ZBIRKA ZADATAKA IZ DISKRETNIH MATEMATIQKIH STRUKTURA
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2 Vesna Manojlovi Mirjana Qangalovi Vladimir Balti ZBIRKA ZADATAKA IZ DISKRETNIH MATEMATIQKIH STRUKTURA Fakultet organizacionih nauka B E O G R A D 2014.
3 Autori: dr Vesna Manojlovi, dr Mirjana Qangalovi, dr Vladimir Balti, DISKRETNE MATEMATIQKE STRUKTURE Izdavaq: Fakultet organizacionih nauka, Beograd, Jove Ili a 154 Recenzent: dr Vera Vujqi Crteжi i slog: dr Vladimir Balti CIP Katalogizacija u publikaciji Narodna biblioteka Srbije, Beograd 51-74:004(075.8)(076) MANOJLOVI, Vesna, 1974 Zbirka zadataka iz diskretnih matematiqkih struktura / Vesna Manojlovi, Mirjana Qangalovi, Vladimir Balti [crteжi Vladimir Balti ]. Beograd : Fakultet organizacionih nauka, 2014 (VrƬaqka BaƬa : SaTCIP). 168 str. : ilustr. ; 297 mm. Tiraж 400. Bibliografija: str ISBN Qangalovi, Mirjana, 1954 [autor] 2. Balti, Vladimir, 1973 [autor] [ilustrator] a) Diskretna matematika Zadaci COBISS.SR ID XtampaƬe ove kƭige podrжano je od strane Tempus projekta INCOMING (Interdisciplinary Curricula in Computing to Meet Labor Market Needs, TEMPUS EE-TEMPUS-JPCR). Project INCOMING (Interdisciplinary Curricula in Computing to Meet Labor Market Needs, TEMPUS EE- TEMPUS-JPCR) has been funded with support from the European Commisson. This publication reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the information contained therein. Fakultet organizacionih nauka, Tiraж: 400 primeraka Xtampa:,,SaTCIP DOO, VrƬaqka BaƬa
4 Sadrжaj PREDGOVOR 5 1. ISKAZNI RAQUN OdređivaƬe istinitosti iskazne formule Eliminacija logiqkih veznika i Eliminacija duplih negacija i pomeraƭe negacije ka,,unutra Normalne forme Disjunktivna normalna forma Konjunktivna normalna forma Primene iskaznog raquna Razni zadaci PREDIKATSKI RAQUN OdređivaƬe vrednosti predikatske formule Zamena promenʃivih izrazom Osobine kvantifikatora RaqunaƬe istinitosne vrednosti predikatske formule Razni zadaci RELACIJSKE STRUKTURE Relacije na konaqnim skupovima Relacije na beskonaqnim skupovima Razni zadaci TEORIJA GRAFOVA Grafovi Stabla Korenska stabla i primene Razni zadaci KONAQNI AUTOMATI I REGULARNE GRAMATIKE Konaqni automati Minimizacija konaqnih automata KombinovaƬe automata Regularne gramatike Razni zadaci LITERATURA 167 3
5 5 Predgovor Ova zbirka zadataka napisana je prema vaжe em Nastavnom planu i programu predmeta Diskretne matematiqke strukture na Fakultetu organizacionih nauka u Beogradu, qiju nastavu autori izvode. KƬiga predstavʃa rezultat vixegodixƭeg iskustva u radu sa studentima i жeʃe da se studentima olakxa pra eƭe nastave i omogu i temeʃnije pripremaƭe ispita. Zbirka je nastala od istoimene kƭige Vesne Manojlovi i Mirjane Qangalovi dodavaƭem delova zbirke,,diskretne matematiqke strukture zbirka ispitnih i doma ih zadataka iz i autora Vladimira Balti a, i to na slede i naqin: Glava 1. Predikatski raqun je podeʃena na dve glave 1. Iskazni raqun i 2. Predikatski raqun, dodata je Glava 4. Teorija grafova, sadrжaj glava 2. Relacijske strukture i 3. Konaqni automati delimiqno je modifikovan u okviru sadaxƭih glava 2. Relacijske strukture i 5. Konaqni automati i regularne gramatike. Gradivo je izloжeno postupno, od jednostavnijih primera ka sloжenijim zadacima. KƬiga se sastoji od pet glava, koje se mogu qitati potpuno nezavisno jedna od druge. U kƭizi su skoro svi zadaci rexeni, a obuhva eni su i mnogi od zadataka sa do sada odrжanih kolokvijuma i ispita iz istog predmeta na Fakultetu organizacionih nauka. Nadamo se da e ova kƭiga pomo i studentima u pripremi ispita i motivisati neke studente da daʃe proxiruju znaƭe iz ove oblasti. Izuzetno nam je zadovoʃstvo da se zahvalimo recezentu prof. dr Veri Vujqi, koja je detaʃno proqitala rukopis