... 352 5HFDSLWXODQGR...................................................................................................... 354 ([HUFtFLRV ............................................................................................................ 354 36 Máximo divisor comum .................................................................... 355 &iOFXOR GR PGF ................................................................................................... 356 &RUUHomR .............................................................................................................. 358 4XmR UiSLGR" ....................................................................................................... 359 8P WHRUHPD LPSRUWDQWH ...................................................................................... 361 5HFDSLWXODQGR...................................................................................................... 364 ([HUFtFLRV ............................................................................................................ 364 37 Aritmética modular ........................................................................... 366 8P QRYR FRQWH[WR SDUD RSHUDo}HV EiVLFDV ......................................................... 366 $GLomR H PXOWLSOLFDomR PRGXODUHV ...................................................................... 367 6XEWUDomR PRGXODU .............................................................................................. 368 'LYLVmR PRGXODU .................................................................................................. 370 8PD REVHUYDomR VREUH D QRWDomR ........................................................................ 375 5HFDSLWXODQGR...................................................................................................... 375 ([HUFtFLRV ............................................................................................................ 376 38 O teorema do resto chinês ................................................................ 378 5HVROXomR GH XPD HTXDomR ................................................................................. 378 5HVROXomR GH GXDV HTXDo}HV ............................................................................... 380 5HFDSLWXODQGR...................................................................................................... 382 ([HUFtFLRV ............................................................................................................ 382

Book 1.indb 12

04/04/2016 16:30:25

Sumário

xiii

39 Fatoração .......................................................................................... 383 ,Q¿QLWRV Q~PHURV SULPRV .................................................................................... 385 8PD IyUPXOD SDUD R Pi[LPR GLYLVRU FRPXP ..................................................... 386 Irracionalidade de .......................................................................................... 387 $SHQDV SRU GLYHUVmR ............................................................................................ 389 5HFDSLWXODQGR...................................................................................................... 389 ([HUFtFLRV ............................................................................................................ 390 &$3Ë78/2 8

ÈOJHEUD .................................................................................................. 395 40 Grupos .............................................................................................. 395 2SHUDo}HV ............................................................................................................ 395 3URSULHGDGHV GH RSHUDo}HV .................................................................................. 396 *UXSRV ................................................................................................................. 399 ([HPSORV............................................................................................................. 401 5HFDSLWXODQGR...................................................................................................... 404 ([HUFtFLRV ............................................................................................................ 404 41 Isomorfismo de grupos ...................................................................... 407 2 PHVPR" ........................................................................................................... 407 *UXSRV FtFOLFRV.................................................................................................... 409 5HFDSLWXODQGR...................................................................................................... 412 ([HUFtFLRV ............................................................................................................ 412 42 Subgrupos ......................................................................................... 414 2 WHRUHPD GH /DJUDQJH ....................................................................................... 417 5HFDSLWXODQGR...................................................................................................... 421 ([HUFtFLRV ............................................................................................................ 421 43 O pequeno teorema de Fermat .......................................................... 424 3ULPHLUD SURYD ..................................................................................................... 424 6HJXQGD SURYD ..................................................................................................... 425 7HUFHLUD SURYD...................................................................................................... 428 2 WHRUHPD GH (XOHU ............................................................................................. 429 Teste de primalidade ........................................................................................... 430 5HFDSLWXODQGR...................................................................................................... 431 ([HUFtFLRV ............................................................................................................ 431 44 Criptografia de chave pública I: introdução ....................................... 432 2 SUREOHPD FRPXQLFDomR SULYDGD HP S~EOLFR................................................... 432 )DWRUDomR ............................................................................................................. 433 'H SDODYUDV SDUD Q~PHURV ................................................................................... 434 $ FULSWRJUD¿D H D OHL ............................................................................................ 436 5HFDSLWXODQGR...................................................................................................... 436 ([HUFtFLRV ............................................................................................................ 436

Book 1.indb 13

04/04/2016 16:30:25

xiv

Matemática discreta

45 Criptografia de chave pública II: o método de Rabin ......................... 437 5Dt]HV TXDGUDGDV PyGXOR n ................................................................................ 437 2V SURFHVVRV GH FULSWRJUD¿D H GHFLIUDomR ........................................................... 442 5HFDSLWXODQGR...................................................................................................... 442 ([HUFtFLRV ............................................................................................................ 443 46 Criptografia de chave pública III: RSA ............................................... 444 $V IXQo}HV 56$ GH FRGL¿FDomR H GHFRGL¿FDomR ................................................ 445 6HJXUDQoD ............................................................................................................ 447 5HFDSLWXODQGR...................................................................................................... 448 ([HUFtFLRV ............................................................................................................ 448

&$3Ë78/2 9

Grafos .................................................................................................... 453 46 Fundamentos da teoria dos grafos .................................................... 453 &RORUDomR GH PDSDV ........................................................................................... 453 7UrV VHUYLoRV ....................................................................................................... 455 2 SUREOHPD GDV VHWH SRQWHV ................................................................................ 456 2 TXH p XP JUDIR" ............................................................................................... 457 $GMDFrQFLD ........................................................................................................... 458 8PD TXHVWmR GH JUDX ........................................................................................... 459 1RWDomR H YRFDEXOiULR DGLFLRQDLV ....................................................................... 461 5HFDSLWXODQGR...................................................................................................... 463 ([HUFtFLRV ............................................................................................................ 463 48 Subgrafos .......................................................................................... 465 6XEJUDIRV LQGX]LGRV H JHUDGRUHV ................................................................ 466 &OLTXHV H FRQMXQWRV LQGHSHQGHQWHV ..................................................................... 468 Complementos .................................................................................................... 470 5HFDSLWXODQGR...................................................................................................... 471 ([HUFtFLRV ............................................................................................................ 472

49 Conexão............................................................................................ 474 Passeios ............................................................................................................... 474 Caminhos ............................................................................................................ 476 'HVFRQH[mR ......................................................................................................... 480 5HFDSLWXODQGR...................................................................................................... 481 ([HUFtFLRV ............................................................................................................ 481 50 Árvores .............................................................................................. 483 Ciclos .................................................................................................................. 483 )ORUHVWDV H iUYRUHV ............................................................................................... 484 3URSULHGDGHV GDV iUYRUHV ..................................................................................... 485 Folhas .................................................................................................................. 487 ÈUYRUHV JHUDGRUDV ............................................................................................... 489 5HFDSLWXODQGR...................................................................................................... 490 ([HUFtFLRV ............................................................................................................ 490

Book 1.indb 14

04/04/2016 16:30:25

Sumário

xv

51 Grafos eulerianos .............................................................................. 493 &RQGLo}HV QHFHVViULDV ......................................................................................... 493 7HRUHPDV IXQGDPHQWDLV....................................................................................... 495 1HJyFLR LQDFDEDGR .............................................................................................. 497 5HFDSLWXODQGR...................................................................................................... 498 ([HUFtFLRV ............................................................................................................ 498 52 Coloração ......................................................................................... 499 &RQFHLWRV IXQGDPHQWDLV ...................................................................................... 500 *UDIRV ELSDUWLGRV ................................................................................................. 502 $ IDFLOLGDGH GH FRORULU FRP GXDV FRUHV H D GL¿FXOGDGH GH FRORULU FRP WUrV FRUHV ....

506 5HFDSLWXODQGR...................................................................................................... 507 ([HUFtFLRV ............................................................................................................ 507 53 Grafos planares ................................................................................. 509 &XUYDV SHULJRVDV ................................................................................................. 509 ,QFOXVmR ............................................................................................................... 511 )yUPXOD GH (XOHU................................................................................................. 511 *UDIRV QmR SODQDUHV ............................................................................................. 515 &RORUDomR GH JUDIRV SODQDUHV .............................................................................. 516 5HFDSLWXODQGR...................................................................................................... 520 ([HUFtFLRV ............................................................................................................ 520 &$3Ë78/2 10

&RQMXQWRV SDUFLDOPHQWH RUGHQDGRV ........................................................ 525 54 Fundamentos dos conjuntos parcialmente ordenados ........................ 525 2 TXH p XP FRQMXQWR 32" ................................................................................... 525 1RWDomR H OLQJXDJHP ........................................................................................... 528 5HFDSLWXODQGR...................................................................................................... 530 ([HUFtFLRV ............................................................................................................ 530 55 Max e min ......................................................................................... 532 5HFDSLWXODQGR...................................................................................................... 534 ([HUFtFLRV ............................................................................................................ 534 56 Ordens lineares ................................................................................. 535 5HFDSLWXODQGR...................................................................................................... 538 ([HUFtFLRV ............................................................................................................ 538 57 Extensões lineares ............................................................................. 539 2UGHQDomR ........................................................................................................... 543 ([WHQV}HV OLQHDUHV GH FRQMXQWRV 32 LQ¿QLWRV ..................................................... 545 5HFDSLWXODQGR...................................................................................................... 546 ([HUFtFLRV ............................................................................................................ 546 58 Dimensão.......................................................................................... 547 &DUDFWHUL]DGRUHV .................................................................................................. 547

Book 1.indb 15

04/04/2016 16:30:25

xvi

Matemรกtica discreta

'LPHQVmR ............................................................................................................ 550 ,PHUVmR ................................................................................................................ 552 5HFDSLWXODQGR...................................................................................................... 555 ([HUFtFLRV ............................................................................................................ 555

59 Reticulados ....................................................................................... 556 ,QI H VXS ............................................................................................................... 556 5HWLFXODGRV .......................................................................................................... 559 5HFDSLWXODQGR...................................................................................................... 561 ([HUFtFLRV ............................................................................................................ 561

Glossรกrio ...................................................................................................................... 567 ร QGLFH 5HPLVVLYR.......................................................................................................... 577

Book 1.indb 16

04/04/2016 16:30:25

O que hรก de novo nesta terceira ediรงรฃo 3UREOHPDV SUREOHPDV SUREOHPDV H PDLV SUREOHPDV (VWD QRYD HGLomR WHP FHQWHQDV GH QRYRV SUREOHPDV LQFOXtGRV DR ORQJR GDV VH o}HV GH H[HUFtFLRV H GH FDStWXORV GH DXWRDYDOLDomR $OJXQV GHVVHV QRYRV SUREOHPDV HVWmR LQWHUUHODFLRQDGRV SDUD GHVHQYROYLPHQWR GH LGHLDV HQWUH RV FDStWXORV 3RU H[HPSOR R Q~PHUR GH GLYLVRUHV SR VLWLYRV GH XP Q~PHUR LQWHLUR SRVLWLYR p HVWUDQKR VH H VRPHQWH VH R Q~PHUR LQWHLUR p XP TXDGUDGR SHUIHLWR 2V DOXQRV VmR LQGLUHWDPHQWH OHYDGRV D FRQMHFWXUDU LVVR H D SURYDU ([HUFtFLR H QRYDPHQWH SRU XPD HQXPHUDomR H[SOtFLWD GRV GLYLVRUHV 3UREOHPD GH $XWR WHVWH

2XWURV SHTXHQRV WySLFRV VmR GHVHQYROYLGRV FRPR D H[WHQVmR GR WHRUHPD ELQiULR QHJDWLYR H[SRHQWH FRQWDQGR FDPLQKRV GH WUHOLoD XVDQGR sudoku FRPR SURYD SRU FRQWUDGLomR H GHVLJXDOGDGHV GH %RQ IHUURQL SDUD DSUR[LPDU D LQFOXVmR H[FOXVmR H DVVLP SRU GLDQWH (P PXLWRV FDVRV HVVHV QRYRV WySLFRV VmR DSUHVHQWDGRV H[FOXVLYDPHQWH por meio de exercรญcios.

8PD QRYD VHomR LQWURGXWyULD VREUH D HVFULWD QD PDWHPiWLFD IRL DGLFLRQDGD DR &DStWXOR 2EULJDGR D WRGRV TXH HVFUHYHUDP UHSRUWDQGR HUURV SRLV DVVLP FRQVHJXLPRV DFHUWi ORV QHVWD QRYD HGLomR

Book 1.indb 25

04/04/2016 16:30:27

C A P ├НT U L O

1

Fundamentos ╞Р ╞Й─Ю─Ъ╞М─В╞Р ─В┼╢┼Р╞╡┼п─В╞М─Ю╞Р ─Ъ─В ┼╡─В╞Ъ─Ю┼╡─Д╞Я─Р─В ╞Р─Ж┼╜ ─В ─Ъ─Ю─о┼╢┼Э─Х─Ж┼╜═Х ┼╜ ╞Ъ─Ю┼╜╞М─Ю┼╡─В ─Ю ─В ╞Й╞М┼╜╟А─В═Ш ╞Р ─Ъ─Ю─о┼╢┼Э─Х╞Б─Ю╞Р ─Ю╞Р╞Й─Ю─Р┼Э─о─Р─В┼╡ ─Р┼╜┼╡ ╞Й╞М─Ю─Р┼Э╞Р─Ж┼╜ ┼╜╞Р ─Р┼╜┼╢─Р─Ю┼Э╞Ъ┼╜╞Р ─Ю┼╡ ╞Л╞╡─Ю ─Ю╞Р╞Ъ─В┼╡┼╜╞Р ┼Э┼╢╞Ъ─Ю╞М─Ю╞Р╞Р─В─Ъ┼╜╞Р═Х ┼╜╞Р teoremas ─В─о╞М┼╡─В┼╡ ─Ю╟Ж─В╞Ъ─В┼╡─Ю┼╢╞Ъ─Ю ┼╜ ╞Л╞╡─Ю ─а ╟А─Ю╞М─Ъ─В─Ъ─Ю┼Э╞М┼╜ ╞Р┼╜─П╞М─Ю ─Ю╞Р╞Р─Ю╞Р ─Р┼╜┼╢─Р─Ю┼Э╞Ъ┼╜╞Р═Х ─Ю ─В╞Р provas ─Ъ─Ю┼╡┼╜┼╢╞Р╞Ъ╞М─В┼╡═Х ─Ъ─Ю ┼╡─В┼╢─Ю┼Э╞М─В ┼Э╞М╞М─Ю─и╞╡╞Ъ─Д╟А─Ю┼п═Х ─В ╟А─Ю╞М─Ъ─В─Ъ─Ю ─Ъ─Ю╞Р╞Р─В╞Р ─В╞Р╞Р─Ю╞М─Х╞Б─Ю╞Р═Ш р┤кр┤йE┼╜ ─Ю┼╢╞Ъ─В┼╢╞Ъ┼╜═Х ─В┼╢╞Ъ─Ю╞Р ─Ъ─Ю ┼Э┼╢┼Э─Р┼Э─В╞М┼╡┼╜╞Р═Х ─и─В─Х─В┼╡┼╜╞Р ╞╡┼╡─В ╞Й─Ю╞М┼Р╞╡┼╢╞Ъ─В═Ч ╞Й┼╜╞М ╞Л╞╡─б═Н

┬Д 1 Alegria Por qu├к? >─Ю┼Э─В ╞Ъ─В┼╡─П─а┼╡ ┼╜ ╞Й╞М─Ю─и─Д─Р┼Э┼╜ ═Ю ┼╜ ─Ю╞Р╞Ъ╞╡─Ъ─В┼╢╞Ъ─Ю═Я═Х ┼╢┼╜ ╞Л╞╡─В┼п ─В─П┼╜╞М─Ъ─В┼╡┼╜╞Р ─П╞М─Ю╟А─Ю┼╡─Ю┼╢╞Ъ─Ю ─В╞Р ╞Л╞╡─Ю╞Р╞Ъ╞Б─Ю╞Р ╞Р┼╜─П╞М─Ю ┼╜ ╞Л╞╡─Ю ─а ┼╡─В╞Ъ─Ю┼╡─Д╞Я─Р─В ─Ю ┼╜ ╞Л╞╡─Ю ╞Р┼Э┼Р┼╢┼Э─о─Р─В ┼╡─В╞Ъ─Ю┼╡─Д╞Я─Р─В ─Ъ┼Э╞Р─Р╞М─Ю╞Ъ─В═Ш d─В┼╡─П─а┼╡ ─и┼╜╞М┼╢─Ю─Р─Ю┼╡┼╜╞Р ╞╡┼╡─В ┼╜╞М┼Э─Ю┼╢╞Ъ─В─Х─Ж┼╜ ┼Э┼╡╞Й┼╜╞М╞Ъ─В┼╢╞Ъ─Ю ╞Р┼╜─П╞М─Ю ─Р┼╜┼╡┼╜ ┼п─Ю╞М ╞╡┼╡ ┼п┼Э╟А╞М┼╜ ─Ъ─Ю ┼╡─В╞Ъ─Ю┼╡─Д╞Я─Р─В═Ш

