UNIVERSITY OF PRETORIA / UNIVERSITEIT VAN PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS
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1 UNIVERSITY OF PRETORIA / UNIVERSITEIT VAN PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS WTW 285 DISCRETE STRUCTURES DISKRETE STRUKTURE EXAMINATION EKSAMEN Internal Examiner / Interne Ekaminatore: Prof. L.M. Pretoriu, R.J. de Beer 4//2006 TIME/TYD: 20 min MARKS / PUNTE: 50 SURNAME/VAN: FIRST NAMES/VOORNAME: STUDENT NUMBER/STUDENTENOMMER: SIGNATURE/HANDTEKENING: SEL NR / CELL NO: PUNTE MARKS LEES DIE VOLGENDE INSTRUK- SIES. Die vraetel betaan uit bladye tot 9 (vrae tot ). Kontroleer of jou vraetel volledig i. 2. Toon al jou werk en motiveer elke bewering. Toon in die beonder aan hoe telling toegepa word. 3. Doen alle krapwerk op die teenblad. Dit word nie nageien nie. 4. A jy meer a die bekikbare ruimte vir n antwoord nodig het, gebruik dan ook die teenblad en dui dit aeblief duidelik aan. 5. Geen potloodwerk of enige iet wat in rooi ink gedoen i, word nageien nie. 6. A jy korrigeerink ( Tipp-Ex ) gebruik, verbeur jy die reg om te kla oor werk wat nie nageien i nie of wat verkeerd nageien i. 7. Enige navrae oor die naienwerk moet binne drie dae nadat die toet uitgedeel i, gedoen word. Daarna word aanvaar dat alle korrek i. READ THE FOLLOWING INSTRUC- TIONS. The paper conit of page to 9 (quetion to ). Check whether your paper i complete. 2. Show all work and jutify each tatement. In particular how how theorem are applied. 2. Do all cribbling on the facing page. It will not be marked. 3. If you need more than the available pace for an anwer, ue the facing page and pleae indicate it clearly. 4. No pencil work or any work in red ink will be marked. 5. If you ue correcting fluid ( Tipp-Ex ), you loe the right to quetion the marking or to indicate work that had not been marked. 7. Any querie about the marking mut be done within three day after the tet have been handed out. After that we aume that everything i in order. Outeurreg voorbehou Copyright reerved
2 QUESTION VRAAG In the Tower of Hanoi Problem, there are 3 pole, named A, B and C. On pole A there are n dik of different ize, uch that each dik i maller than the one beneath it. Let n be the minimum number of move required to move thee n dik from pole A to C if you may not put a larger dik on top of a maller one and you may only move a dik to an adjacent pole. In die Toring van Hanoi Probleem, i daar 3 pale, genoem A, B en C. Op paal A i daar n kywe van verkillende grootte, ó dat elke kyf kleiner a die een onder hom i. Laat n die minimum aantal tappe wee benodig om die n kywe van paal A tot paal C te kuif a jy nie n groter kyf op n kleiner een mag plaa nie, en elke kyf net na n aangrenende paal gekuif mag word.. Find a recurrence relation for n. Vind n rekurrenierelaie vir n. 2. Ue the method of iteration to olve the recurrence relation. [3] Gebruik die metode van iteraie om die rekurrenierelaie op te lo.