i svojim izuzetno korisnim primedbama doprinela da kƭiga bude jox boʃa. ZahvaƩujemo se i NedeƩku Stefanovi u koji je obavio celokupnu pripremu za xtampu istoimene kƭige Vesne Manojlovi i Mirjane Qangalovi i neki od Ƭegovih crteжa su odatle preuzeti u Glavi 5. Sve ostale slike u ovoj zbirci su nacrtane pomo u programskog paketa WinGCLC koji je razvio Predrag Janiqi (na qemu mu se autori svesrdno zahvaʃuju), a moжe se na i na adresi janicic/gclc/index.html Zahvalni smo i svim onim studentima koji su nam ukazali na grexke u obe prethodne kƭige, koje su ovde sada otkloƭene. U Beogradu novembra Autori
6 Literatura [1] James A. Anderson, Diskretna matematika sa kombinatorikom, Računarski fakultet, Beograd, [2] Vladimir Balti, Diskretne matematiqke strukture zbirka ispitnih i doma ih zadataka iz i 2009., FON, Beograd, [3] Vladimir Balti, Duxan uki, orđe Krtini, Ivan Mati, Pripremni zadaci za takmiqeƭa iz matematike uqenika sredƭih xkola, DMS, Beograd, [4] Ken Bogart, Cliff Stein, Discrete Math in Computer Science, web draft, [5] Miklós Bóna, A Walk Through Combinatorics, World Scientific, [6] Dragoš Cvetković, Teorija grafova i njene primene, Naučna knjiga, Beograd, [7] Dragoš Cvetković, Mirjana Čangalović, Djordje Dugošija, Vera Kovačević-Vujčić, Slobodan Simić, Jovo Vuleta, Kombinatorna optimizacija, Beograd [8] Dragoš Cvetković, Slobodan Simić Diskretna matematika, Matematika za kompjuterske nauke, Libra, Beograd, [9] Dragoš Cvetković, Slobodan Simić Odabrana poglavlja iz diskretne matematike, Akademska misao, Beograd, [10] Mirjana Qangalovi, Vesna Manojlovi, Vladimir Balti, Diskretne matematiqke strukture, Fakultet organizacionih nauka, Beograd, [11] John A. Dossey, Albert D. Otto, Lawrence E. Spence, Charles Vanden Eynden, Discrete Mathematics, Pearson, Addison Wesley, Boston, [12] Martin Gardner, The last recreations: hydras, eggs, and other mathematical mystifications, Copernicus Springer-Verlag, New York, [13] Martin Gardner, The 2nd Scientific American book of mathematical puzzles and diversions, The University of Chicago Press, Chicago, [14] Martin Gardner, Wheels, life, and other mathematical amusements, W.H. Freeman and Company, New York, 1983 [15] Ronald L. Graham, Martin Grötschel, László Lovász (editors), Handbook of Combinatorics, Volume 1 & 2, North-Holland, [16] Ralph P. Grimaldi, Discrete and combinatorial mathematics: an applied introduction 3rd ed, Addison-Wesley Publishing Company, [17] Frank Harary, Graph Theory, Narosa Publishing House, [18] Predrag Janiqi, Matematiqka logika u raqunarstvu, Matematiqki fakultet, Beograd,
7 [19] Vesna Manojlovi, Mirjana Qangalovi Zbirka zadataka iz diskretnih matematiqkih struktura, Fakultet organizacionih nauka, Beograd, [20] László Lovász, József Pelikán, Katalin Vesztergombi, Discrete Mathematics, Elementary and Beyond, Springer, [21] Vojislav Petrović, Teorija grafova, Novi Sad, [22] Kenet H. Rosen, Discrete Mathematics and Its Applications, McGraw Hill, Boston, [23] Kenet H. Rosen, Handbook of Discrete and Combinatorial Mathematics, CRC Press, Boston, [24] Neil J.A. Sloane, Online Encyclopedia of Integer Sequences, [25] Richard P. Stanley, Enumerative combinatorics, Volume 2, Cambridge University Press, Cambridge, [26] Dragan Stevanovi, Vladimir Balti, Slobodan Simi, Miroslav iri, Diskretna matematika osnove kombinatorike i teorije grafova, DMS, Beograd, [27] Dragan Stevanović, Marko Milošević, Vladimir Baltić, Diskretna matematika, Zbirka rešenih zadataka, DMS, Beograd, [28] George J. Summers, The great book of mind teasers and mind puzzlers, Sterling Publishing Co., New York, [29] Darko Veljan, Kombinatorika sa teorijom grafova, Školska knjiga, Zagreb, 1989.
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