$QWHV GH DUUHJDoDUPRV DV PDQJDV H FRPHoDUPRV D WUDEDOKDU D VpULR JRVWDULD GH GLYLGLU FRP YRFr DOJXPDV FRQVLGHUDo}HV VREUH D TXHVWmR SRU TXH HVWXGDU PDWHPiWLFD" $ PDWHPiWLFD p LQFULYHOPHQWH ~WLO $ PDWHPiWLFD p GH LPSRUWkQFLD YLWDO SDUD FDGD IDFHWD GD WHFQRORJLD PRGHUQD D GHVFREHUWD GH QRYRV PHGLFDPHQWRV HVFDORQDPHQWR GH OLQKDV DpUHDV FRQ┬┐DELOLGDGH GD FRPXQLFDomR FRGL┬┐FDomR GH P~VLFDV H ┬┐OPHV HP &'V H '9'V H┬┐FLrQFLD GRV PRWRUHV GH DXWRPyYHLV H DVVLP SRU GLDQWH (OD DOFDQoD OLPLWHV PXLWR DOpP GDV FLrQFLDV WpFQLFDV $ PDWHPiWLFD WDPEpP p IXQGDPHQWDO SDUD WRGDV DV FLrQFLDV VRFLDLV GHVGH D FRPSUHHQVmR GDV ├АXWXDo}HV GD HFRQRPLD DWp D PRGHODJHP GH UHGHV VR FLDLV HP HVFRODV RX HPSUHVDV &DGD UDPR GDV EHODV DUWHV ┬▒ LQFOXLQGR OLWHUDWXUD P~VLFD HV FXOWXUD SLQWXUD H WHDWUR ┬▒ WDPEpP VH EHQH┬┐FLRX GD PDWHPiWLFD RX IRL SRU HOD LQVSLUDGR

&RPR D PDWHPiWLFD p WDQWR ├АH[tYHO QRYRV HOHPHQWRV HP PDWHPiWLFD VmR LQYHQWD GRV GLDULDPHQWH TXDQWR ULJRURVD SRGHPRV SURYDU VHP FRQWURYpUVLDV TXH QRVVDV DVVHU o}HV HVWmR FRUUHWDV HOD p D PHOKRU IHUUDPHQWD DQDOtWLFD TXH D KXPDQLGDGH GHVHQYROYHX 2 VXFHVVR VHP SDUDOHORV GD PDWHPiWLFD FRPR IHUUDPHQWD SDUD UHVROXomR GH SUREOH PDV HP FLrQFLD HQJHQKDULD VRFLHGDGH H DUWHV p UD]mR VX┬┐FLHQWH SDUD HVWXGDU HVWD PDUD

Book 1.indb 1

04/04/2016 16:30:29

2

Matemática discreta

YLOKRVD PDWpULD 1yV PDWHPiWLFRV VRPRV LPHQVDPHQWH RUJXOKRVRV GDV FRQTXLVWDV TXH VmR LPSXOVLRQDGDV SHOD DQiOLVH PDWHPiWLFD (QWUHWDQWR SDUD PXLWRV GH QyV HVVD QmR p D SULQFLSDO PRWLYDomR SDUD R HVWXGR GD PDWHPiWLFD

A agonia e o êxtase 3RU TXH RV PDWHPiWLFRV GHYRWDP VXDV YLGDV DR HVWXGR GD PDWHPiWLFD" 3DUD D PDLRULD GH QyV p SRU FDXVD GD DOHJULD TXH VHQWLPRV DR WUDEDOKDU FRP D PDWHPiWLFD $ PDWHPiWLFD p GLItFLO SDUD WRGR PXQGR 1mR LPSRUWD TXH QtYHO GH UHDOL]Do}HV RX FR QKHFLPHQWRV YRFr RX VHX SURIHVVRU WHQKD GHVWD PDWpULD Ki VHPSUH XP SUREOHPD PDLV GLItFLO H IUXVWUDQWH HVSHUDQGR DGLDQWH 'HVHQFRUDMDGRU" 'L¿FLOPHQWH 4XDQWR PDLRU R GHVD¿R PDLRU D QRVVD VHQVDomR GH UHDOL]DomR TXDQGR R YHQFHPRV $ PHOKRU SDUWH GD PDWHPiWLFD p D DOHJULD TXH YLYHQFLDPRV DR SUDWLFDU HVWD DUWH $ PDLRULD GDV IRUPDV GH DUWH SRGH VHU GHVIUXWDGD SRU HVSHFWDGRUHV 3RVVR GHOHLWDU PH FRP XP FRQFHUWR UHDOL]DGR SRU P~VLFRV WDOHQWRVRV ¿FDU ERTXLDEHUWR GLDQWH GH XPD EHOD SLQWXUD RX SURIXQGDPHQWH FRPRYLGR SHOD OLWHUDWXUD $ PDWHPiWLFD SRUpP OLEHUD HVVD FDUJD HPRFLRQDO VRPHQWH VREUH DTXHOHV TXH UHDOPHQWH WUDEDOKDP FRP HOD 4XHUR TXH YRFr VLQWD HVVD DOHJULD WDPEpP 'HVVH PRGR QR ¿QDO GHVWD EUHYH VHomR Ki XP ~QLFR SUREOHPD D VHU VROXFLRQDGR 3DUD TXH YRFr H[SHULPHQWH HVVD DOHJULD não deixe sob hipótese alguma que ninguém o ajude a resolver este problema. (VSHUR TXH DR REVHUYDU R SUREOHPD SHOD SULPHLUD YH] YRFr QmR YHMD VXD VROXomR LPHGLDWDPHQWH PDV HP YH] GLVVR TXHEUH D FDEHoD XP SRXFR 1mR VH VLQWD PDO PRVWUHL HVWH SUREOHPD SDUD PDWHPiWLFRV H[WUHPDPHQWH WDOHQWRVRV TXH QmR HQFRQWUDUDP XPD VROXomR LPHGLDWDPHQWH 7UDEDOKH H SHQVH FRQWLQXDPHQWH ± D VROXomR DSDUHFHUi (VSHUR TXH DR VROXFLRQDU HVWH TXHEUD FDEHoD YRFr VRUULD GLDQWH GD UHVROXomR $TXL HVWi R TXHEUD FDEHoD

1 Exercícios 1.1 6LPSOL¿TXH D VHJXLQWH H[SUHVVmR DOJpEULFD x – a x – b x – c x – z

WŽƌ ŽƵƚƌŽ ůĂĚŽ͕ ĐĂƐŽ ǀŽĐġ ƚĞŶŚĂ ƐŽůƵĐŝŽŶĂĚŽ ĞƐƚĞ ƉƌŽďůĞŵĂ͕ ŶĆŽ ŽĨĞƌĞĕĂ ĂũƵĚĂ ĂŽƐ ŽƵƚƌŽƐ͗ ǀŽĐġ ŶĆŽ ǀĂŝ ƋƵĞƌĞƌ ĞƐƚƌĂŐĂƌ Ă ĚŝǀĞƌƐĆŽ ĚĞůĞƐ͘

2 Falando (e escrevendo) sobre matemática Precisamente! *RVWHPRV RX QmR GH PDWHPiWLFD WRGRV DGPLUDPRV XPD GH VXDV FDUDFWHUtVWLFDV ~QLFDV Ki UHVSRVWDV GH¿QLWLYDV 3RXFRV RXWURV HPSUHHQGLPHQWRV GD HFRQRPLD j DQiOLVH OLWHUi ULD H GD KLVWyULD j SVLFRORJLD SRGHP FRQWDU FRP HVVD YDQWDJHP $OpP GLVVR HP PDWH PiWLFD SRGHPRV IDODU H HVFUHYHU FRP H[WUHPD SUHFLVmR (QTXDQWR RV OLYURV FDQo}HV H SRHPDV IRUDP HVFULWRV VREUH R DPRU p PXLWR PDLV IiFLO ID]HU D¿UPDo}HV SUHFLVDV H YHUL¿FDU VXD YHUGDGH VREUH D PDWHPiWLFD GR TXH VREUH DV UHODo}HV KXPDQDV /LQJXDJHP SUHFLVD p YLWDO SDUD R HVWXGR GD PDWHPiWLFD ,QIHOL]PHQWH RV DOXQRV jV YH]HV YHHP D PDWHPiWLFD FRPR XPD VpULH LQWHUPLQiYHO GH FiOFXORV QXPpULFRV H DOJp EULFRV HP TXH DV OHWUDV VmR XVDGDV VRPHQWH SDUD QRPHDU YDULiYHLV 1D YHUGDGH SDUD VH FRPXQLFDU FRP FODUH]D H SUHFLVmR PDWHPiWLFD SUHFLVDPRV GH PXLWR PDLV GR TXH Q~PHURV YDULiYHLV RSHUDo}HV H VtPERORV SUHFLVDPRV GH SDODYUDV FRPSRVWDV HP VHQWHQoDV VLJQL¿

Book 1.indb 2

04/04/2016 16:30:29

Capítulo 1

Fundamentos

3

FDWLYDV TXH WUDQVPLWDP H[DWDPHQWH R VLJQL¿FDGR TXH SUHWHQGHPRV 6HQWHQoDV PDWHPiWL FDV PXLWDV YH]HV LQFOXHP D QRWDomR WpFQLFD PDV DV UHJUDV GD JUDPiWLFD DSOLFDP VH D HODV SOHQDPHQWH ,QGLVFXWLYHOPHQWH DWp TXH DOJXpP H[SULPD LGHLDV HP XPD IUDVH FRHUHQWH HODV HVWmR DSHQDV SHOD PHWDGH $OpP GLVVR R HVIRUoR PHQWDO SDUD FRQYHUWHU LGHLDV PDWHPiWLFDV HP OLQJXDJHP p YLWDO SDUD DSUHHQGHU HVVHV FRQFHLWRV $SURYHLWH R WHPSR SDUD H[SUHVVDU VXDV LGHLDV FRP FODUH]D WDQWR YHUEDO PHQWH FRPR SRU HVFULWR $SUHQGHU PDWHPiWLFD UHTXHU HQYROYLPHQWR GH WRGDV DV URWDV DWp R VHX Fp UHEUR VXDV PmRV ROKRV ERFD H RXYLGRV WRGRV SUHFLVDP HQWUDU HP DomR 'LJD DV LGHLDV HP YR] DOWD H DQRWH DV 9RFr DSUHQGHUi D VH H[SUHVVDU GH IRUPD PDLV FODUD H DSUHHQGHUi PHOKRU RV FRQFHLWRV ĞƌƟĮƋƵĞͲƐĞ ĚĞ ǀĞƌŝĮĐĂƌ ĐŽŵ ƐĞƵ ŝŶƐƚƌƵƚŽƌ ƋƵĂŝƐ ƟƉŽƐ ĚĞ ĐŽůĂďŽƌĂĕĆŽ ƐĆŽ ƉĞƌŵŝƟĚĂƐ Ğŵ ƐƵĂƐ ƚĂƌĞĨĂƐ Ğ ƚƌĂďĂůŚŽƐ͘

Um pouco de ajuda (VFUHYHU p GLItFLO $ PHOKRU PDQHLUD GH DSUHQGHU p SUDWLFDU HVSHFLDOPHQWH FRP D DMX GD GH XP SDUFHLUR $ PDLRULD GDV SHVVRDV DFKD GLItFLO HGLWDU VXD SUySULD HVFULWD QRVVR FpUHEUR VDEH R TXH TXHUHPRV GL]HU H QRV ID] DFUHGLWDU TXH R TXH FRORFDPRV QR SDSHO p H[DWDPHQWH R TXH SUHWHQGtDPRV 6H YRFr GL] ³ERP YRFr HQWHQGHX R TXH HX TXLV GL]HU´ HQWmR SUHFLVD WHQWDU QRYDPHQWH 1HVWD EUHYH VHomR IRUQHFHPRV DOJXPDV GLFDV H DYLVRV VREUH DOJXQV HUURV FRPXQV 8PD OLQJXDJHP SUySULD 3RU WRGR R OLYUR YRFr HQFRQWUDUi DV QRWDV Linguagem matemática! TXH H[SOLFDP DOJXPDV IRUPDV LGLRVVLQFUiWLFDV HP TXH RV PDWHPiWLFRV XVDP SDODYUDV FRPXQV 3DODYUDV FRPXQV WDLV FRPR IXQomR RX primo) VmR XVDGDV GH IRUPD GLIHUHQWH QD PDWHPiWLFD $ ERD QRWtFLD p TXH TXDQGR QyV FRRSWDPRV SDODYUDV D VHU YLoR GD PDWHPiWLFD RV VLJQL¿FDGRV TXH OKHV GDPRV VmR SUHFLVRV FRPR XP ¿R GH QDYDOKD YHMD D SUy[LPD VHomR GHVWH OLYUR SDUD VDEHU PDLV VREUH LVVR

6HQWHQoDV FRPSOHWDV (VWD p D UHJUD PDLV EiVLFD GD JUDPiWLFD H DSOLFD VH j PDWHPiWLFD WDQWR TXDQWR D TXDOTXHU GLVFLSOLQD 1RWDomR PDWHPiWLFD GHYH VHU SDUWH GH XPD VHQWHQoD Ruim: 3x + 5.

,VVR QmR p XPD VHQWHQoD 2 TXH VLJQL¿FD x " 2 TXH R HVFULWRU HVWi WHQWDQGR GL]HU" Boa: 4XDQGR VXEVWLWXtPRV x ± SRU x R UHVXOWDGR p ,QFRPSDWLELOLGDGH GH FDWHJRULDV (VVH p XP GRV HUURV PDLV FRPXQV TXH DV SHVVRDV FRPHWHP DR HVFUHYHU H IDODU GH PDWHPiWLFD 8P VHJPHQWR GH OLQKD QmR p XP Q~PHUR XPD IXQomR QmR p XPD HTXDomR XP FRQMXQWR QmR p XPD RSHUDomR H DVVLP SRU GLDQWH &RQVLGHUH HVWD IUDVH O Air Force One é o presidente dos Estados Unidos.

,VVR QDWXUDOPHQWH p XP GLVSDUDWH 1HQKXP ³ERP YRFr HQWHQGHX R TXH HX TXLV GL]HU´ RX ³YRFr HQWHQGHX D LGHLD JHUDO´ SRGH GHVID]HU R HUUR GH HVFUHYHU TXH XP DYLmR p XP VHU KXPDQR 1R HQWDQWR HVWH p H[DWDPHQWH R WLSR GH HUUR TXH RV HVFULWRUHV GH PDWH PiWLFD QRYDWRV FRPHWHP FRP IUHTXrQFLD $VVLP QmR HVFUHYD ³D IXQomR p LJXDO D ´ TXDQGR YRFr TXHU GL]HU ³TXDQGR D IXQ omR p DYDOLDGD HP x R UHVXOWDGR p ´ 1RWH TXH QyV QmR SUHFLVDPRV VHU SUROL[RV 1mR HVFUHYD ³f ´ PDV ³f ´ Ruim: 6H RV ODGRV GH XP WULkQJXOR UHWkQJXOR T WrP FRPSULPHQWRV H HQWmR T = 30.

Boa: 6H RV ODGRV GH XP WULkQJXOR UHWkQJXOR T WrP FRPSULPHQWRV H HQWmR D iUHD GH 7 p Evite pronomes e IiFLO HVFUHYHU XPD IUDVH FKHLD GRV SURQRPHV TXH YRFr ± R HVFULWRU ± HQWHQGH PDV TXH p LQFRPSUHHQVtYHO SDUD TXDOTXHU RXWUD SHVVRD

Book 1.indb 3

04/04/2016 16:30:29

4

Matemática discreta

Ruim: 6H PRYHUPRV WXGR HQWmR ¿FD PDLV VLPSOHV H HVVD p D QRVVD UHVSRVWD 'r QRPHV DR TXH YRFr HVWi HVFUHYHQGR WDLV FRPR OHWUDV LQGLYLGXDLV SDUD Q~PHURV H Q~PHURV GH OLQKD SDUD HTXDo}HV

Boa: 4XDQGR PRYHPRV WRGRV RV WHUPRV TXH HQYROYHP x j HVTXHUGD QD HTXDomR

GHVFREULPRV TXH HVVHV WHUPRV VH FDQFHODP H LVVR QRV SHUPLWH GHWHUPLQDU R YDORU GH y.

