离散数学模拟题及部分答案英文.docx

离散数学模拟题及部分答案英文.docx

《离散数学模拟题及部分答案英文.docx》由会员分享，可在线阅读，更多相关《离散数学模拟题及部分答案英文.docx（28页珍藏版）》请在冰豆网上搜索。

离散数学模拟题及部分答案英文

DiscreteMathematicTest

Editor:

JinPeng

Date:

2008.5.6

Contents:

DiscreteMathematicTest（Unit1）

PartI（T/Fquestions,15Scores）

Inthispart,youwillhave15statements.Makeyourownjudgment,andthenputT（True）orF（False）aftereachstatement.

1.LetA,B,andCbesetssuchthatA∪B=A∪C,thenB=C.（）

2.LetAandBbesubsetsofasetU,andAB,thenA△B=AB

andA∩B’=.（）

3.Letpandqandrbethreestatements.If~pÚ~q≡~pÚ~r,thenqandrhavethesamevalue.（）

4.LetA,BbesetssuchthatbothAÍBandAÎBispossible.（）

5.Letpandqbetwostatements,then（p®～q）®（（～pÚ～q）（p®～q））isatautology.（）

6.LetA,Bbesets,P（A）isthepowersetofA,thenP（AB）=P（A）P（B）.（）

7.LetA,B,andCbesets,thenifAÎB，BÍC，thenAÍC.（）

8.LetA,Bbesets,ifA={Æ},B=P（P（A））,then{Æ}ÎBand{Æ}ÍB.（）

9.Letxberealnumber,thenxÎ{x}{{x}}and{x}Í{x}{{x}}.（）

10.LetA,B,andCbesets,thenA（B∪C）=（AB）∪（AC）.（）

11.IfA={x}∪x,thenxÎAandxÍA.（）

12.（x）（P（x）∧Q（x））and（x）P（x）∧（x）Q（x）areequivalent.（）

13.LetAandBbesets,thenA×（BC）=（A×B）（A×C）.（）

14.Theargumentformula（pÚq）®（rs）,（sÚt）®w╞p®wisvalid.（）

15.（x）（P（x）®Q（x））and（x）P（x）®（x）Q（x）areequivalent.（）

PartII（1Foundations:

SetsLogic,andAlgorithms,85Scores）

1.（8points）WhatsetssoeachoftheVenndiagramsinfollowingFigurerepresent?

2.（8points）LetU={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}.LetA={1,5,6,9,10,15}andB={5,6,8,9,12,13}.Determinethefollowing:

Finda.SAb.SA’c.SBd.SA∩B.

3.（8points）Aclassof45studentshas3minorsforoptions,respectivelyA,BandC.Aisthesetofstudentstakingalgebra,Bisthesetofstudentswhoplaybasketball,Cisthesetofstudentstakingthecomputerprogrammingcourse.Amongthe45students,12choosesubjectA,8chooseBandanother6chooseC.Additionally,9studentschooseallofthethreesubjects.

Whatistheatleastnumberofstudentsdonottakingthealgebracourseandthecomputerprogrammingcourseandplayingbasketball?

4.（8points）FindaformulaAthatusesthevariablespandqsuchthatAistrueonlywhenexactlyoneofpandqistrue.

5.（8points）Provethevalidityofthelogicalconsequences.

AnneplaysgolforAnneplaysbasketball.Therefore,Anneplaysgolf.

6.（9points）Provethevalidityofthelogicalconsequences.

Ifthebudgetisnotcutthenpricesremainstableifandonlyiftaxeswillberaised.Ifthebudgetisnotcut,thentaxeswillberaised.Ifpriceremainstable,thentaxeswillnotberaised.Therefore,taxeswillnotberaised.

7.（8points）

（1）Whatistheuniversalquantificationofthesentence:

x2+xisaneveninteger,wherexisaneveninteger?

Istheuniversalquantificationatruestatement?

（2）Whatistheexistentialquantificationofthesentence:

xisaprimeinteger,wherexisanoddinteger?

