Name: ID# (last 4 digits): PIN: CSE Discrete Structures

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1 CSE - Discrete Structures Exm : Comprehensive Nme: ID# (lst digits): PIN: CSE - Discrete Structures Finl Exm - Fll 00 Closed book, closed notes Dte: Dec. 9 00, :00 pm - :0 pm. Indicte if the following functions re one-to one, onto, or bijective. ( points) ) b) c)!#" Mnfred Huber Pge

2 CSE - Discrete Structures Exm : Comprehensive. Prove by giving pproprite constnts tht the following functions hve the indicted orders of mgnitude. ( points) ) $&%(')+*,-('$+"$./ 0&+* b) " %'7,8.& 09: * Mnfred Huber Pge

3 ; CSE - Discrete Structures Exm : Comprehensive. Give the forml definition for the following grph. Also indicte if it is loop-free, simple, connected, or cyclic. ( points) 7 Mnfred Huber Pge

4 J J " " * * " " * * CSE - Discrete Structures Exm : Comprehensive. Drw the directed or undirected grphs corresponding to the following forml definitions. List ll cycles in the grph. ( points) ) < >=@? BAC BDE BF BGIH K=@?MLN O? O? B? S T?UL#$VDE BGE S T? B?QPRH WXVAC BFYN S V? TF B?Y S V? $V? BA S T?ZPO XV? BD[ b) < >=Z\I N%] BM ^.U N_YH K=@?MLN O? O? B? S T?UL#$K\`\ S V? B?QPRH /%(8%Y S V? &b8_] S T?.c %] S T?ZPO /_b Mnfred Huber Pge

5 CSE - Discrete Structures Exm : Comprehensive. For the following pir of directed grphs, indicte if they re isomorphic. If they re isomorphic give the corresponding isomorphism. ( points) e e e e e e Mnfred Huber Pge

6 CSE - Discrete Structures Exm : Comprehensive. For the following grphs, indicte if they re plnr. If they re, drw n euivlent version without crossing rcs. Otherwise justify why they re not plnr. (8 points) ) b) Mnfred Huber Pge

7 e CSE - Discrete Structures Exm : Comprehensive 7. Give the preorder, postorder nd inorder trversls for the following trees. ( points) ) c b f g i d h f k d e l m b) h l e b c d f m g h i n o p g k Mnfred Huber Pge 7

8 CSE - Discrete Structures Exm : Comprehensive 8. Determine if the following grphs contin Hmilton Circuit. If such cycle exists, list it. ( points) ) 7 b) 7 Mnfred Huber Pge 8

9 j CSE - Discrete Structures Exm : Comprehensive 9. Use Dijkstr s lgorithm to determine the shortest pth between nodes nd i. List ll the intermedite distnce nd predecessor functions F]j nd k, s well s the finl pth nd its length. ( points) x 9 y Mnfred Huber Pge 9

10 CSE - Discrete Structures Exm : Comprehensive 0. Give minimum spnning tree for the following grphs. (8 points) ) 9 l b) 8 m Mnfred Huber Pge 0

11 CSE - Discrete Structures Exm : Comprehensive. Construct truth tbles for the following expressions nd determine if they re tutologies or contrdictions. ( points) ) J-n To&p J b) J p o7 o Mnfred Huber Pge

12 i i CSE - Discrete Structures Exm : Comprehensive. Use predicte logic to prove the following rguments. (All steps in the proof seuence hve to be nnotted by the rule used to derive it. Only the rules listed on the pges t the end of this exm cn be used.) (0 points) ) V?Yrpsutv[ nxw y Vz]{ w y b) Vz]{ t i [ } nxw nx~ yp Vz] w $ Vz]{ ~ T?Y$ } B?Y^ Mnfred Huber Pge

13 . " CSE - Discrete Structures Exm : Comprehensive. Prove or disprove the following conjectures nd indicte which proof techniue you used. You cn use ny of the proof techniues introduced in clss (counterexmple, exhustive proof, direct proof, proof by contrposition, proof by contrdiction, proof by induction). (8 points) ) The conctention of two strings of odd length results in string of even length. b) For every positive integer, % ƒ. Mnfred Huber Pge

14 CSE - Discrete Structures Exm : Comprehensive. Prove the subset reltion J8 o7r J8 ˆ ŠKVo ˆ. ( points). Solve the following counting problem nd indicte if it is permuttion or combintion problem. (You only hve to give the formul. The finl number is not reuired.) ( points) How mny different sets of single digit numbers cn be formed if every number cn occur rbitrrily often? Mnfred Huber Pge

15 CSE - Discrete Structures Exm : Comprehensive. Show tht the set =@ŒM R\ HŽ is countble. ( points) 7. Form the closures reuired to form n euivlence reltion from the reltion >=]V? B?YN EVAC BD[ ETD@ OD{N EVF B?Y CVAC BGE CVG BGEBH K=@? BAC BDE BF BGIH over the set. ( points) Mnfred Huber Pge

16 n n CSE - Discrete Structures Exm : Comprehensive Proofs using Forml Logic For ll your forml logic proofs you cn use only the rules given in the following tbles. In ddition you re llowed to pply the deduction method nd to use the method of temporry hypotheses. All other rules hve to be proven first. Euivlence Rules Rule Nme Expression Euivlent Expression Commuttivity (comm) Associtivity (ss) Distributivity (dis) De Morgn s Lws (De Morgn) Impliction (imp) Double negtion (dn) Self-reference (self) Negtion (neg) utv p w w p p w ~ w p ~ ~ p ~ n w p ~ p w p w p w n w w nxw wn nxw nx~ wn0~ rp ps nxw rp nx~ n ~ }p wnx~ nxw nxw P z] Mnfred Huber Pge

17 i i CSE - Discrete Structures Exm : Comprehensive Inference Rules Rule Nme From Cn Derive Conjunction (con), w p w p w Simplifiction (sim), w w w Modus ponens (mp), w Modus tollens (mt), w n w Addition (dd) utv Universl instntition (ui) utv T?] (Be creful with the rule s restrictions) z] Existentil Instntition (ei) z] T?] (Be creful with the rule s restrictions) utv Universl generliztion (ug) (Be creful with the rule s restrictions) Existentil generliztion (eg) (Be creful with the rule s restrictions) V?Y)Vz] z] For ll proofs the steps hve to be nnotted such s to indicte the rule nd which elements of the proof seuence it ws pplied to. Mnfred Huber Pge 7