5HHVFUHYD e TXDVH LPSRVVtYHO HVFUHYHU EHP HP XP SULPHLUR HVERoR $OpP GR PDLV DOJXQV SUREOHPDV PDWHPiWLFRV SRGHP VHU UHVROYLGRV FRUUHWDPHQWH FRP PDLV UDSLGH] ,QIHOL]PHQWH DOJXQV HVWXGDQWHV QmR YRFr p FODUR FRPHoDP D UHVROYHU XP SUREOHPD ULVFDU HUURV GHVHQKDU VHWDV QDV QRYDV SDUWHV GD VROXomR H HP VHJXLGD HQYLDP HVVD WHU UtYHO EDJXQoD FRPR XP SURGXWR DFDEDGR &UHGR 7DO FRPR DFRQWHFH FRP WRGDV DV RX WUDV IRUPDV GH HVFULWD HODERUH XP SULPHLUR HVERoR HGLWH R H HP VHJXLGD UHHVFUHYD R Aprenda a usar o /AT(; 2 SURFHVVR GH HGLomR H UHHVFULWD p IHLWR GH IRUPD PXL WR PDLV IiFLO SRU SURFHVVDGRUHV GH WH[WR ,QIHOL]PHQWH p PXLWR PDLV GLItFLO GLJLWDU PDWHPiWLFD GR TXH SURVD FRPXP $OJXQV GRV SURJUDPDV GH SURFHVVDPHQWR R TXH YR Fr Yr p R TXH YRFr FRQVHJXH >:<6,:<*@ FRPR R 0LFURVRIW :RUG LQFOXHP XP HGLWRU GH HTXDo}HV TXH SHUPLWH GLJLWDU H LQVHULU IyUPXODV PDWHPiWLFDV HP GRFXPHQ WRV 5HDOPHQWH PXLWRV FLHQWLVWDV H HQJHQKHLURV XVDP R :RUG SDUD HVFUHYHU WUDEDOKRV WpFQLFRV UHSOHWRV GH IyUPXODV FRPSOH[DV ƉĂůĂǀƌĂ /AT(X ĞƐƚĄ ĞƐĐƌŝƚĂ ĐŽŵ ůĞƚƌĂƐ ĚĞ ǀĄƌŝŽƐ ƚĂŵĂŶŚŽƐ Ğŵ ŶşǀĞŝƐ ĚŝĨĞƌĞŶƚĞƐ͕ Ğŵ ƉĂƌƚĞ ƉĂƌĂ ĚŝƐƟŶŐƵŝͲůĂ ĚĞ ůĄƚĞdž͕ Ƶŵ ƟƉŽ ĚĞ ďŽƌƌĂĐŚĂ͘ /ŶŝĐŝĂůŵĞŶƚĞ͕ ĞƐƚĞ ůŝǀƌŽ ĨŽŝ ĐŽŵƉŽƐƚŽ ƵƐĂŶĚŽ /AT(X͘ 1R HQWDQWR R SDGUmR SDUD D GLJLWDomR PDWHPiWLFD p R /AT(X $SUHQGHU D HVFUHYHU GRFXPHQ tos no /AT(X H[LJH XP LQYHVWLPHQWR GH WHPSR LQLFLDO VLJQL¿FDWLYR PDV QHQKXP LQYHVWLPHQWR HP GLQKHLUR SRLV Ki PXLWDV LPSOHPHQWDo}HV GR /AT(X TXH VmR JUDWXLWDV H IXQFLRQDP QD PDLRULD GRV FRPSXWDGRUHV :LQGRZV 0DF26 /LQX[ 2V GRFXPHQWRV SURGX]LGRV QR /AT(X VmR YL VXDOPHQWH PDLV DWUDHQWHV GR TXH RV GH VLVWHPDV :<6,:,* H PDLV IiFHLV GH HGLWDU 1R /AT(X EDVWD GLJLWDU FRPDQGRV HVSHFLDLV SDUD SURGX]LU QRWDomR PDWHPiWLFD 3RU H[HPSOR SDUD HVFUHYHU D IyUPXOD TXDGUiWLFD p b ˙ b 2 4ac xD 2a

p Vy GLJLWDU x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} +i PXLWRV JXLDV H OLYURV GLVSRQtYHLV SDUD DSUHQGHU D XVDU R /AT(; LQFOXLQGR DOJXQV GLVSRQtYHLV JUDWXLWDPHQWH QD web.

2 Exercícios 2.1 $V VHLV SHoDV DEDL[R SRGHP VHU GLVSRVWDV GH PRGR D IRUPDU XP TXDGUDGR GH q FRP R TXDGUDGR GR PHLR GH q YD]LR FRPR QD ¿JXUD j HVTXHUGD

Book 1.indb 4

04/04/2016 16:30:30

Capítulo 1

Fundamentos

5

'HWHUPLQH FRPR UHVROYHU HVVH TXHEUD FDEHoD H HP VHJXLGD HVFUHYD LQVWUXo}HV FODUDV VHP GLDJUDPDV SDUD TXH RXWUD SHVVRD DV SRVVD OHU FRUUHWDPHQWH H HQFDL[DU DV SHoDV SDUD FKHJDU j VROXomR 9RFr SRGH EDL[DU XPD YHUVmR SDUD LPSULPLU DV SHoDV GR TXHEUD FDEHoD DVVLP SRGH FRUWi ODV HP PHGLGDV PDLRUHV QR site GR DXWRU ZZZ DPV MKX HGX HUV ZS FRQWHQW XSORDGV VLWHV SX]]OH SGI

3 Definição $ PDWHPiWLFD H[LVWH DSHQDV QDV PHQWHV GDV SHVVRDV 1mR H[LVWH WDO ³FRLVD´ FRPR R Q~ PHUR 3RGHPRV GHVHQKDU R VtPEROR SDUD R Q~PHUR HP XP SHGDoR GH SDSHO PDV QmR SRGHPRV ¿VLFDPHQWH VHJXUDU XP HP QRVVDV PmRV 2V Q~PHURV DVVLP FRPR WRGRV RV RXWURV REMHWRV PDWHPiWLFRV VmR SXUDPHQWH FRQFHLWXDLV 2V REMHWRV PDWHPiWLFRV DGTXLUHP H[LVWrQFLD SRU GH¿QLo}HV 3RU H[HPSOR XP Q~PHUR p FKDPDGR primo RX par GHVGH TXH VDWLVIDoD FRQGLo}HV SUHFLVDV VHP DPELJXLGDGH (VVDV FRQGLo}HV DOWDPHQWH HVSHFt¿FDV FRQVWLWXHP D GH¿QLomR GR FRQFHLWR 'HVVD IRUPD HVWDPRV DWXDQGR FRPR OHJLVODGRUHV TXH GH¿QHP FULWpULRV HVSHFt¿FRV WDLV FRPR TXDOL¿FDomR SDUD XP SURJUDPD GH JRYHUQR $ GLIHUHQoD p TXH DV OHLV SRGHP SHUPLWLU FHUWD DPELJXLGDGH HQTXDQWR XPD GH¿QLomR PDWHPiWLFD GHYH VHU DEVROXWDPHQWH FODUD &RQVLGHUHPRV XP H[HPSOR ŵ ƵŵĂ ĚĞĮŶŝĕĆŽ͕ ĂƐ ƉĂůĂǀƌĂƐ ƐĞŶĚŽ ĚĞĮŶŝĚĂƐ ƐĆŽ͕ Ğŵ ŐĞƌĂů͕ ĚĞƐƚĂĐĂĚĂƐ Ğŵ ŝƚĄůŝĐŽ͘

z

DEFINIÇÃO 3.1 (Par) 8P LQWHLUR p FKDPDGR par VH IRU GLYLVtYHO SRU &ODUR" 1mR WRWDOPHQWH 2 SUREOHPD p TXH HVVD GH¿QLomR FRQWpP WHUPRV TXH DLQGD QmR IRUDP GH¿QLGRV HP SDUWLFXODU inteiro e GLYLVtYHO 6H TXLVHUPRV VHU H[WUHPDPHQWH GHWD OKLVWDV SRGHPRV DOHJDU TXH DLQGD QmR GH¿QLPRV R WHUPR &DGD XP GHVVHV WHUPRV ± inteiro GLYLVtYHO e 2 ± SRGH VHU GH¿QLGR HP WHUPRV GH FRQFHLWRV PDLV VLPSOHV PDV HVVH p XP MRJR TXH QmR SRGHPRV JDQKDU LQWHLUDPHQWH 6H FDGD WHUPR IRU GH¿QLGR HP WHUPRV PDLV VLPSOHV HVWDUHPRV FRQWLQXDPHQWH HP EXVFD GH GH¿QLo}HV 'HYH FKHJDU XP PRPHQWR HP TXH GLJDPRV ³(VWH WHUPR QmR HVWi GH¿QLGR PDV FUHPRV HQWHQGHU R TXH HOH VLJQL¿FD´ $ VLWXDomR p FRPR D GD FRQVWUXomR GH XPD FDVD &DGD SDUWH GD FDVD p FRQVWUXtGD D SDUWLU GDV SDUWHV DQWHULRUHV $QWHV GR WHOKDGR H GDV SDUHGHV GHYHPRV FRQVWUXLU D HV WUXWXUD $QWHV GH HULJLUPRV D HVWUXWXUD GHYH KDYHU XP DOLFHUFH &RPR FRQVWUXWRUHV GD FDVD FRQVLGHUDPRV D FRQVWUXomR GR DOLFHUFH FRPR R SULPHLUR SDVVR ± PDV HVVH QmR p QD YHUGDGH R SULPHLUR SDVVR 3UHFLVDPRV WHU R WHUUHQR OLJDU D iJXD H D HOHWULFLGDGH QD SURSULHGDGH 3DUD TXH KDMD iJXD GHYH KDYHU SRoRV H HQFDQDPHQWRV QR VROR 3$5( &KHJDPRV D XP QtYHO GR SURFHVVR TXH UHDOPHQWH SRXFR WHP D YHU FRP D FRQVWUXomR GD FDVD 2V UHFXUVRV VmR YLWDLV SDUD D FRQVWUXomR PDV QmR p QRVVD IXQomR FRPR FRQVWUX WRUHV QRV SUHRFXSDUPRV FRP R WLSR GH WUDQVIRUPDGRUHV XVDGRV QD VXEHVWDomR HOpWULFD 9ROWHPRV j PDWHPiWLFD H j 'H¿QLomR e SRVVtYHO GH¿QLUPRV DV SDODYUDV inteiro 2 e GLYLVtYHO HP FRQFHLWRV PDLV EiVLFRV ([LJH JUDQGH WUDEDOKR GH¿QLUPRV LQWHLURV

Book 1.indb 5

04/04/2016 16:30:30

6

Matemática discreta

PXOWLSOLFDomR HWF HP FRQFHLWRV PDLV VLPSOHV 2 TXH GHYHPRV ID]HU" ,GHDOPHQWH GH YHUtDPRV FRPHoDU GR REMHWR PDWHPiWLFR PDLV EiVLFR ± R conjunto ± H SHUFRUUHU QRVVR FDPLQKR DWp RV LQWHLURV (PERUD VH WUDWH GH XP SURFHGLPHQWR SOHQDPHQWH MXVWL¿FiYHO QHVWH OLYUR YDPRV FRQVWUXLU QRVVR HGLItFLR PDWHPiWLFR VXSRQGR Mi IRUPDGR R DOLFHUFH 3RU RQGH GHYHPRV FRPHoDU" 2 TXH SRGHPRV VXSRU" 1HVWH OLYUR FRQVLGHUDPRV RV LQWHLURV FRPR QRVVR SRQWR GH SDUWLGD 2V inteiros VmR RV Q~PHURV LQWHLURV SRVLWLYRV RV LQWHLURV QHJDWLYRV H R ]HUR 2X VHMD R FRQMXQWR GRV LQWHLURV GHQRWDGR SHOD OHWUD ' p ' ^ ± ± ± ` K ƐşŵďŽůŽ ' Ġ ƵƐĂĚŽ ƉĂƌĂ ŝŶƚĞŝƌŽƐ Ğ Ġ ĨĄĐŝů ĚĞ ƚƌĂĕĂƌ͕ ŵĂƐ ĨƌĞƋƵĞŶƚĞŵĞŶƚĞ ĂƐ ƉĞƐƐŽĂƐ ŶĆŽ ĐŽŶƐĞŐƵĞŵ͘ WŽƌ ƋƵġ͍ ůĂƐ ĐĂĞŵ ŶĂ ƐĞŐƵŝŶƚĞ ĂƌŵĂĚŝůŚĂ͘ WƌŝŵĞŝƌŽ ƚƌĂĕĂŵ Ƶŵ = Ğ Ğŵ ƐĞŐƵŝĚĂ ƉƌŽĐƵƌĂŵ ĂĐƌĞƐĐĞŶƚĂƌ Ƶŵ ƚƌĂĕŽ ĂĚŝĐŝŽŶĂů͘ /ƐƐŽ ŶĆŽ ĨƵŶĐŝŽŶĂ͊ ĞǀĞͲƐĞ ƚƌĂĕĂƌ Ƶŵ ϳ Ğ ĞŶƚĆŽ ŽƵƚƌŽ ϳ ĞŶƚƌĞůĂĕĂĚŽ͕ ĚĞ ĐĂďĞĕĂ ƉĂƌĂ ďĂŝdžŽ͕ ƉĂƌĂ ƐĞ ŽďƚĞƌ Ƶŵ '͘

$GPLWLUHPRV WDPEpP TXH VDEHPRV VRPDU VXEWUDLU H PXOWLSOLFDU QmR SUHFLVDPRV SURYDU IDWRV EiVLFRV VREUH RV Q~PHURV WDLV FRPR q 2 = 6. Admitiremos as proprie GDGHV DOJpEULFDV EiVLFDV GH DGLomR VXEWUDomR H PXOWLSOLFDomR H IDWRV EiVLFRV VREUH UH ODo}HV GH RUGHP d ! H t &RQVXOWH R $SrQGLFH ' GLVSRQtYHO QR VLWH GD &HQJDJH /HDUQLQJ SDUD PDLV GDGRV TXH TXLVHU XWLOL]DU $VVLP QD 'H¿QLomR QmR SUHFLVDPRV GH¿QLU inteiro nem 2 7RGDYLD DLQGD GH YHPRV GH¿QLU R TXH TXHUHPRV GL]HU SRU GLYLVtYHO 3DUD VDOLHQWDU R IDWR TXH DLQGD QmR WRUQDPRV FODUR HVVH SRQWR FRQVLGHUHPRV D TXHVWmR p GLYLVtYHO SRU " 3UHWHQGHPRV GL]HU TXH D UHVSRVWD D HVVD SHUJXQWD p QmR PDV WDOYH] HOD SRVVD VHU sim SRLV u 2 = 1 21 $VVLP VH DGPLWLUPRV IUDo}HV p SRVVtYHO GLYLGLU SRU 1RWH VH DLQGD TXH QR SDUiJUDIR DQWHULRU JDUDQWLUDP VH SURSULHGDGHV EiVLFDV GD DGLomR VXEWUDomR H PXOWLSOL FDomR PDV QmR ± H HYLGHQWH SRU VXD DXVrQFLD ± GD GLYLVmR 1HFHVVLWDPRV DVVLP GH XPD GH¿QLomR FXLGDGRVD GH GLYLVtYHO. z

DEFINIÇÃO 3.2 (Divisível) 6HMDP a e b LQWHLURV 'L]HPRV TXH a p GLYLVtYHO por b VH H[LVWLU XP LQWHLUR c GH PRGR TXH bc = D 'L]HPRV WDPEpP TXH b divide a RX TXH b p XP fator de a RX TXH b p XP divisor de a $ QRWDomR FRUUHVSRQGHQWH p b|a.

(VWD GH¿QLomR LQWURGX] YiULRV WHUPRV GLYLVtYHO fator divisor e divide DVVLP FRPR D QRWDomR b|a &RQVLGHUHPRV XP H[HPSOR

EXEMPLO 3.3 Vejamos: p GLYLVtYHO SRU " 3DUD UHVSRQGHU D HVVD SHUJXQWD H[DPLQHPRV D GH¿QLomR TXH GL] TXH a p GLYLVtYHO SRU b VH H[LVWLU XP LQWHLUR c WDO TXH c 2EYLDPHQ WH HVVH LQWHLUR H[LVWH H p c = 3.

1HVVDV FRQGLo}HV GL]HPRV WDPEpP TXH GLYLGH RX HTXLYDOHQWHPHQWH TXH p XP IDWRU GH RX DLQGD TXH p XP GLYLVRU GH ([SUHVVD VH HVVH IDWR SHOD QRWDomR _ 1R HQWDQWR QmR p GLYLVtYHO SRU SRUTXH QmR Ki LQWHLUR x SDUD R TXDO x DVVLP _ p IDOVR

Book 1.indb 6

04/04/2016 16:30:30

Capítulo 1

Fundamentos

7

$JRUD D 'H¿QLomR HVWi SURQWD SDUD XVR 2 Q~PHUR p SDU SRUTXH _ H VDEH PRV TXH _ SRUTXH q (QWUHWDQWR QmR p SDU SRUTXH QmR p GLYLVtYHO SRU ± QmR Ki LQWHLUR x SDUD R TXDO x 1RWH TXH QmR GLVVHPRV TXH p tPSDU SRLV DLQGD SUHFLVDPRV GH¿QLU R WHUPR ímpar 1DWXUDOPHQWH VDEHPRV TXH p XP Q~PHUR tPSDU PDV VLPSOHVPHQWH DLQGD QmR ³FULDPRV´ RV Q~PHURV tPSDUHV PHGLDQWH HVSHFL¿FDomR GH XPD GH¿QLomR SDUD HOHV 7XGR R TXH SRGHPRV GL]HU D HVWD DOWXUD p TXH QmR p SDU $VVLP YDPRV GH¿QLU R WHUPR ímpar. z

DEFINIÇÃO 3.4 (Ímpar) 8P LQWHLUR a p FKDPDGR ímpar GHVGH TXH KDMD XP LQWHLUR x GH PRGR TXH a = 2x + 1. $VVLP p tPSDU SRUTXH SRGHPRV HVFROKHU x QD GH¿QLomR REWHQGR q 6 + 1.

1RWH VH TXH D GH¿QLomR IRUQHFH XP FULWpULR FODUR VHP DPELJXLGDGH SDUD GHWHUPLQDU VH XP LQWHLUR p tPSDU 3RU IDYRU REVHUYH R TXH D GH¿QLomR GH ímpar QmR GL] HOD QmR D¿UPD TXH XP LQWHLUR p tPSDU GHVGH TXH QmR VHMD SDU ,VVR QDWXUDOPHQWH p YHUGDGHLUR FRQIRUPH SURYDUHPRV HP XP FDStWXOR VXEVHTXHQWH 4XH ³WRGR LQWHLUR p tPSDU RX SDU PDV QmR DPERV´ p XP IDWR TXH provamos.