3 2 QUESTION 2 VRAAG 2 Define a et A of binary tring a follow: BASE: ɛ A (ɛ i the empty word). RECURSION: if, t A then. 0 A 2. 0 A 3. t A. RESTRICTION: No binary tring i in A other than thoe defined in the BASE and RECURSION.. Which of 000 and 00 are in A? Motivate. Definieer A n verameling binêre tringe oo volg: BASIS: ɛ A (ɛ i die leë woord). REKURSIE: a, t A dan i. 0 A 2. 0 A 3. t A. BEPERKING: Geen binêre tring i in A behalwe dié wat deur die BASIS en REKURSIE gegee word nie. Watter een van 000 en 00 i in A? Motiveer. 2. Ue tructural induction to prove that for each tring in A the number of i le than or equal to the number of 0. Gebruik trukturele indukie om aan te toon dat vir elke tring in A die aantal e kleiner of gelyk aan die aantal 0 e i. [3]
4 3 QUESTION 3 VRAAG 3 Solve the following recurrence relation.. n = 2 n 35 n 2 ; 0 = 0; = 2. Lo die volgende rekurrenierelaie op. 2. n = 4 n 49 n 2 ; 0 = ; = 4. [3] [3]
5 4 QUESTION 4 VRAAG 4 Let R be an equivalence relation on a et A. Define a new relation R 2 on A by ar 2 b there i a c uch that arc and crb. Laat R n ekwivalenierelaie op n verameling A wee. Definieer n nuwe relaie R 2 deur. I R 2 reflexive? Motivate. I R 2 reflekief? Motiveer. 2. I R 2 ymmetric? Motivate. I R 2 immetrie? Motiveer. 3. I R 2 tranitive? Motivate. I R 2 tranitief? Motiveer. QUESTION 5 VRAAG 5 Find an invere for 2 modulo 3. Vind n invere vir 2 modulo 3.
6 5 QUESTION 6 VRAAG 6 Conider the finite tate automaton defined Bekou die eindige toetand outomaton by the tranition diagram gedefinieer deur die diagram 0. Are binary tring with three accepted by A? Explain. 3 Word binêre woorde met drie e aanvaar deur A? Verduidelik. 2. Derive the language defined by the regular expreion ω = Lei die taal af gedefinieer deur die reguliere uitdrukking ω = Decribe L(A), the language accepted by A. Bekryf L(A), die taal wat deur A aanvaar word.
7 6 4. Find a regular expreion that define L(A). Vind n reguliere uitdruking wat L(A) definieer. QUESTION 7 VRAAG 7 [] Conider the finite tate automaton A defined by the tranition diagram Find the 0-equivalence clae, the - equivalence clae, the 2-equivalence clae and the *-equivalence clae of A. Bekou die eindige toetand outomaton A gedefinieer deur die diagram 3 5 Vind die 0-ekwivalenieklae, die - ekwivalenieklae, die 2-ekwivalenieklae en die *-ekwivalenieklae van A.
8 7 2. Contruct the quotient automaton A of A. Kontrueer die kwoientoutomaton A van A. 3. Decribe the language L(A) accepted by A. Bekryf die taal L(A) wat deur A aanvaar word. QUESTION 8 VRAAG 8 [] Recall that K n denote the complete graph on n vertice.. Show that for all integer n the number n(n ) of edge i. 2 Onthou dat K n die volledige grafiek op n punte aandui. Toon aan dat vir alle heelgetalle n die n(n ) aantal lyne i. 2
9 8 2. Show that the number of edge of a imple graph with n vertice i le than or equal n(n ) to. 2 Bewy dat die aantal lyne in n eenvoudige n(n ) grafiek i kleiner of gelyk aan. 2 QUESTION 9 VRAAG 9 Find all non-iomorphic tree with 5 vertice. Explain why your lit i complete and why none of the graph in the lit are iomorphic. (Hint: Focu on the maximum vertex degree of each tree.) Vind alle nie-iomorfe bome met 5 punte. Verduidelik hoekom u ly volledig i en hoekom geen grafieke in die ly iomorf aan mekaar i. (Wenk: Foku op die makimum graad van die punte van elke boom.) [4]
10 9 QUESTION 0 VRAAG 0 Ue Krukal algorithm and Prim algorithm tarting at v 0 to find a minimum pan- algoritme waar u by v 0 begin om n min- Gebruik Krukal e algoritme en Prim e ning tree for the graph below. In both cae imum panboom in die grafiek hieronder indicate the order in which edge are added te vind. In beide gevalle dui die orde aan to form the tree. v waarin die lyne gekie word om die boom 0 2 v te vorm v 5 v4 v 3 v v 6 3 v 7 QUESTION VRAAG [4] Determine all the non-iomorphic panning tree of the graph below. v v 5 v 2 Bepaal al die nie-iomorfe panbome van die grafiek hieronder. v 4 v 3
UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS
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