Istheexistentialquantificationatruestatement?

8.（12points）Symbolizethefollowingsentencesbyusingpredicates,quantifiers,andlogicalconnectives.

（1）Anynaturenumberhasonlyonesuccessornumber.

（2）Forallx,yN,x+y=xifandonlyify=0.

（3）NotallnaturenumberxN,itexistanaturenumberyN,suchthatx≤y.

9.（8points）Showthatx（～F（x）∨A（x）），x（A（x）→B（x）），xF（x）

|=xB（x）

10.（8points）Inthebubblesortalgorithm,ifsuccessiveelementsL[j]andL[j+1]aresuchthatL[j]>L[j+1],thentheyareinterchanged,thatis,swapped.Therefore,thebubblesortalgorithmmayrequireelementstobeswapped.Showhowbubblesortsortstheelements7563142inincreasingorder.Drawfigures.

DiscreteMathematicTest（Unit2）

PartI（T/Fquestions,15Scores）

Inthispart,youwillhave15statements.Makeyourownjudgment,andthenputT（True）orF（False）aftereachstatement.

1.LetAandBbesetssuchthatanysubsetsofABisarelationfromAtoB.（）

2.LetR={（1,1）,（1,2）,,（3,3）,（3,1）,（1,3）}berelationsonthesetA={1,2,3}thenRistransitive.（）

3.LetR={（1,1）,（2,2）,（2,3）,（3,3）}berelationsonthesetA={1,2,3}thenRissymmetric.（）

4.LetRbeasymmetricrelation.thenRnissymmetricforallpositiveintegersn.（）

5.LetRandSarereflexiverelationsonasetAthenmaybenotreflexive.（）

6.LetR={（a,a）,（b,b）,（c,c）,（a,b）,（b,c）}berelationsonthesetA={a,b,c}thenRisequivalencerelation.（）

7.IfRisequivalencerelation,thenthetransitiveclosureofRisR.（）

8.LetRberelationsonasetA,thenRmaybesymmetricandantisymmetic.（）

9.IfandarepartitionofagivensetA,then∪isalsoapartitionofA.（）

10.LetRandSareequivalencerelationsonasetA,LetψbethesetofallequivalenceclassofR,andϖbethesetofallequivalenceclassofS,ifR≠S,thenψ∩ϖ=Φ.（）

11.Let（S,）beaposetsuchthatSisafinitenonemptyset,thenShasninimalelement,andtheelementsisunique.（）

12.LetRandSarerelationsonasetA,thenMR∩SMR∧MS.（）

13.IfarelationRissymmetric.thenthereisloopateveryvertexofitsdirectedgraph.（）

14.AdirectedgraphofapartialorderrelationRcannotcontainacloseddirectedpathotherthanloops.（）

15.Theposet,whereP（S）isthepowersetofasetSisnotachain.（）

PartII（1Foundations:

SetsLogic,andAlgorithms,85Scores）

1.（8points）LetRbetherelation{（1,2）,（1,3）,（2,3）,（2,4）,（3,1）},andletSberelation{（2,1）,（3,1）,（3,2）,（4,2）}.FindSR.andR3.

2.（8points）Determinewhethertherelationsrepresentedbythefollowingzero-onematricesarepartialorders.

3.（8points）Determinethenumberofdifferentequivalencerelationsonasetwiththreeelementsbylistingthem.

4.（8points）LetR={（a,b）∈A|adividesb},whereA={1,2,3,4}.FindthematrixMRofR.ThendeterminewhetherRisreflexive,symmetric,ortransitive.

5.（8points）DeterminewhethertherelationRonthesetofallpeopleisreflexive,symmetric,antisymmetric,and/ortransitive,where（a,b）Rifandonlyif

a）aistallerthanb.

b）aandbwerebornonthesameday.

c）ahasthesamefirstnameasb.