(LV XPD GH¿QLomR SDUD RXWUR FRQFHLWR IDPLOLDU

z

DEFINIÇÃO 3.5 (Primo) 8P LQWHLUR p p primo se p ! H VH RV ~QLFRV GLYLVRUHV SRVLWLYRV GH p VmR H p.

3RU H[HPSOR p SULPR SRUTXH VDWLVID] DPEDV DV FRQGLo}HV GD GH¿QLomR SULPHLUR p PDLRU TXH H VHJXQGR RV ~QLFRV GLYLVRUHV SRVLWLYRV GH VmR H (QWUHWDQWR QmR p SULPR SRUTXH WHP XP GLYLVRU SRVLWLYR GLIHUHQWH GH H HOH PHV PR SRU H[HPSOR _ 1 e 3 12.

2 Q~PHUR p SULPR" 1mR 3DUD VDEHU R SRUTXr WRPHPRV p H YHMDPRV VH p sa WLVID] D GH¿QLomR GH SULPR +i GXDV FRQGLo}HV 3ULPHLUR GHYHPRV WHU p ! H VHJXQGR RV ~QLFRV GLYLVRUHV SRVLWLYRV GH p VmR H p $ VHJXQGD FRQGLomR p VDWLVIHLWD 2V ~QLFRV GLYLVRUHV GH VmR H HOH SUySULR 0DV p QmR VDWLVID] D SULPHLUD FRQGLomR SRUTXH ! p IDOVD 3RUWDQWR QmR p SULPR 5HVSRQGHPRV D SHUJXQWD p SULPR" $ UD]mR SHOD TXDO QmR p SULPR p TXH D GH¿QL omR IRL HODERUDGD HVSHFL¿FDPHQWH SDUD WRUQDU QmR SULPR 7RGDYLD D SHUJXQWD UHDO TXH JRVWDUtDPRV GH UHVSRQGHU p SRU TXH IRUPXODPRV D 'H¿QLomR GH IRUPD D H[FOXLU " 3URFXUDUHL UHVSRQGHU D HVVD SHUJXQWD HP XP PRPHQWR PDV Ki XP SRQWR ¿ORVy¿FR TXH GHYH VHU VDOLHQWDGR $ GHFLVmR GH H[FOXLU R Q~PHUR QD GH¿QLomR IRL GHOLEHUDGD H FRQVFLHQWH &RP HIHLWR D UD]mR GH QmR VHU SULPR p ³SRUTXH HX DVVLP GLVVH´ (P SULQFt SLR SRGHUtDPRV GH¿QLU D SDODYUD primo GH XPD IRUPD GLIHUHQWH SHUPLWLQGR TXH R Q~PHUR IRVVH SULPR 2 SUREOHPD SULQFLSDO FRP D XWLOL]DomR GH XPD GH¿QLomR GLIHUHQWH SDUD SULPR p TXH R FRQFHLWR GH número primo HVWi EHP ¿UPDGR QD FRPXQLGDGH PDWHPiWLFD 6H OKH IRVVH ~WLO DGPLWLU FRPR SULPR HP VHX WUDEDOKR YRFr GHYHULD HVFROKHU XP WHUPR GLIHUHQWH SDUD VHX FRQFHLWR WDO FRPR SULPR UHOD[DGR RX SULPR DOWHUQDWLYR. $ERUGHPRV DJRUD D TXHVWmR SRU TXH IRUPXODPRV D 'H¿QLomR GH PRGR D H[FOXLU " $ LGHLD p TXH RV Q~PHURV SULPRV FRQVWLWXHP RV ³EORFRV GH VXVWHQWDomR´ GD PXOWL SOLFDomR 0DLV j IUHQWH SURYDUHPRV TXH WRGR LQWHLUR SRVLWLYR SRGH VHU GHFRPSRVWR GH

Book 1.indb 7

04/04/2016 16:30:30

8

Matemática discreta

PDQHLUD ~QLFD HP IDWRUHV SULPRV 3RU H[HPSOR SRGH VHU IDWRUDGR FRPR q 2 q 1mR Ki RXWUD PDQHLUD GH GHFRPSRU HP IDWRUHV SULPRV D QmR VHU WURFDQGR D RUGHP GRV IDWRUHV 2V IDWRUHV SULPRV GH VmR SUHFLVDPHQWH H 6H I{VVHPRV DGPLWLU FRPR Q~PHUR SULPR HQWmR SRGHUtDPRV GHFRPSRU HP IDWRUHV ³SULPRV´ FRPR 1 q 2 q 2 q XPD IDWRUDomR GLIHUHQWH $VVLP FRPR GH¿QLPRV Q~PHURV SULPRV p DSURSULDGR GH¿QLUPRV WDPEpP Q~PHURV compostos. z

DEFINIÇÃO 3.6 (Composto) 8P Q~PHUR SRVLWLYR a p FKDPDGR composto VH H[LVWH XP LQWHLUR b de modo TXH b a e b|a.

3RU H[HPSOR R Q~PHUR p FRPSRVWR SRUTXH VH YHUL¿FD D FRQGLomR GD GH¿QLomR Ki XP Q~PHUR b FRP b H b_ QD YHUGDGH b p HVVH Q~PHUR ~QLFR

'D PHVPD IRUPD R Q~PHUR p FRPSRVWR 1HVVH FDVR Ki YiULRV Q~PHURV b WDLV TXH b H b|360.

2V Q~PHURV SULPRV QmR VmR FRPSRVWRV 6H p p SULPR HQWmR SRU GH¿QLomR QmR SRGH KDYHU GLYLVRU GH p entre 1 e p OHLD DWHQWDPHQWH D 'H¿QLomR

$OpP GLVVR R Q~PHUR QmR p FRPSRVWR 2EYLDPHQWH QmR H[LVWH XP Q~PHUR b com b 3REUH Q~PHUR 2 Q~PHUR QmR p SULPR QHP FRPSRVWR Ki HQWUHWDQWR XP WHUPR HVSHFLDO TXH VH DSOLFD DR Q~PHUR HOH p FKDPDGR unidade

Recapitulando 1HVWD VHomR LQWURGX]LPRV R FRQFHLWR GH GH¿QLomR PDWHPiWLFD $V GH¿QLo}HV WLSLFD PHQWH WrP D IRUPD ³8P REMHWR X p FKDPDGR R WHUPR D VHU GH¿QLGR GHVGH TXH VDWLVIDoD FRQGLo}HV HVSHFt¿FDV´ $SUHVHQWDPRV R FRQMXQWR GRV LQWHLURV ' H GH¿QLPRV RV WHUPRV divisíveis ímpares pares primos e compostos.

3 Exercícios 3.1. 'HWHUPLQH TXDLV GDV DVVHUo}HV VHJXLQWHV VmR YHUGDGHLUDV H TXDLV VmR IDOVDV 8WLOL]H D 'H¿QLomR SDUD H[SOLFDU VXDV UHVSRVWDV a. _

H ± _± b. 3|99 f. 0|4 c. ± _

J _ d. ± _±

K _ 3.2. (LV XPD DOWHUQDWLYD SRVVtYHO SDUD D 'H¿QLomR 'L]HPRV TXH a p GLYLVtYHO por b se for LQWHLUR ([SOLTXH SRU TXH HVVD GH¿QLomR DOWHUQDWLYD p GLIHUHQWH GD 'H¿QLomR

$TXL diferente VLJQL¿FD TXH D 'H¿QLomR H D GH¿QLomR DOWHUQDWLYD HVSHFL¿FDP conceitos diferentes $VVLP SDUD UHVSRQGHU D HVVD TXHVWmR GHYHPRV HQFRQWUDU LQWHLURV a e b WDLV TXH a VHMD GLYLVtYHO SRU b GH DFRUGR FRP XPD GH¿QLomR PDV a QmR VHMD GLYLVtYHO SRU b de acordo com D RXWUD GH¿QLomR

3.3. 1HQKXP GRV Q~PHURV VHJXLQWHV p SULPR ([SOLTXH SRU TXH HOHV QmR VDWLVID]HP D 'H¿QLomR 4XDLV GHVVHV Q~PHURV VmR FRPSRVWRV" a. 21. d. . b.

H ± c. Q

I ±

Book 1.indb 8

04/04/2016 16:30:31

Capítulo 1

Fundamentos

9

3.4. 2V números naturais VmR RV LQWHLURV QmR QHJDWLYRV LVWR p

^ ` K ƐşŵďŽůŽ Ġ ƵƐĂĚŽ ƉĂƌĂ ŶƷŵĞƌŽƐ ŶĂƚƵƌĂŝƐ͘

$SOLTXH R FRQFHLWR GH Q~PHURV QDWXUDLV SDUD FULDU GH¿QLo}HV SDUD DV VHJXLQWHV UHODo}HV GH LQWHLURV menor que PHQRU TXH RX LJXDO D d maior que !

H PDLRU TXH RX LJXDO D t

Nota 0XLWRV DXWRUHV GH¿QHP RV Q~PHURV QDWXUDLV DSHQDV FRPR RV LQWHLURV SRVLWLYRV SDUD HOHV ]HUR QmR p XP Q~PHUR QDWXUDO 3DUD PLP LVVR QmR SDUHFH QDWXUDO - 2V FRQFHLWRV inteiros positivos e LQWHLURV QmR QHJDWLYRV QmR WrP DPELJXLGDGH H VmR UHFRQKHFLGRV XQLYHUVDOPHQWH HQWUH RV PDWHPiWLFRV -i R WHUPR Q~PHUR QDWXUDO QmR p SDGURQL]DGR

3.5. 8P Q~PHUR UDFLRQDO p XP Q~PHUR IRUPDGR SHOD GLYLVmR GH GRLV LQWHLURV a/b FRP b z 2 FRQMXQWR GH WRGRV RV UDFLRQDLV p GHQRWDGR SRU .

([SOLTXH SRU TXH WRGR LQWHLUR p XP Q~PHUR UDFLRQDO PDV QHP WRGRV RV UDFLRQDLV VmR LQWHLURV K ƐşŵďŽůŽ Ġ ƵƐĂĚŽ ƉĂƌĂ ŶƷŵĞƌŽƐ ƌĂĐŝŽŶĂŝƐ͘

3.6. 'H¿QD R TXH VLJQL¿FD XP LQWHLUR VHU XP quadrado perfeito 3RU H[HPSOR RV LQWHLURV H VmR TXDGUDGRV SHUIHLWRV 6XD GH¿QLomR GHYH FRPHoDU

8P LQWHLUR x p FKDPDGR quadrado perfeito GHVGH TXH

3.7. 'H¿QD R TXH VLJQL¿FD XP Q~PHUR VHU D raiz quadrada GH RXWUR 3.8. 'H¿QD R perímetro GH XP SROtJRQR 3.9. 6XSRQKD Mi GH¿QLGR R FRQFHLWR GH GLVWkQFLD HQWUH GRLV SRQWRV GH XP SODQR )RUPXOH FXLGDGRVDPHQWH D FRQGLomR SDUD TXH XP SRQWR HVWHMD entre RXWURV GRLV SRQWRV 6XD GH¿QLomR GHYH FRPHoDU Suponhamos que A, B, C sejam pontos do plano. Dizemos que C está entre A e B desde que...

2EVHUYDomR FRPR YRFr HVWi HODERUDQGR HVVD GH¿QLomR YRFr WHP FHUWD ÀH[LELOLGDGH &RQVLGHUH D SRVVLELOLGDGH GH R SRQWR C VHU R PHVPR TXH R SRQWR A RX R SRQWR B RX DLQGD TXH A e B SRVVDP VHU R PHVPR SRQWR 3HVVRDOPHQWH VH A e C IRVVHP R PHVPR SRQWR HX GLULD TXH C está entre A e B LQGHSHQGHQWHPHQWH GH RQGH B SRVVD HVWDU PDV YRFr SRGH HVFROKHU SODQHMDU VXD GH¿QLomR GH PRGR D H[FOXLU HVVD SRVVLELOLGDGH 4XDOTXHU TXH VHMD VXD GHFLVmR HOD p ERD PDV FHUWL¿TXH VH GH TXH VXD GH¿QLomR DWHQGD DR TXH WHP HP YLVWD

2XWUD REVHUYDomR 1mR p QHFHVViULR R FRQFHLWR GH FROLQHDULGDGH SDUD GH¿QLU D QRomR entre.

8PD YH] GH¿QLGR entre XVH D QRomR SDUD GH¿QLU R TXH VLJQL¿FD WUrV SRQWRV VHUHP FROLQHDUHV 6XD GH¿QLomR GHYH FRPHoDU

Sejam A, B e C pontos do plano. Dizemos que eles são colineares desde que...

0DLV XPD REVHUYDomR DJRUD VH A e B VmR R PHVPR SRQWR R OHLWRU FHUWDPHQWH GHVHMD TXH VXD GH¿QLomR LPSOLTXH TXH A B e C VmR FROLQHDUHV

3.10. 'H¿QD R ponto médio GH XP VHJPHQWR GH OLQKD 3.11. $OJXPDV SDODYUDV VmR GLItFHLV GH GH¿QLU FRP SUHFLVmR PDWHPiWLFD SRU H[HPSOR amor PDV DOJXPDV SRGHP VHU EHP GH¿QLGDV 7HQWH HVFUHYHU GH¿QLo}HV SDUD HVWDV a. adolescente. d. GH] FHQWDYRV b. DYy

e. ௘SDOtQGURPR c. DQR ELVVH[WR

f. ௘ ௘KRPyIRQR 9RFr SRGH DVVXPLU TXH RV FRQFHLWRV PDLV EiVLFRV FRPR moeda RX pronúncia Mi HVWHMDP GH¿QLGRV

Book 1.indb 9

04/04/2016 16:30:31

10

Matemática discreta

3.12. 2V PDWHPiWLFRV GLVFUHWRV JRVWDP HVSHFLDOPHQWH GH SUREOHPDV GH FRQWDJHP SUREOHPDV TXH TXHVWLRQDP quantos" 9DPRV FRQVLGHUDU D TXHVWmR TXDQWRV GLYLVRUHV SRVLWLYRV XP Q~PHUR WHP" 3RU H[HPSOR WHP TXDWUR GLYLVRUHV SRVLWLYRV H

4XDQWRV GLYLVRUHV SRVLWLYRV WrP FDGD XP GRV Q~PHURV VHJXLQWHV" a. 8 b. 32 c. 2n HP TXH n p XP LQWHLUR SRVLWLYR d. 10 e. 100 f. 1.000.000 g. 10n HP TXH n p XP LQWHLUR SRVLWLYR h. 30 = 2 q 3 q 5 i. 42 = 2 q 3 q 3RU TXH H WrP R PHVPR Q~PHUR GH GLYLVRUHV SRVLWLYRV"

j. 2310 = 2 q 3 q 5 q 7 q 11 k. 1 q 2 q 3 q 4 q 5 q 6 q 7 q 8 l. 0 3.13. 8P LQWHLUR n p FKDPDGR perfeito VH IRU LJXDO j VRPD GH WRGRV RV VHXV GLYLVRUHV TXH VmR VLPXOWDQHDPHQWH SRVLWLYRV H LQIHULRUHV D n 3RU H[HPSOR p SHUIHLWR SRUTXH RV GLYLVRUHV SRVLWLYRV GH VmR H 1RWH TXH a. +i XP Q~PHUR SHUIHLWR LQIHULRU D $FKH R b. (VFUHYD XP SURJUDPD GH FRPSXWDGRU SDUD DFKDU R Q~PHUR SHUIHLWR LPHGLDWDPHQWH VXSHULRU a 28.

3.14. (P XP MRJR GD /LJD ,QIDQWLO Ki WUrV MXt]HV 8P p HQJHQKHLUR R RXWUR p ItVLFR H R WHUFHLUR PDWHPiWLFR +i XPD MRJDGD SUy[LPD j EDVH H RV WUrV MXt]HV FRQFRUGDP TXH R FRUUHGRU HVWi IRUD

)XULRVR R SDL GR FRUUHGRU JULWD SDUD RV MXt]HV ³3RU TXH YRFrV GL]HP TXH HOH HVWi IRUD"´ 2 HQJHQKHLUR UHVSRQGH ³(OH HVWi IRUD SRUTXH HX GLJR FRPR HOH HVWi´.

2 ItVLFR UHVSRQGH ³(OD HVWi IRUD SRUTXH p FRPR HX R YHMR´. ( R PDWHPiWLFR UHVSRQGH ³(OD HVWi IRUD SRUTXH HX GLJR TXH HVWi´.

([SOLTXH R SRQWR GH YLVWD GR PDWHPiWLFR

4 Teorema 8P teorema p XPD D¿UPDomR GHFODUDWLYD VREUH PDWHPiWLFD SDUD D TXDO H[LVWH XPD SURYD $ QRomR GH SURYD p R DVVXQWR GD SUy[LPD VHomR ± QD YHUGDGH p XP WHPD FHQWUDO GHVWH OLYUR %DVWD GL]HUPRV SRU RUD TXH XPD prova p XPD GLVVHUWDomR TXH PRVWUD GH PDQHLUD LUUHIXWiYHO TXH XPD D¿UPDomR p YHUGDGHLUD 1HVWD VHomR HQIRFDPRV D QRomR GH WHRUHPD 5HLWHUDQGR XP teorema p XPD D¿UPD omR GHFODUDWLYD VREUH PDWHPiWLFD SDUD D TXDO H[LVWH XPD SURYD 2 TXH p XPD D¿UPDomR GHFODUDWLYD" 1D OLQJXDJHP FRWLGLDQD H[SUHVVDPRV PXLWRV WLSRV GH VHQWHQoD $OJXPDV GHODV VmR SHUJXQWDV 2QGH HVWi R MRUQDO" 2XWUDV VHQWHQoDV VmR RUGHQV SDUH ( WDOYH] R WLSR PDLV FRPXP GH VHQWHQoD VHMD XPD D¿UPDomR GHFODUDWLva ± XPD VHQWHQoD TXH H[SUHVVD XPD LGHLD VREUH D QDWXUH]D RX HVWDGR GH DOJXPD FRPR YDL FKRYHU DPDQKm RX RV <DQNHHV JDQKDUDP QD QRLWH SDVVDGD 2V SUDWLFDQWHV GH WRGD GLVFLSOLQD ID]HP D¿UPDo}HV GHFODUDWLYDV VREUH VXD DWLYLGDGH 2 HFRQRPLVWD GL] ³VH D RIHUWD GH XP SURGXWR FDL HQWmR VHX SUHoR DXPHQWD´ 2 ItVLFR D¿UPD ³TXDQGR GHL[DPRV FDLU XP REMHWR SHUWR GD VXSHUItFLH GD 7HUUD HOH DFHOHUD j UD]mR GH m/s2´

Book 1.indb 10

04/04/2016 16:30:31

Capítulo 1

Fundamentos

11

2V PDWHPiWLFRV WDPEpP ID]HP D¿UPDo}HV ± TXH DFUHGLWDPRV VHUHP YHUGDGHLUDV ± VREUH PDWHPiWLFD 7DLV D¿UPDo}HV VH HQTXDGUDP HP WUrV FDWHJRULDV D¿UPDo}HV TXH VDEHPRV VHUHP YHUGDGHLUDV SRUTXH SRGHPRV SURYi ODV R TXH FKDPD mos teoremas D¿UPDo}HV FXMD YHUDFLGDGH QmR SRGHPRV JDUDQWLU R TXH GHQRPLQDPRV conjecturas D¿UPDo}HV IDOVDV R TXH DV FKDPDPRV erros. +i PDLV XPD FDWHJRULD GH D¿UPDo}HV PDWHPiWLFDV &RQVLGHUHPRV D VHQWHQoD ³D UDL] TXDGUDGD GH XP WULkQJXOR p XP FtUFXOR´ &RPR D RSHUDomR GH H[WUDomR GH XPD UDL] TXDGUDGD VH DSOLFD D Q~PHURV H QmR D ¿JXUDV JHRPpWULFDV D VHQWHQoD QmR WHP VHQWLGR 7DLV D¿UPDo}HV VmR absurdas! ^ĞŵƉƌĞ ǀĞƌŝĮƋƵĞ ƐĞƵƐ ƉƌſƉƌŝŽƐ ƚƌĂďĂůŚŽƐ ƋƵĂŶƚŽ Ă ƐĞŶƚĞŶĕĂƐ ƐĞŵ ƐĞŶƟĚŽ͘ ƐƐĞ ƟƉŽ ĚĞ ĞƌƌŽ Ġ ŵƵŝƚŽ ĐŽŵƵŵ͘ ŽŶƐŝĚĞƌĞ ĐĂĚĂ ƉĂůĂǀƌĂ Ğ ƐşŵďŽůŽ ƋƵĞ ĞƐĐƌĞǀĞƌ͘ WĞƌŐƵŶƚĞͲƐĞ Ă ƌĞƐƉĞŝƚŽ ĚŽ ƐŝŐŶŝĮĐĂĚŽ ĚĞ ĐĂĚĂ ƚĞƌŵŽ͕ Ğ ƐĞ ĂƐ ĞdžƉƌĞƐƐƁĞƐ ă ĞƐƋƵĞƌĚĂ Ğ ă ĚŝƌĞŝƚĂ Ğŵ ƐƵĂƐ ĞƋƵĂĕƁĞƐ ƌĞƉƌĞƐĞŶƚĂŵ ŽďũĞƚŽƐ ĚĞ ŵĞƐŵŽ ƟƉŽ͘

A natureza da verdade 'L]HU TXH XPD D¿UPDomR p verdadeira DVVHYHUD TXH D D¿UPDomR p FRUUHWD H PHUHFH FRQ ¿DQoD 0DV D QDWXUH]D GD YHUGDGH p PXLWR PDLV UtJLGD QD PDWHPiWLFD GR TXH HP TXDO TXHU RXWUD GLVFLSOLQD &RQVLGHUHPRV SRU H[HPSOR R VHJXLQWH IDWR PHWHRUROyJLFR EHP FRQKHFLGR ³HP MXOKR R WHPSR HP %DOWLPRUH p TXHQWH H ~PLGR´ 3RVVR DVVHJXUDU SHOD PLQKD H[SHULrQFLD SHVVRDO TXH HVVD D¿UPDomR p YHUGDGHLUD ,VVR VLJQL¿FD TXH WRGR GLD HP WRGR PrV GH MXOKR p TXHQWH H ~PLGR" 2EYLDPHQWH QmR 1mR p UD]RiYHO HVSHUDUPRV LQWHUSUHWDomR WmR UtJLGD GH XPD D¿UPDomR JHUDO VREUH R WHPSR &RQVLGHUHPRV D D¿UPDomR GR ItVLFR TXH DFDEDPRV GH DSUHVHQWDU ³4XDQGR GHL [DPRV FDLU XP REMHWR SUy[LPR j VXSHUItFLH GD 7HUUD HOH DFHOHUD j UD]mR GH P V2´ (VVD D¿UPDomR WDPEpP p YHUGDGHLUD H p H[SUHVVD FRP PDLRU SUHFLVmR GR TXH QRVVD DVVHUomR VREUH R FOLPD HP %DOWLPRUH 0DV HVVD OHL ItVLFD QmR p DEVROXWDPHQWH FRUUHWD 3ULPHLUR R YDORU p DSUR[LPDGR 6HJXQGR R WHUPR SUy[LPR p YDJR 'H XPD SHUV SHFWLYD JDOiFWLFD D OXD HVWi ³SUy[LPD´ GD 7HUUD PDV HVWH QmR p R VLJQL¿FDGR GH proximidade TXH WHPRV HP YLVWD 3RGHPRV DGPLWLU TXH SUy[LPR VLJQL¿TXH ³D PHQRV GH PHWURV GD VXSHUItFLH GD WHUUD´ PDV LVVR QRV GHL[D FRP XP SUREOHPD 0HVPR D XPD DOWLWXGH GH PHWURV D JUDYLGDGH p OLJHLUDPHQWH LQIHULRU j JUDYLGDGH QD VXSHUItFLH 3LRU DLQGD D JUDYLGDGH QD VXSHUItFLH GD WHUUD QmR p FRQVWDQWH D DWUDomR JUDYLWDFLRQDO QR FXPH GR 0RQWH (YHUHVW p OLJHLUDPHQWH PHQRU TXH DR QtYHO GR PDU $ GHVSHLWR GHVVDV YiULDV REMHo}HV H TXDOL¿FDo}HV D D¿UPDomR GH TXH RV REMHWRV OLEHUDGRV SUy[LPR j VXSHUItFLH GD WHUUD DFHOHUDP j UD]mR GH P V2 p YHUGDGHLUD &RPR HVWXGLRVRV GR FOLPD RX ItVLFRV FRQKHFHPRV DV OLPLWDo}HV GH QRVVD QRomR GH YHUGDGH 4XDVH WRGDV DV D¿UPDo}HV VmR OLPLWDGDV HP VHX REMHWLYR H VDEHPRV TXH VXD YHUGDGH QmR SRGH VHU FRQVLGHUDGD FRPR DEVROXWD H XQLYHUVDO 7RGDYLD HP PDWHPiWLFD D SDODYUD verdadeiro GHYH VHU FRQVLGHUDGD DEVROXWD LQFRQ GLFLRQDO H VHP H[FHomR &RQVLGHUHPRV XP H[HPSOR 7DOYH] R PDLV FpOHEUH WHRUHPD GD JHRPHWULD VHMD R VH JXLQWH UHVXOWDGR FOiVVLFR GH 3LWiJRUDV

Book 1.indb 11

04/04/2016 16:30:31

12

Matemática discreta

TEOREMA 4.1 (Pitagórico) 6H a e b VmR RV FRPSULPHQWRV GRV FDWHWRV GH XP WULkQJXOR UHWkQJXOR H c p R FRPSULPHQWR GD KLSRWHQXVD HQWmR a

c

a2 + b2 = c2.

b

$ UHODomR a2 + b2 = c2 YDOH SDUD RV FDWHWRV H D KLSRWHQXVD GH WRGR WULkQJXOR UHWkQJXOR GH PDQHLUD DEVROXWD H VHP H[FHomR 6DEHPRV LVVR SRUTXH SRGHPRV SURYDU HVVH WHRUHPD IDODUHPRV PDLV VREUH SURYDV DGLDQWH

2 WHRUHPD GH 3LWiJRUDV p QD YHUGDGH DEVROXWDPHQWH YHUGDGHLUR" 3RGHUtDPRV FRJLWDU VH WUDoiVVHPRV XP WULkQJXOR UHWkQJXOR HP XP SHGDoR GH SDSHO H PHGtVVHPRV RV FRPSUL PHQWRV GRV ODGRV D PHQRV GH XP ELOLRQpVLPR GH XPD SROHJDGD WHUtDPRV H[DWDPHQWH a2 + b2 = c2" 3URYDYHOPHQWH QmR SRUTXH R WUDoDGR GH XP WULkQJXOR UHWkQJXOR QmR p XP WULkQJXOR UHWkQJXOR 8P GHVHQKR RX WUDoDGR p XPD DMXGD YLVXDO SDUD HQWHQGHUPRV XP FRQFHLWR PDWHPiWLFR PDV XP GHVHQKR p DSHQDV WLQWD QR SDSHO 8P WULkQJXOR UHWkQJXOR ³UHDO´ H[LVWH apenas em nossas mentes.

(P FRQWUDSDUWLGD FRQVLGHUHPRV R VHJXLQWH HQXQFLDGR ³2V Q~PHURV SULPRV VmR tPSD UHV´ (OD p YHUGDGHLUD" 1mR 2 Q~PHUR p SULPR PDV QmR p tPSDU 3RUWDQWR D D¿UPDomR p IDOVD 3RGHUtDPRV GL]HU TXH HOD p TXDVH YHUGDGHLUD SRLV WRGRV RV Q~PHURV SULPRV H[FHWR VmR tPSDUHV 1D UHDOLGDGH Ki PXLWR PDLV H[FHo}HV j UHJUD ³RV GLDV GH MXOKR HP %DOWLPRUH VmR TXHQWHV H ~PLGRV´ XPD VHQWHQoD WLGD FRPR YHUGDGHLUD GR TXH j D¿UPDomR ³RV Q~PH URV SULPRV VmR tPSDUHV´ 2V PDWHPiWLFRV DGRWDUDP D FRQYHQomR GH TXH XPD D¿UPDomR p verdadeira desde TXH HOD VHMD DEVROXWDPHQWH YHUGDGHLUD VHP H[FHomR 8PD D¿UPDomR TXH QmR p DEVROX WDPHQWH YHUGDGHLUD QHVVH VHQWLGR HVWULWR p FKDPDGD IDOVD. 8P HQJHQKHLUR XP ItVLFR H XP PDWHPiWLFR HVWmR ID]HQGR XP SDVVHLR GH WUHP SHOD (VFyFLD H REVHUYDP XPDV RYHOKDV QHJUDV HP XPD FROLQD ³2OKH´ GL] R HQJHQKHLUR ³DV RYHOKDV QHVWD SDUWH GD (VFyFLD VmR QHJUDV ´ ³1D YHUGDGH´ UHVSRQGH R ItVLFR ³YRFr QmR GHYH WLUDU FRQFOXV}HV SUHFLSLWDGDV 7XGR R TXH SRGHPRV GL]HU p TXH QHVWD SDUWH GD (VFyFLD Ki DOJXPDV RYHOKDV QHJUDV´ ³%HP DR PHQRV GH XP ODGR´ GL] R PDWHPiWLFR

Se-então 2V PDWHPiWLFRV XVDP D OLQJXDJHP FRWLGLDQD GH PDQHLUD OLJHLUDPHQWH GLIHUHQWH GDV SHV VRDV HP JHUDO $WULEXtPRV D FHUWDV SDODYUDV VLJQL¿FDGRV HVSHFLDLV GLIHUHQWHV GR XVR SDGUmR 2V PDWHPiWLFRV WRPDP DV SDODYUDV GR LGLRPD H XVDP QDV FRPR WHUPRV WpFQLFRV $WULEXtPRV QRYR VHQWLGR D SDODYUDV FRPR FRQMXQWR JUXSR e JUDIR 7DPEpP FULDPRV QRVVDV SUySULDV SDODYUDV FRPR ELMHomR e SDUFLDOPHQWH RUGHQDGR WRGDV HVVDV SDODYUDV VHUmR GH¿QLGDV PDLV j IUHQWH

ŽŶƐŝĚĞƌĞ Ž ƵƐŽ ŵĂƚĞŵĄƟĐŽ Ğ ĐŽŵƵŵ ĚĂ ƉĂůĂǀƌĂ ƉƌŝŵŽ͘ YƵĂŶĚŽ Ƶŵ ĞĐŽŶŽŵŝƐƚĂ Ěŝnj ƋƵĞ Ă ƚĂdžĂ ĚĞ ũƵƌŽƐ ƉƌŝŵĂ Ġ ĂŐŽƌĂ ĚĞ ϴй͕ ŶĆŽ ĞƐƚĂŵŽƐ ƉƌĞŽĐƵƉĂĚŽƐ ƐĞ ϴ ŶĆŽ Ġ Ƶŵ ŶƷŵĞƌŽ ƉƌŝŵŽ͊

1yV PDWHPiWLFRV QmR DSHQDV WRPDPRV QRPHV H DGMHWLYRV H DWULEXtPRV D HOHV QRYR VLJQL¿FDGR PDV WDPEpP PRGL¿FDPRV VXWLOPHQWH R VHQWLGR GH SDODYUDV FRPXQV FRPR

Book 1.indb 12

04/04/2016 16:30:31

Capítulo 1

Fundamentos

13

ou SDUD DWHQGHU DRV QRVVRV SURSyVLWRV HVSHFt¿FRV (PERUD SRVVDPRV VHU FXOSDGRV GH YLRODU R XVR SDGUmR VRPRV SOHQDPHQWH FRQVLVWHQWHV QD PDQHLUD FRPR R ID]HPRV &KD mamos OLQJXDJHP PDWHPiWLFD PDWHPDWLTXrV HVVH XVR DOWHUDGR GD OLQJXDJHP SDGUmR H R H[HPSOR PDLV LPSRUWDQWH GLVWR p D FRQVWUXomR VH HQWmR $ JUDQGH PDLRULD GRV WHRUHPDV SRGH VHU H[SUHVVD QD IRUPD ³VH A HQWmR B´ 3RU H[HPSOR R WHRUHPD ³D VRPD GH GRLV Q~PHURV LQWHLURV SDUHV p SDU´ SRGH VHU UHIRUPXODGR FRPR ³VH x e y VmR LQWHLURV SDUHV HQWmR x + y WDPEpP p SDU´ EĂ ĂĮƌŵĂĕĆŽ ͞ƐĞ A͕ ĞŶƚĆŽ B͕͟ A Ġ ĐŚĂŵĂĚŽ hipótese, Ğ B͕ conclusão͘

1D FRQYHUVDomR FRWLGLDQD XPD D¿UPDomR GR WLSR ³VH HQWmR´ SRGH WHU YiULDV LQWHU SUHWDo}HV 3RU H[HPSOR SRVVR GL]HU j PLQKD ¿OKD ³VH YRFr FRUWDU D JUDPD HQWmR HX OKH SDJDUHL ´ 6H HOD ¿]HU R WUDEDOKR QDWXUDOPHQWH HVSHUDUi R SDJDPHQWR &HUWDPHQWH HOD QmR GLVFRUGDULD VH HX OKH GHVVH PHVPR TXH HOD QmR ¿]HVVH R WUDEDOKR PDV FHU WDPHQWH QmR R HVSHUDULD $SHQDV XPD FRQVHTXrQFLD p DVVHJXUDGD 7RGDYLD VH GLJR D PHX ¿OKR ³VH QmR FRPHU VHX IHLMmR YRFr QmR WHUi VREUHPHVD´ HOH HQWHQGH TXH D PHQRV TXH HOH FRPD WRGRV RV YHJHWDLV QmR KDYHUi GRFH 0DV HOH WDPEpP HQWHQGH TXH VH HOH FRPHU WRGRV RV YHJHWDLV WHUi D VREUHPHVD 1HVVH FDVR SURPHWHP VH GXDV FRQVHTXrQ FLDV XPD QR FDVR GH HOH FRPHU WRGRV RV YHJHWDLV H RXWUD HP FDVR QHJDWLYR 2 HPSUHJR PDWHPiWLFR GH ³VH HQWmR´ p HTXLYDOHQWH DR GH ³VH YRFr FRUWDU D JUDPD HX OKH SDJDUHL ´ $ D¿UPDomR ³6H A HQWmR B´ VLJQL¿FD VHPSUH TXH D FRQGLomR A for YHUGDGHLUD D FRQGLomR B WDPEpP R VHUi &RQVLGHUHPRV D VHQWHQoD ³VH x e y VmR SDUHV HQWmR x + y p SDU´ 7XGR R TXH HVVD VHQWHQoD DVVHJXUD p TXH TXDQGR x e y VmR DPERV SDUHV x + y WDPEpP R p SDU D VHQWHQoD QmR H[FOXL D SRVVLELOLGDGH GH x + y ser par a despeito de x RX y QmR R VHUHP QD YHUGDGH VH x e y VmR DPERV tPSDUHV VDEHPRV TXH x + y WDPEpP p SDU