6.（8points）Defineaequivalencerelationsonthesetofstudentsinyourdiscretemathematicsclass.Determinetheequivalenceclassesfortheseequivalencerelations.

7.（10points）LetRbetherelationonthesetoforderedpairsofpositiveintegerssuchthatifandonlyif.ShowthatRisanequivalencerelation.

8.（8points）AnswerthefollowingquestionsforthepartialorderrepresentedbythefollowingHassediagram.

9.（9points）LetRbetherelationonthesetA={a,b,c,d}suchthatthematrixofRis

find

（1）reflexiveclosureofR.

（2）symmetricclosureofR.

（3）transitiveclosureofR.

10.（10points）

（1）Showthatthereisexactlyonegreatestelementofaposet,ifsuchanelementexists.

（2）Showthattheleastupperboundofasetinposetisuniqueifitexists.

DiscreteMathematicTest（Unit3）

PartI（T/Fquestions,15Scores）

Inthispart,youwillhave15statements.Makeyourownjudgment,andthenputT（True）orF（False）aftereachstatement.

1.Thereexistasimplegraphwithfouredgesanddegreesequence1,2,3,4.（）

2.Thereareatleasttwopeoplewhithexactlythesamenumberoffriendsinanygatheringofn>1people.

.（）

3.Thenumberofedgesinacompletegraphwithnverticesisn（n-1）.（）

4.ThecomplementofgraphGisnotpossibleasubgraphofG.（）

5.Tthatanycycle-freegraphcontainsavertexofdegree0or1.（）

6.ThegraphG,eitherGoritscomplementG’,isaconnectedgraph.（）

7.AnygraphGanditscomplementG’cannotbeisomorphic（）

8.AnEulerianisaHamiltoniangraph,butaHamiltoniangraphisnotAnEulerian.（）

9.Ifeverymemberofapartyofsixpeopleknowsatleastthreepeople,provethattheycansitaroundatableinsuchawaythateachofthemknowsbothhisneighbors.（）

10.Acircuiteitherisacycleorcanbereducedtoacycle.（）

11.AgraphGwithnvertices.GisconnectedifandonlyifGisatree.（）

12.Aconnectedgraphisacircuitifthedegreeofeachvertexis2.（）

13.Acircuiteitherisacycleorcanbereducedtoacycle.（）

14.ForanysimpleconnectedplanargraghGthatX（G）6.（）

15.．Thesumoftheodddegreesofallverticesofagraphiseven.（）

PartII（1Foundations:

SetsLogic,andAlgorithms,85Scores）

1.（10points）Doesthereexistasimplegraphwithdegreesequence1,2,3,5?

Justifyyouanswer.

2.（10points）Supposethereare90smalltownsinacountry.Fromeachtownthereisadirectbusroutetoaleast50towns.Isitpossibletogofromonetowntoantothertownbybuspossiblychangingfromonebusandthentakinganotherbustoanothertown?

3.10points）Findthenumberofdistinctpathsoflength2ingraphsK5.

4.（5pointsDrawalldifferentgraphswithtwoverticesandtwoedges.

5.（10points）DeterminewherethegraphsinFigure1haveEulertrails.IfthegraphhasanEulertrail,exhibitone.

6．（10points）UseaK-maptofindtheminimizedsum-of-productBooleanexpressionsoftheexpressions.

xyzw+xyzw’+xyx’w’+xy’zw’+x’yzw+x’yzw’+x’y’z’w’+x’y’z’w

7.（10points）Insert5,10,and20,inthisorder,inthebinarysearchtreeoffollowingFigure.Drawthebinarysearchtreeaftereachinsertion.

8．（8points）Doesthereexistasimpleconnectedplanargraphwith35verticesand100edges?

9.（10points）LetGbeasimpleconnectedgraphwithnvertices.Supposethedegreeofeachvertexisatleasen1.DoesitimplytheexistenceofaHamiltoniancycleinG?

DiscreteMathematicTest1

PartI（T/Fquestions）

Directions:

inthispart,youwillhave15stateme