1D D¿UPDomR ³VH A HQWmR B´ SRGHPRV WHU D FRQGLomR A YHUGDGHLUD RX IDOVD H D FRQGLomR B YHUGDGHLUD RX IDOVD 5HVXPDPRV HVVHV IDWRV HP XP TXDGUR 6H D D¿UPDomR ³VH A HQWmR B´ p YHUGDGHLUD WHPRV R VHJXLQWH Condição A

Condição B

Verdadeira

Verdadeira

3RVVtYHO

Verdadeira

Falsa

,PSRVVtYHO

Falsa

Verdadeira

3RVVtYHO

Falsa

Falsa

3RVVtYHO

7XGR R TXH VH D¿UPD p TXH VHPSUH TXH A IRU YHUGDGHLUD B GHYH Vr OD WDPEpP 6H A QmR p YHUGDGHLUD HQWmR QHQKXPD DOHJDomR VREUH B p VXVWHQWDGD SRU ³VH A HQWmR B´ (LV XP H[HPSOR ,PDJLQH TXH HX VHMD XP SROtWLFR FRQFRUUHQGR D XP FDUJR HOHWLYR H DQXQFLH HP S~EOLFR ³VH IRU HOHLWR GLPLQXLUHL RV LPSRVWRV´ (P TXH FRQGLo}HV SRVVR VHU FRQVLGHUDGR XP PHQWLURVR" 6XSRQKD TXH HX VHMD HOHLWR H UHGX]D RV LPSRVWRV &HUWDPHQWH QmR VHUHL FKDPDGR GH PHQWLURVR ± PDQWLYH PLQKD SURPHVVD 6XSRQKD TXH HX VHMD HOHLWR H QmR UHGX]D RV LPSRVWRV 2 FLGDGmR WHUi WRGR GLUHLWR GH FKDPDU PH PHQWLURVR ± QmR FXPSUL PLQKD SURPHVVD 6XSRQKD DJRUD TXH HX QmR VHMD HOHLWR PDV PHGLDQWH XP OREE\ FRQVLJD ID]HU TXH RV LPSRVWRV VHMDP UHGX]LGRV 2 SRYR FHUWDPHQWH QmR PH FKDPDUi GH PHQWLURVR ± QmR TXHEUHL PLQKD SURPHVVD

Book 1.indb 13

04/04/2016 16:30:31

14

Matemática discreta

3RU ¿P VXSRQKD TXH HX QmR VHMD HOHLWR H RV LPSRVWRV QmR VHMDP UHGX]LGRV 1RYD PHQWH HX QmR SRGHULD VHU DFXVDGR GH PHQWLU ± SURPHWL UHGX]LU RV LPSRVWRV DSHQDV VH HX IRVVH HOHLWR $ ~QLFD FLUFXQVWkQFLD HP TXH ³VH IRU HOHLWR A HQWmR B EDL[DUHL RV LPSRVWRV´ QmR p YHUGDGHLUD p VH A IRU YHUGDGHLUD H B falsa.

(P UHVXPR D D¿UPDomR ³VH $ HQWmR B´ DVVHJXUD TXH D FRQGLomR B p YHUGDGHLUD VHPSUH TXH A R IRU PDV QmR ID] TXDOTXHU UHIHUrQFLD D B TXDQGR A for falsa. &ƌĂƐĞĂĚŽƐ ĂůƚĞƌŶĂƟǀŽƐ ƉĂƌĂ ͞ƐĞ A, ĞŶƚĆŽ B͘͟

$V D¿UPDo}HV GR WLSR ³VH HQWmR´ SHUPHLDP WRGD D PDWHPiWLFD 6HULD FDQVDWLYR HP SUHJDU DV PHVPDV IUDVHV UHSHWLGDPHQWH QD HVFULWD PDWHPiWLFD &RQVHTXHQWHPHQWH Ki XPD GLYHUVLGDGH GH PDQHLUDV DOWHUQDWLYDV SDUD H[SUHVVDU ³VH A HQWmR B´ 7RGDV DV IUDVHV TXH VHJXHP H[SUHVVDP H[DWDPHQWH D PHVPD D¿UPDomR TXH ³VH A HQWmR B´ ³A implica B´ 1D YR] SDVVLYD SRGH H[SUHVVDU VH FRPR ³B p LPSOLFDGR SRU A´ ³6HPSUH TXH A WHPRV B´ 7DPEpP ³B VHPSUH TXH A´ ³A p VX¿FLHQWH SDUD B´ 7DPEpP ³A p FRQGLomR VX¿FLHQWH SDUD B´ (VWH p XP H[HPSOR GH OLQJXDJHP PDWHPiWLFD $ SDODYUD VX¿FLHQWH SRGH WHU QD OLQ JXDJHP FRUUHQWH D FRQRWDomR GH ³DSHQDV VX¿FLHQWH´ $TXL QmR VH DGPLWH WDO FRQR WDomR 2 VLJQL¿FDGR p ³GHVGH TXH A VHMD YHUGDGHLUR HQWmR B WDPEpP GHYH Vr OR´ ³3DUD TXH B VHMD YHUGDGHLUR p VX¿FLHQWH TXH WHQKDPRV A´ ³B p QHFHVViULR SDUD A´ (VWH p RXWUR H[HPSOR GH OLQJXDJHP PDWHPiWLFD $ PDQHLUD FRPR GHYHPRV HQWHQGHU HVVH IUDVHDGR p SDUD TXH A VHMD YHUGDGHLUR p necessário TXH B WDPEpP VHMD YHUGDGHLUR ³A VRPHQWH VH B ´ 2 VLJQL¿FDGR p TXH A pode ocorrer somente se B WDPEpP RFRUUHU ³A B ´ 2 VtPEROR HVSHFLDO Or VH ³LPSOLFD´ ³B $ ´ 2 VtPEROR Or VH ³p LPSOLFDGR SRU´

Se e somente se $ JUDQGH PDLRULD GRV WHRUHPDV p RX SRGH VHU IDFLOPHQWH H[SUHVVD QD IRUPD VH HQWmR $OJXQV WHRUHPDV YmR XP SDVVR DGLDQWH VmR GD IRUPD ³VH A HQWmR B H VH B HQWmR A´ 3RU H[HPSOR VDEHPRV TXH p YHUGDGHLUD D D¿UPDomR 6H XP LQWHLUR x p SDU HQWmR x p tPSDU H VH x p tPSDU HQWmR x p SDU (VVD D¿UPDomR p SUROL[D +i PDQHLUDV FRQFLVDV GH H[SUHVVDU D¿UPDo}HV GD IRUPD ³A implica B H B implica A´ QDV TXDLV QmR SUHFLVDPRV HVFUHYHU DV FRQGLo}HV A e B GXDV YH]HV FDGD XPD $ H[SUHVVmR FKDYH p se e somente se $ D¿UPDomR ³VH A HQWmR B H VH B HQWmR A´ SRGH UHHVFUHYHU VH FRPR ³A se e somente se B´ 2 H[HPSOR GDGR VH HVFUHYH PDLV DGHTXDGDPHQWH FRPR VHJXH 8P LQWHLUR [ p SDU VH H VRPHQWH VH [ IRU tPSDU 2 TXH VLJQL¿FD XPD D¿UPDomR GR WLSR ³VH H VRPHQWH VH´" &RQVLGHUHPRV D D¿UPD omR ³A se e somente se B´ $V FRQGLo}HV A e B SRGHP VHU FDGD XPD GHODV YHUGDGHLUD

Book 1.indb 14

04/04/2016 16:30:31

Capítulo 1

Fundamentos

15

RX IDOVD KDYHQGR DVVLP TXDWUR SRVVLELOLGDGHV TXH SRGHPRV UHVXPLU HP XP TXDGUR 6H D D¿UPDomR ³A se e somente se B´ p YHUGDGHLUD WHPRV R VHJXLQWH Condição A

Condição B

Verdadeira

Verdadeira

3RVVtYHO

Verdadeira

Falsa

,PSRVVtYHO

Falsa

Verdadeira

,PSRVVtYHO

Falsa

Falsa

3RVVtYHO

e LPSRVVtYHO D FRQGLomR A VHU YHUGDGHLUD TXDQGR B p IDOVD SRUTXH A B. Da mesma IRUPD p LPSRVVtYHO D FRQGLomR B VHU YHUGDGHLUD TXDQGR A p IDOVD SRUTXH B A. As VLP DV GXDV FRQGLo}HV A e B GHYHP VHU DPEDV YHUGDGHLUDV RX DPEDV IDOVDV 9ROWHPRV j D¿UPDomR 8P LQWHLUR [ p SDU VH H VRPHQWH [ IRU tPSDU $ FRQGLomR A p ³x p SDU´ H D FRQGLomR B p ³x p tPSDU´ 3DUD DOJXQV LQWHLURV SRU H[HPSOR x A e B VmR DPEDV YHUGDGHLUDV p SDU H p tPSDU PDV SDUD RXWURV LQ WHLURV SRU H[HPSOR x DPEDV DV FRQGLo}HV VmR IDOVDV QmR p SDU H QmR p tPSDU

&ƌĂƐĞĂĚŽƐ ĂůƚĞƌŶĂƟǀŽƐ ƉĂƌĂ ͞ ƐĞ Ğ ƐŽŵĞŶƚĞ ƐĞ ͘͟

$VVLP FRPR Ki YiULDV PDQHLUDV GH H[SUHVVDU XPD D¿UPDomR GR WLSR ³VH HQWmR´ Ki WDPEpP YiULDV IRUPDV GH H[SUHVVDU XPD D¿UPDomR GR WLSR ³VH H VRPHQWH VH´ ³A sse B´ &RPR D H[SUHVVmR ³VH H VRPHQWH VH´ RFRUUH FRP IUHTXrQFLD D DEUHYLDWXUD ³VVH´ p EDVWDQWH XVDGD ³A p QHFHVViULR H VX¿FLHQWH SDUD B´ ³A p HTXLYDOHQWH D B ´ $ UD]mR SDUD R HPSUHJR GD SDODYUD HTXLYDOHQWH p TXH D FRQGLomR A p YiOLGD H[DWDPHQ WH QDV PHVPDV FLUFXQVWkQFLDV VRE DV TXDLV HP TXH D FRQGLomR B VH PDQWpP ³$ p YHUGDGH H[DWDPHQWH TXDQGR % p YHUGDGH´ 2 WHUPR exatamente VLJQL¿FD TXH DV FLUFXQVWkQFLDV SDUD TXH D FRQGLomR A VHMD YHU GDGH VmR SUHFLVDPHQWH DV PHVPDV FLUFXQVWkQFLDV SDUD TXH D FRQGLomR B VHMD YHUGDGH ³A B´ 2 VtPEROR p XP DPiOJDPD GRV VtPERORV e .

E, ou e não hƐŽ ŵĂƚĞŵĄƟĐŽ ĚĞ e.

2V PDWHPiWLFRV XWLOL]DP DV SDODYUDV e ou e QmR HP VHQWLGRV PXLWR SUHFLVRV 2 XVR PD temático de e e QmR p HVVHQFLDOPHQWH R PHVPR TXH QD OLQJXDJHP FRWLGLDQD 2 HPSUHJR de ou p PDLV LGLRVVLQFUiWLFR hƐŽ ŵĂƚĞŵĄƟĐŽ ĚĞ não.

Book 1.indb 15

04/04/2016 16:30:31

16

MatemĂĄtica discreta

$ D¿UPDomR ³A e B´ VLJQL¿FD TXH DPEDV DV D¿UPDo}HV A e B VmR YHUGDGHLUDV 3RU H[HPSOR ³7RGR LQWHLUR FXMR DOJDULVPR GDV XQLGDGHV p p GLYLVtYHO SRU e SRU ´ ,VVR VLJQL¿FD TXH XP Q~PHUR TXH WHUPLQD HP ]HUR WDO FRPR p GLYLVtYHO WDQWR SRU FRPR SRU 2 HPSUHJR GH e SRGH VHU UHVXPLGR QD WDEHOD D VHJXLU A

B

AeB

Verdadeira

Verdadeira

Verdadeira

Verdadeira

Falsa

Falsa

Falsa

Verdadeira

Falsa

Falsa

Falsa

Falsa

$ D¿UPDomR ³QmR A´ p YHUGDGHLUD VH H VRPHQWH VH A p IDOVD 3RU H[HPSOR D D¿UPD omR ³7RGRV RV SULPRV VmR tPSDUHV´ p IDOVD $VVLP D D¿UPDomR ³1HP WRGRV RV SULPRV VmR tPSDUHV´ p YHUGDGHLUD 1RYDPHQWH SRGHPRV UHVXPLU R XVR GH QmR HP XPD WDEHOD A

nĂŁo A

Verdadeira

Falsa

Falsa

Verdadeira

hĆ?Ĺ˝ ĹľÄ&#x201A;Ć&#x161;Ä&#x17E;ĹľÄ&#x201E;Ć&#x;Ä?Ĺ˝ Ä&#x161;Ä&#x17E; ou.

&RQVHTXHQWHPHQWH R XVR PDWHPiWLFR GH e e QmR FRUUHVSRQGH PXLWR DSUR[LPDGD PHQWH DR XVR FRUUHQWH 2 PHVPR QmR DFRQWHFH FRP R XVR GH ou 1D OLQJXDJHP SDGUmR ou HP JHUDO VXJHUH HVFROKD GH XPD RSomR RX RXWUD PDV QmR GH DPEDV &RQVLGHUHPRV D SHUJXQWD ÂłKRMH TXDQGR VDLUPRV SDUD MDQWDU JRVWDULD GH SL]]D RX FRPLGD FKLQHVD"´ $ LPSOLFDomR p FRPHUHPRV XP RX RXWUR SUDWR PDV QmR DPERV (P FRQWUDSRVLomR R ou PDWHPiWLFR DGPLWH D SRVVLELOLGDGH GH ambos $ DÂżUPDomR ÂłA RX B´ VLJQLÂżFD TXH A p YHUGDGHLUR RX B p YHUGDGHLUR RX DPERV A e B VmR YHUGD GHLURV 3RU H[HPSOR FRQVLGHUHPRV R VHJXLQWH 6XSRQKDPRV [ H \ LQWHLURV FRP D SURSULHGDGH [_\ H \_[ (QWmR [ \ RX [ Âą\ $ FRQFOXVmR GHVVH UHVXOWDGR QRV GL] TXH SRGHPRV WHU XP GRV VHJXLQWHV FDVRV Â&#x192; x = y PDV QmR x Âąy SRU H[HPSOR x = 3 e y

Â&#x192; x Âąy PDV QmR x = y SRU H[HPSOR x Âą H y

Â&#x192; x = y e x = â&#x20AC;&#x201C;y R TXH p SRVVtYHO DSHQDV VH x = 0 e y = 0.

(LV XPD WDEHOD SDUD DÂżUPDo}HV ou A

B

A ou B

Verdadeira

Verdadeira

Verdadeira

Verdadeira

Falsa

Verdadeira

Falsa

Verdadeira

Verdadeira

Falsa

Falsa

Falsa

Designaçþes para um teorema $OJXQV WHRUHPDV VmR PDLV LPSRUWDQWHV RX PDLV LQWHUHVVDQWHV TXH RXWURV +i GHVLJQD o}HV DOWHUQDWLYDV TXH RV PDWHPiWLFRV XVDP HP OXJDU GH teorema &DGD XPD WHP XPD

Book 1.indb 16

04/04/2016 16:30:31

Capítulo 1

Fundamentos

17

FRQRWDomR OLJHLUDPHQWH GLIHUHQWH $ SDODYUD teorema WHP D FRQRWDomR GH LPSRUWkQFLD H JHQHUDOLGDGH 2 WHRUHPD GH 3LWiJRUDV FHUWDPHQWH PHUHFH VHU FKDPDGR XP teorema. $ D¿UPDomR ³R TXDGUDGR GH XP LQWHLUR SDU WDPEpP p SDU´ WDPEpP p XP WHRUHPD PDV WDOYH] QmR PHUHoD XPD GHVLJQDomR WmR SURIXQGD ( D D¿UPDomR ³ ´ p WHFQLFD PHQWH XP WHRUHPD PDV QmR MXVWL¿FD XPD GHVLJQDomR WmR SUHVWLJLRVD ƉĂůĂǀƌĂ teorema ŶĆŽ ĚĞǀĞ ƐĞƌ ĐŽŶĨƵŶĚŝĚĂ ĐŽŵ ƚĞŽƌŝĂ͘ hŵ teorema Ġ ƵŵĂ ĂĮƌŵĂĕĆŽ ĞƐƉĞĐşĮĐĂ ƋƵĞ ƉŽĚĞ ƐĞƌ ƉƌŽǀĂĚĂ͘ hŵĂ teoria Ġ Ƶŵ ĐŽŶũƵŶƚŽ ŵĂŝƐ ĂŵƉůŽ ĚĞ ŝĚĞŝĂƐ ƐŽďƌĞ Ƶŵ ƉƌŽďůĞŵĂ Ğŵ ƉĂƌƟĐƵůĂƌ͘

$ VHJXLU OLVWDPRV SDODYUDV TXH FRQVWLWXHP DOWHUQDWLYDV D teorema H RIHUHFHPRV XPD RULHQWDomR SDUD VHX XVR Resultado. 8PD H[SUHVVmR PRGHVWD JHQpULFD SDUD XP WHRUHPD +i XP DU GH KXPLO GDGH DR FKDPDUPRV XP WHRUHPD VLPSOHVPHQWH GH ³UHVXOWDGR´ 7DQWR WHRUHPDV LPSRU WDQWHV FRPR WHRUHPDV VHP LPSRUWkQFLD SRGHP VHU FKDPDGRV UHVXOWDGRV Fato. 8P WHRUHPD GH LPSRUWkQFLD EDVWDQWH OLPLWDGD $ D¿UPDomR ³ ´ p XP fato.

Proposição. 8P WHRUHPD GH LPSRUWkQFLD VHFXQGiULD 8PD SURSRVLomR p PDLV LP SRUWDQWH RX PDLV JHUDO GR TXH XP IDWR PDV QmR WHP WDQWR SUHVWtJLR TXDQWR XP WHRUHPD Lema. 8P WHRUHPD FXMR REMHWLYR SULQFLSDO p DMXGDU D SURYDU RXWUR WHRUHPD PDLV LPSRUWDQWH $OJXQV WHRUHPDV H[LJHP GHPRQVWUDo}HV FRPSOLFDGDV )UHTXHQWHPHQWH SR GHPRV GHFRPSRU HP SDUWHV PHQRUHV R WUDEDOKR GH SURYDU XP WHRUHPD FRPSOLFDGR 2V OHPDV VmR DV SDUWHV RX LQVWUXPHQWRV XVDGRV SDUD HODERUDU XPD SURYD PDLV FRPSOLFDGD Corolário. 5HVXOWDGR FRP XPD SURYD UiSLGD FXMR SDVVR SULQFLSDO p R XVR GH RXWUR WHRUHPD SURYDGR DQWHULRUPHQWH Alegação. $QiORJR D OHPD 8PD DOHJDomR p XP WHRUHPD FXMD D¿UPDomR HP JHUDO DSDUHFH QD SURYD GH XP WHRUHPD 2 REMHWLYR GH XPD DOHJDomR p DMXGDU D RUJDQL]DU RV SDVVRV FKDYH GH XPD SURYD 7DPEpP D IRUPXODomR GH XPD DOHJDomR SRGH HQYROYHU WHUPRV TXH WrP VHQWLGR DSHQDV QR FRQWH[WR GD SURYD

Afirmação verdadeira por vacuidade 2 TXH GHYHPRV SHQVDU GH XPD D¿UPDomR GR WLSR ³VH HQWmR´ HP TXH D KLSyWHVH p LPSRV VtYHO" &RQVLGHUHPRV R VHJXLQWH

AFIRMAÇÃO 4.2 (Vazia) 6H XP LQWHLUR p VLPXOWDQHDPHQWH XP TXDGUDGR SHUIHLWR H SULPR HQWmR p QHJDWLYR (VWD D¿UPDomR p YHUGDGHLUD RX IDOVD" $ D¿UPDomR QmR p VHP VHQWLGR 2V WHUPRV quadrado perfeito FRQVXOWH R ([HUFtFLR

primo e negativo DSOLFDP VH DGHTXDGDPHQWH D LQWHLURV 3RGHUtDPRV VHU WHQWDGRV D GL]HU TXH D D¿UPDomR p IDOVD SRUTXH RV Q~PHURV TXDGUD GRV H RV Q~PHURV SULPRV QmR SRGHP VHU QHJDWLYRV (QWUHWDQWR SDUD TXH XPD D¿UPDomR GD IRUPD ³VH A HQWmR B´ VHMD GHFODUDGD IDOVD GHYHPRV HQFRQWUDU XPD VLWXDomR HP TXH D FOiXVXOD A VHMD YHUGDGHLUD H D FOiXVXOD B VHMD IDOVD 1R FDVR GD $¿UPDomR D FRQ GLomR A p LPSRVVtYHO QmR Ki Q~PHUR TXH VHMD VLPXOWDQHDPHQWH XP TXDGUDGR SHUIHLWR H

Book 1.indb 17

04/04/2016 16:30:31

18

Matemática discreta

SULPR $VVLP QXQFD SRGHUHPRV DFKDU XP LQWHLUR TXH WRUQH D FRQGLomR A YHUGDGHLUD H D FRQGLomR B IDOVD 3RU FRQVHJXLQWH D $¿UPDomR p YHUGDGHLUD $¿UPDo}HV GD IRUPD ³VH A HQWmR B´ HP TXH D FRQGLomR A p LPSRVVtYHO VmR FKD madas vazias H RV PDWHPiWLFRV FRQVLGHUDP YHUGDGHLUDV WDLV D¿UPDo}HV SRUTXH HODV QmR DGPLWHP H[FHo}HV

Recapitulando 1HVWD VHomR IRL LQWURGX]LGD D QRomR GH teorema XPD D¿UPDomR GHFODUDWLYD VREUH PD WHPiWLFD TXH DGPLWH XPD SURYD 'LVFXWLPRV D QDWXUH]D DEVROXWD GD SDODYUD verdadeiro HP PDWH PiWLFD 'LVFXWLPRV H[WHQVDPHQWH DV IRUPDV ³VH HQWmR´ H ³VH H VRPHQWH VH´ GH WHRUHPDV DVVLP FRPR XPD OLQJXDJHP DOWHUQDWLYD SDUD H[SUHVVDU WDLV UHVXOWDGRV ([SOL FDPRV D PDQHLUD FRPR RV PDWHPiWLFRV XWLOL]DP DV SDODYUDV e ou e QmR. Apresentamos YiULRV VLQ{QLPRV GH teorema H H[SOLFDPRV VXDV FRQRWDo}HV 3RU ¿P GLVFXWLPRV D¿U PDo}HV ³VH HQWmR´ YD]LDV H QRWDPRV TXH RV PDWHPiWLFRV FRQVLGHUDP WDLV D¿UPDo}HV YHUGDGHLUDV

4 Exercícios 4.1. &DGD XPD GDV D¿UPDo}HV VHJXLQWHV SRGH VHU IRUPXODGD QD IRUPD ³VH HQWmR´ 5HHVFUHYD FDGD XPD GDV VHQWHQoDV VHJXLQWHV QD IRUPD ³VH A HQWmR B´ a. 2 SURGXWR GH XP LQWHLUR tPSDU H XP LQWHLUR SDU p SDU b. 2 TXDGUDGR GH XP LQWHLUR tPSDU p tPSDU c. 2 TXDGUDGR GH XP Q~PHUR SULPR QmR p SULPR d. 2 SURGXWR GH GRLV LQWHLURV QHJDWLYRV p QHJDWLYR 1DWXUDOPHQWH LVVR p IDOVR

e. $V GLDJRQDLV GH XP ORVDQJR VmR SHUSHQGLFXODUHV f. 7ULkQJXORV FRQJUXHQWHV WrP D PHVPD iUHD g. $ VRPD GH WUrV LQWHLURV FRQVHFXWLYRV p GLYLVtYHO SRU WUrV $EDL[R YRFr HQFRQWUDUi SDUHV GH D¿UPDo}HV $ H % 3DUD FDGD SDU LQGLTXH TXDLV GDV WUrV IUDVHV VHJXLQWHV VmR YHUGDGHLUDV H TXDLV VmR IDOVDV

6H A HQWmR B.

6H B HQWmR A.

A se e somente se B. 1RWD 9RFr QmR SUHFLVD SURYDU VXDV D¿UPDo}HV a. A 2 SROtJRQR 3456 p XP UHWkQJXOR B 2 SROtJRQR 3456 p XP TXDGUDGR b. A 2 SROtJRQR 3456 p XP UHWkQJXOR B 2 SROtJRQR 3456 p XP SDUDOHORJUDPR c. A -RH p DY{ B -RH p GR VH[R PDVFXOLQR d. A (OOHQ UHVLGH HP /RV $QJHOHV B (OOHQ UHVLGH QD &DOLIyUQLD e. A (VWH DQR p GLYLVtYHO SRU B (VWH p XP DQR ELVVH[WR f. A $V OLQKDV ᐉ1 e ᐉ2 VmR SDUDOHODV B $V OLQKDV ᐉ1 e ᐉ2 VmR SHUSHQGLFXODUHV 3DUD RV GHPDLV LWHQV [ H \ UHIHUHP VH D Q~PHURV UHDLV g. A x ! B x2 ! h. A x B x3 i. A [\ 0. B x RX y = 0. j. A xy = 0. B x = 0 e y = 0. k. A x + y = 0. B x = 0 e y = 0. ĂĮƌŵĂĕĆŽ ͞ƐĞ B͕ ĞŶƚĆŽ A͟ Ġ ĚŝƚĂ inversa ĚĂ ĂĮƌŵĂĕĆŽ ͞ƐĞ A͕ ĞŶƚĆŽ B͘͟

Book 1.indb 18

04/04/2016 16:30:31

Capítulo 1

Fundamentos

19

4.3. e XP HUUR FRPXP FRQIXQGLU DV GXDV D¿UPDo}HV VHJXLQWHV a. 6H A HQWmR % b. 6H B HQWmR A.

(QFRQWUH GXDV FRQGLo}HV A e B GH PRGR TXH D D¿UPDomR D VHMD YHUGDGHLUD PDV D D¿UPDomR E VHMD IDOVD 4.4. &RQVLGHUH DV GXDV D¿UPDo}HV a. 6H A HQWmR B. b. QmR A RX B.

(P TXH FLUFXQVWkQFLD HVVDV D¿UPDo}HV VmR YHUGDGHLUDV" (P TXH FLUFXQVWkQFLDV HODV VmR IDOVDV" ([SOLTXH SRU TXH HVVDV D¿UPDo}HV VmR HP HVVrQFLD LGrQWLFDV 4.5. &RQVLGHUH DV GXDV D¿UPDo}HV a. 6H $ HQWmR B. b. 6H QmR B HQWmR QmR A

ĂĮƌŵĂĕĆŽ ͞ƐĞ ;ŶĆŽ Ϳ͕ ĞŶƚĆŽ ;ŶĆŽ Ϳ͟ Ġ ĚŝƚĂ ĐŽŶƚƌĂƉŽƐŝƟǀĂ ă ĂĮƌŵĂĕĆŽ ͞^Ğ ͕ ĞŶƚĆŽ ͘͟

(P TXH FLUFXQVWkQFLDV HVVDV D¿UPDo}HV VmR YHUGDGHLUDV" 4XDQGR VmR IDOVDV" ([SOLTXH SRU TXH HVVDV D¿UPDo}HV VmR HP HVVrQFLD LGrQWLFDV

4.6. &RQVLGHUH DV GXDV D¿UPDo}HV a. A se e somente se B. b. QmR A VH H VRPHQWH VH QmR %

6RE TXH FLUFXQVWkQFLDV HVVDV D¿UPDo}HV VmR YHUGDGHLUDV" 4XDQGR VmR IDOVDV" ([SOLTXH SRU TXH HVVDV D¿UPDo}HV VmR HVVHQFLDOPHQWH LGrQWLFDV 4.7. &RQVLGHUH XP WULkQJXOR HTXLOiWHUR FXMRV ODGRV WrP FRPSULPHQWRV a = b = c 1RWH TXH QHVVH FDVR a2 + b2 z c2 ([SOLTXH SRU TXH LVVR QmR FRQVWLWXL XPD YLRODomR GR WHRUHPD GH 3LWiJRUDV 4.8. ([SOLTXH FRPR WUDoDU QD VXSHUItFLH GH XPD HVIHUD XP WULkQJXOR TXH WHQKD WUrV kQJXORV UHWRV 2V FDWHWRV H D KLSRWHQXVD GH GHWHUPLQDGR WULkQJXOR VDWLVID]HP D FRQGLomR a2 + b2 = c2" ([SOLTXH SRU TXH QmR VH WUDWD GH XPD YLRODomR GR WHRUHPD GH 3LWiJRUDV hŵ ůĂĚŽ ĚĞ Ƶŵ ƚƌŝąŶŐƵůŽ ĞƐĨĠƌŝĐŽ Ġ Ƶŵ ĂƌĐŽ ĚĞ Ƶŵ ĐşƌĐƵůŽ ĚĂ ĞƐĨĞƌĂ ƐŽďƌĞ Ă ƋƵĂů ĞůĞ ĞƐƚĄ ĚĞƐĞŶŚĂĚŽ͘ 4.9. &RQVLGHUH D IUDVH ³XPD OLQKD p D GLVWkQFLD PDLV FXUWD HQWUH GRLV SRQWRV´ )DODQGR HVWULWDPHQWH HVVD IUDVH QmR ID] VHQWLGR

(QFRQWUH GRLV HUURV QHVVD IUDVH H D UHHVFUHYD DSURSULDGDPHQWH

4.10. &RQVLGHUH D VHJXLQWH D¿UPDomR XP WDQWR TXDQWR HVWUDQKD ³VH XP SRUTXLQKR GD tQGLD IRU SHJR SHOR UDER VHXV ROKRV VDOWDUmR SDUD IRUD´ ,VVR p YHUGDGH" 4.11 0DLV VREUH FRQMHFWXUDV 'H RQGH YrP RV QRYRV WHRUHPDV" (OHV VmR FULDo}HV GH PDWHPiWLFRV FRPHoDUDP FRPR FRQMHFWXUDV GHFODUDo}HV PDWHPiWLFDV FXMD YHUGDGH DLQGD VH GHYH HVWDEHOHFHU (P RXWUDV SDODYUDV FRQMHFWXUDV VmR VXSRVLo}HV JHUDOPHQWH SDOSLWHV $R ROKDU SDUD PXLWRV H[HPSORV H SURFXUDU SRU SDGU}HV RV PDWHPiWLFRV H[SUHVVDP VXDV REVHUYDo}HV FRPR GHFODUDo}HV TXH HVSHUDP SURYDU 2V VHJXLQWHV LWHQV VmR SURMHWDGRV SDUD OHYDU YRFr DWUDYpV GR SURFHVVR GH ID]HU FRQMHFWXUDV (P FDGD FDVR WHVWDU YiULRV H[HPSORV H WHQWDU IRUPXODU VXDV REVHUYDo}HV FRPR XP WHRUHPD D VHU SURYDGR 9RFr QmR WHP TXH SURYDU HVVDV D¿UPDo}HV SRU RUD TXHUHPRV VLPSOHVPHQWH TXH YRFr H[SUHVVH R TXH HQFRQWUDU HP OLQJXDJHP PDWHPiWLFD a. 2 TXH YRFr SRGH GL]HU VREUH D VRPD GH Q~PHURV tPSDUHV FRQVHFXWLYRV FRPHoDQGR FRP " 2X VHMD DYDOLH H DVVLP SRU GLDQWH H IRUPXOH XPD FRQMHFWXUD

Book 1.indb 19

04/04/2016 16:30:32

20

Matemática discreta

b. 2 TXH YRFr SRGH GL]HU VREUH D VRPD GRV FXERV SHUIHLWRV FRQVHFXWLYRV FRPHoDQGR FRP ,VWR p R TXH YRFr SRGH GL]HU VREUH 3 3 + 33 3 + 33 + 53 3 + 33 + 53 + 73 H DVVLP por diante. c. 6HMD n XP LQWHLUR SRVLWLYR 'HVHQKH n OLQKDV GXDV GDV TXDLV VmR SDUDOHODV QR SODQR 4XDQWDV UHJL}HV VmR IRUPDGDV" d. &RORTXH n SRQWRV XQLIRUPHPHQWH HP WRUQR GH XP FtUFXOR $ SDUWLU GH XP SRQWR GHVHQKH XP FDPLQKR SDUD WRGRV RV RXWURV DR UHGRU GR FtUFXOR DWp YROWDU DR FRPHoR (P DOJXQV FDVRV FDGD SRQWR p YLVLWDGR H HP RXWURV DOJXQV VmR SHUGLGRV (P TXDLV FLUFXQVWkQFLDV WRGRV RV SRQWRV VmR YLVLWDGRV FRPR QD ¿JXUD FRP n " 6XSRQKD TXH HP YH] GH VDOWDU SDUD FDGD VHJXQGR SRQWR VDOWHPRV SDUD FDGD WHUFHLUR SRQWR 3DUD TXDLV YDORUHV GH n R FDPLQKR WRFD FDGD SRQWR" )LQDOPHQWH VXSRQKD TXH YLVLWHPRV WRGRV RV k pVLPRV SRQWRV HP TXH k está entre 1 e n

4XDQGR R FDPLQKR WRFD FDGD SRQWR" e. 8PD HVFROD WHP XP ORQJR FRUUHGRU GH DUPiULRV QXPHUDGRV H DVVLP SRU GLDQWH DWp 1HVWH SUREOHPD UHIHULUHPR QRV D YLUDU XP DUPiULR VLJQL¿FDQGR DEULU XP TXH HVWHMD IHFKDGR RX IHFKDU XP TXH HVWHMD DEHUWR 2X VHMD YLUDU XP DUPiULR p PXGDU VHX HVWDGR IHFKDGR DEHUWR (VWXGDQWH Qo1 caminha pelo corredor e fecha todos os armários.

(VWXGDQWH Qo FDPLQKD SHOR FRUUHGRU H YLUD WRGRV RV DUPiULRV FRP Q~PHURV SDUHV 3RUWDQWR DJRUD RV DUPiULRV tPSDUHV HVWmR IHFKDGRV H RV DUPiULRV SDUHV HVWmR DEHUWRV (VWXGDQWH Qo FDPLQKD SHOR FRUUHGRU H YLUD WRGRV RV DUPiULRV TXH VmR GLYLVtYHLV SRU (VWXGDQWH Qo FDPLQKD SHOR FRUUHGRU H YLUD WRGRV RV DUPiULRV TXH VmR GLYLVtYHLV SRU TXDWUR 'D PHVPD IRUPD RV DOXQRV H DVVLP SRU GLDQWH FDPLQKDP SHOR FRUUHGRU FDGD XP YLUDQGR RV DUPiULRV GLYLVtYHLV SRU VHX SUySULR Q~PHUR DWp TXH ¿QDOPHQWH R HVWXGDQWH YLUD R SULPHLUR H ~QLFR DUPiULR GLYLVtYHO SRU R ~OWLPR DUPiULR

4XDLV DUPiULRV HVWmR DEHUWRV H TXDLV HVWmR IHFKDGRV" *HQHUDOL]H SDUD TXDOTXHU Q~PHUR GH DUPiULRV 1RWD SHGLPRV TXH YRFr SURYH VXD FRQMHFWXUD PDLV WDUGH YHMD ([HUFtFLR

5 Prova &ULDPRV FRQFHLWRV PDWHPiWLFRV SRU PHLR GH GH¿QLo}HV 3RVWXODPRV HQWmR DVVHUo}HV VR EUH QRo}HV PDWHPiWLFDV H HP VHJXLGD SURFXUDPRV SURYDU TXH QRVVDV LGHLDV VmR FRUUHWDV 2 TXH p XPD prova" (P FLrQFLD D YHUGDGH VXUJH GD H[SHULPHQWDomR 1D OHL D YHUGDGH p DYDOLDGD SRU XP MXOJDPHQWR H GHFLGLGD SRU XP MXL] RX XP M~UL 1R HVSRUWH D YHUGDGH p D GHFLVmR GRV MXt]HV FRQVHTXHQWH GH VXD FDSDFLGDGH (P PDWHPiWLFD WHPRV D prova. $ YHUGDGH HP PDWHPiWLFD QmR p GHPRQVWUDGD PHGLDQWH XP H[SHULPHQWR ,VVR QmR TXHU GL]HU TXH R H[SHULPHQWR QmR WHQKD LPSRUWkQFLD SDUD D PDWHPiWLFD ± PXLWR SHOR FRQWUiULR 7HVWDQGR QRVVDV LGHLDV H H[HPSORV SRGHPRV IRUPXODU D¿UPDo}HV TXH FUH PRV VHUHP YHUGDGHLUDV FRQMHFWXUDV SURFXUDPRV HP VHJXLGD SURYDU HVVDV D¿UPDo}HV FRQYHUWHQGR DVVLP FRQMHFWXUDV HP WHRUHPDV

3RU H[HPSOR UHFRUGH D D¿UPDomR ³WRGRV RV Q~PHURV SULPRV VmR tPSDUHV´ 6H SDU WLUPRV GR Q~PHUR HQFRQWUDPRV FHQWHQDV GH PLOKDUHV GH Q~PHURV SULPRV TXH VmR WR GRV tPSDUHV ,VVR VLJQL¿FD TXH WRGRV RV Q~PHURV SULPRV VHMDP tPSDUHV" 1DWXUDOPHQWH QmR 2 IDWR p TXH VLPSOHVPHQWH RPLWLPRV R Q~PHUR

Book 1.indb 20

04/04/2016 16:30:32

Capítulo 1

Fundamentos

21

&RQVLGHUHPRV XP H[HPSOR EHP PHQRV yEYLR CONJECTURA 5.1 (Goldbach) 7RGR LQWHLUR SDU PDLRU GR TXH p D VRPD GH GRLV SULPRV 9HUL¿TXHPRV TXH HVVD D¿UPDomR p YiOLGD SDUD RV SULPHLURV Q~PHURV SDUHV 7HPRV 4=2 + 2 12 = 5 + 7

6=3 + 3 14 = 7 + 7

8=3 + 5 16 = 11 + 5

10 = 3 + 7 18 = 11 + 7.

3RGHUtDPRV HVFUHYHU XP SURJUDPD GH FRPSXWDGRU SDUD FRQ¿UPDU TXH RV SULPHLURV ELOK}HV GH Q~PHURV SDUHV D FRPHoDU GH VmR FDGD XP D VRPD GH GRLV SULPRV ,VVR LPSOLFD TXH D FRQMHFWXUD GH *ROGEDFK VHMD YHUGDGHLUD" 1mR $ HYLGrQFLD QXPpULFD WRUQD D FRQMHFWXUD DGPLVVtYHO PDV QmR SURYD TXH VHMD YHUGDGHLUD $Wp KRMH QmR VH FRQVHJXLX XPD SURYD GD FRQMHFWXUD GH *ROGEDFK H DVVLP VLPSOHVPHQWH QmR VDEHPRV VH HOD p YHUGDGHLUD RX IDOVD >ŝŶŐƵĂŐĞŵ ŵĂƚĞŵĄƟĐĂ͊ hŵĂ ƉƌŽǀĂ Ġ ĨƌĞƋƵĞŶƚĞŵĞŶƚĞ ĐŚĂŵĂĚĂ ĚĞ argumento͘ EĂ ůŝŶŐƵĂŐĞŵ ƵƐƵĂů͕ Ă ƉĂůĂǀƌĂ argumento ƚĞŵ ƵŵĂ ĐŽŶŽƚĂĕĆŽ ĚĞ ĚĞƐĂĐŽƌĚŽ ŽƵ ĐŽŶƚƌŽǀĠƌƐŝĂ͘ EĆŽ ĚĞǀĞŵŽƐ ĂƐƐŽĐŝĂƌ ƚĂů ĐŽŶŽƚĂĕĆŽ ŶĞŐĂƟǀĂ Ă Ƶŵ ĂƌŐƵŵĞŶƚŽ ŵĂƚĞŵĄƟĐŽ͘ EĂ ǀĞƌĚĂĚĞ͕ ŽƐ ŵĂƚĞŵĄƟĐŽƐ ƐĞŶƚĞŵͲƐĞ ŚŽŶƌĂĚŽƐ ƋƵĂŶĚŽ ƐƵĂƐ ƉƌŽǀĂƐ ƐĆŽ ĐŚĂŵĂĚĂƐ ͞ďĞůŽƐ ĂƌŐƵŵĞŶƚŽƐ͘͟

8PD SURYD p XPD DUJXPHQWDomR TXH PRVWUD GH PDQHLUD LQGLVFXWtYHO TXH XPD D¿U PDomR p YHUGDGHLUD $V SURYDV PDWHPiWLFDV VmR HVWUXWXUDGDV FXLGDGRVDPHQWH H HVFULWDV HP XPD IRUPD DVVD] HVWLOL]DGD &HUWDV IUDVHV FKDYH H FRQVWUXo}HV OyJLFDV DSDUHFHP FRP IUHTXrQFLD QDV SURYDV 1HVWD VHomR H HP VHo}HV VXEVHTXHQWHV PRVWUDPRV FRPR DV SURYDV VmR UHGLJLGDV 2V WHRUHPDV TXH YDPRV SURYDU QHVWD VHomR VmR WRGRV EDVWDQWH VLPSOHV 1D YHUGDGH QmR LUHPRV DSUHQGHU TXDLVTXHU IDWRV VREUH Q~PHURV TXH QmR VHMDP GH QRVVR SOHQR FRQKHFLPHQ WR 2 REMHWLYR GHVWD VHomR QmR p REWHU QRYDV LQIRUPDo}HV VREUH Q~PHURV H VLP DSUHQGHU D UHGLJLU SURYDV $VVLP VHP PDLV GHORQJDV YDPRV FRPHoDU D UHGLJLU SURYDV 9DPRV SURYDU R VHJXLQWH X PROPOSIÇÃO 5.2 $ VRPD GH GRLV LQWHLURV SDUHV p SDU 9DPRV HVFUHYHU DTXL D SURYD FRPSOHWD H D VHJXLU GLVFXWLUHPRV FRPR HVVD SURYD IRL FULDGD 1HVVD SURYD QXPHUDPRV FDGD VHQWHQoD GH PRGR TXH SRVVDPRV H[DPLQi OD SDV VR D SDVVR 1RUPDOPHQWH HVFUHYHUtDPRV HVVD EUHYH SURYD HP XP ~QLFR SDUiJUDIR VHP QXPHUDU DV VHQWHQoDV Prova GD 3URSRVLomR

1. 9DPRV PRVWUDU TXH VH x e y VmR LQWHLURV SDUHV HQWmR x + y p XP LQWHLUR SDU 2. 6HMDP x e y inteiros pares.

3. Como x p SDU VDEHPRV SHOD 'H¿QLomR TXH x p GLYLVtYHO SRU LVWR p _x

4. $QDORJDPHQWH FRPR y p SDU _y.

Book 1.indb 21

04/04/2016 16:30:32

22

Matemática discreta

5. Como 2|x VDEHPRV SHOD 'H¿QLomR TXH Ki XP LQWHLUR a WDO TXH x = 2a.

6. $QDORJDPHQWH FRPR _y H[LVWH XP LQWHLUR b GH PRGR TXH y = 2b.

7. 2EVHUYH TXH x + y = 2a + 2b a + b

8. 3RUWDQWR H[LVWH XP LQWHLUR c D VDEHU a + b GH PRGR TXH x + y = 2c.

9. 3RU FRQVHJXLQWH 'H¿QLomR _ x + y

10. 3RUWDQWR 'H¿QLomR x + y p SDU ŽŶǀĞƌƐĆŽ ƉĂƌĂ Ă ĨŽƌŵĂ ƐĞͲĞŶƚĆŽ͘

([DPLQHPRV GHWLGDPHQWH FRPR HVWD SURYD IRL UHGLJLGD 2 SULPHLUR SDVVR p FRQYHUWHU D D¿UPDomR FRQWLGD QD SURSRVLomR SDUD D IRUPD ³VH HQWmR´ $ D¿UPDomR SDVVD D VHU ³D VRPD GH GRLV LQWHLURV SDUHV p SDU´ (VFUHYHPRV D D¿UPDomR QD IRUPD ³VH HQWmR´ FRPR VHJXH ³6H x e y VmR LQWHLURV SDUHV HQWmR x + y p XP LQWHLUR SDU´ 1RWH TXH LQWURGX]LPRV OHWUDV x e y SDUD UHSUHVHQWDU RV GRLV LQWHLURV SDUHV (VVDV OHWUDV VH DGDSWDP EHP QD SURYD 2EVHUYH TXH D SULPHLUD VHQWHQoD GD SURYD DSUHVHQWD D SURSRVLomR QD IRUPD ³VH HQWmR´ $ VHQWHQoD LQGLFD D HVWUXWXUD GHVVD SURYD $ KLSyWHVH D SDUWH ³VH´ LQIRUPD R OHLWRU GH TXH DGPLWLUHPRV TXH x e y VmR LQWHLURV SDUHV H D FRQFOXVmR D SDUWH ³HQWmR´

QRV GL] TXH HVWDPRV WHQWDQGR SURYDU TXH x + y p SDU $ VHQWHQoD SRGH VHU FRQVLGHUDGD XP SUHkPEXOR GD SURYD $ SURYD FRPHoD GH IDWR QD VHQWHQoD ƐĐƌŝƚĂ ĚĂ ƉƌŝŵĞŝƌĂ Ğ ĚĂ ƷůƟŵĂ ƐĞŶƚĞŶĕĂƐ ƵƟůŝnjĂŶĚŽ ĂƐ ŚŝƉſƚĞƐĞƐ Ğ ĐŽŶĐůƵƐĆŽ ĚĂ ĚĞĐůĂƌĂĕĆŽ͘

2 SUy[LPR SDVVR FRQVLVWH HP HVFUHYHU R FRPHoR H[DWR H R ¿P H[DWR GD SURYD $ KLSyWHVH GD VHQWHQoD LQGLFD R TXH HVFUHYHU D VHJXLU $¿UPD ³ VH x e y VmR LQWHLURV SDUHV ´ GH IRUPD TXH HVFUHYHPRV VLPSOHVPHQWH ³6HMDP x e y inteiros pa UHV´ 6HQWHQoD

,PHGLDWDPHQWH DSyV HVFUHYHUPRV D SULPHLUD VHQWHQoD HVFUHYHPRV D ~OWLPD sen WHQoD GD SURYD TXH p RXWUD PDQHLUD GH HVFUHYHU D FRQFOXVmR GD IRUPD ³VH HQWmR´ GD D¿UPDomR ³3RUWDQWR x + y p SDU´ 6HQWHQoD

2 DUFDERXoR GD SURYD HVWi FRQVWUXtGR 6DEHPRV RQGH FRPHoDU x e y VmR SDUHV H VDEHPRV SDUD RQGH QRV GLULJLUPRV x + y p SDU

džƉĂŶƐĆŽ ĚĂƐ ĚĞĮŶŝĕƁĞƐ͘

2 SUy[LPR SDVVR p H[SDQGLU DV GH¿QLo}HV R TXH ID]HPRV HP DPEDV DV H[WUHPLGDGHV GD SURYD $ VHQWHQoD D¿UPD TXH x p SDU 2 TXH VLJQL¿FD LVVR" 3DUD YHUL¿Fi OR FRQIHULPRV RX UHFRUGDPRV D GH¿QLomR GD SDODYUD par 'r XPD UiSLGD ROKDGD QD 'H¿QLomR

(OD D¿UPD TXH XP LQWHLUR p SDU GHVGH TXH VHMD GLYLVtYHO SRU 6DEHPRV DVVLP TXH x p GLYLVtYHO SRU R TXH SRGHPRV HVFUHYHU FRPR _x LVVR QRV Gi D VHQWHQoD

Book 1.indb 22

04/04/2016 16:30:32

UMA INTRODUÇÃO TRADUÇÃO DA 3

a

EDIÇÃO NORTE-AMERICANA

Dirigida a interessados em matemática discreta de maneira geral, esta obra apresenta uma visão geral de toda a matemática a partir da perspectiva da matemática discreta. O livro cobre desde a natureza da matemática (definições, teoremas, provas e contraexemplos), lógica, conjuntos e relações, até técnicas avançadas de prova, probabilidades discretas, teoria dos números, teoria dos grafos e álgebra abstrata, incluindo aplicações em criptografia.

Esta terceira edição conta com muitos problemas e exercícios que exploram as interligações entre os diversos temas tratados.

Aplicações: Esta obra pode ser adotada como livro-texto em disciplinas de matemática discreta em cursos de engenharia, matemática e ciência da computação. Texto didático e agradável, pode ser lido por qualquer interessado na matéria.

ISBN 13 978-85-221-2534-0 ISBN 10 85-221-2534-1

MATEMÁTICA DISCRETA: UMA INTRODUÇÃO

MATEMÁTICA DISCRETA

EDWARD R. SCHEINERMAN

EDWARD R. SCHEINERMAN

OUTRAS OBRAS INTRODUÇÃO À LÓGICA MATEMÁTICA Carlos Alberto F. Bispo, Luiz B. Castanheira e Oswaldo Melo S. Filho LÓGICA PARA COMPUTAÇÃO Flávio Soares Corrêa da Silva, Marcelo Finger e Ana Cristina Vieira de Melo

EDWARD R. SCHEINERMAN

MATEMÁTICA DISCRETA

FUNDAMENTOS DA CIÊNCIA DA COMPUTAÇÃO Tradução da 2a edição Internacional Behrouz Forouzan e Firouz Mosharraf INTRODUÇÃO À TEORIA DA COMPUTAÇÃO Tradução da 2a edição Norte-Americana Michael Sipser

UMA INTRODUÇÃO TRADUÇÃO DA 3a EDIÇÃO NORTE-AMERICANA

Para suas soluções de curso e aprendizado, visite www.cengage.com.br

Matemática_18x26.indd 1

9 788522 125340

4/8/16 10:53 AM