最新数据库系统基础教程第五章答案.docx

最新数据库系统基础教程第五章答案.docx

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最新数据库系统基础教程第五章答案

Exercise5.1.1

Asaset:

speed

2.66

2.10

1.42

2.80

3.20

2.20

2.00

1.86

3.06

Average=2.37

Asabag:

speed

2.66

2.10

1.42

2.80

3.20

3.20

2.20

2.20

2.00

2.80

1.86

2.80

3.06

Average=2.48

Exercise5.1.2

Asaset:

hd

250

80

320

200

300

160

Average=218

Asabag:

hd

250

250

80

250

250

320

200

250

250

300

160

160

80

Average=215

Exercise5.1.3a

Asaset:

bore

15

16

14

18

Asabag:

bore

15

16

14

16

15

15

14

18

Exercise5.1.3b

πbore（Ships

Classes）

Exercise5.1.4a

Forbags:

Ontheleft-handside:

GivenbagsRandSwhereatupletappearsnandmtimesrespectively,theunionofbagsRandSwillhavetupletappearn+mtimes.ThefurtherunionofbagTwiththetupletappearingotimeswillhavetupletappearn+m+otimesinthefinalresult.

Ontheright-handside:

GivenbagsSandTwhereatupletappearsmandotimesrespectively,theunionofbagsRandSwillhavetupletappearm+otimes.ThefurtherunionofbagRwiththetupletappearingntimeswillhavetupletappearm+o+ntimesinthefinalresult.

Forsets:

Thisisasimilarcasewhendealingwithbagsexceptthetupletcanonlyappearatmostonceineachset.Thetupletonlyappearsintheresultifallthesetshavethetuplet.Otherwise,thetupletwillnotappearintheresult.Sincewecannothaveduplicates,theresultonlyhasatmostonecopyofthetuplet.

Exercise5.1.4b

Forbags:

Ontheleft-handside:

GivenbagsRandSwhereatupletappearsnandmtimesrespectively,theintersectionofbagsRandSwillhavetupletappearmin（n,m）times.ThefurtherintersectionofbagTwiththetupletappearingotimeswillproducetupletmin（o,min（n,m））timesinthefinalresult.

Ontheright-handside:

GivenbagsSandTwhereatupletappearsmandotimesrespectively,theintersectionofbagsRandSwillhavetupletappearmin（m,o）times.ThefurtherintersectionofbagRwiththetupletappearingntimeswillproducetupletmin（n,min（m,o））timesinthefinalresult.

TheintersectionofbagsR,SandTwillyieldaresultwheretupletappearsmin（n,m,o）times.

Forsets:

Thisisasimilarcasewhendealingwithbagsexceptthetupletcanonlyappearatmostonceineachset.Thetupletonlyappearsintheresultifallthesetshavethetuplet.Otherwise,thetupletwillnotappearintheresult.

Exercise5.1.4c

Forbags:

Ontheleft-handside:

GiventhattuplerinR,whichappearsmtimes,cansuccessfullyjoinwithtuplesinS,whichappearsntimes,weexpecttheresulttocontainmncopies.AlsogiventhattupletinT,whichappearsotimes,cansuccessfullyjoinwiththejoinedtuplesofrands,weexpectthefinalresulttohavemnocopies.

Ontheright-handside:

GiventhattuplesinS,whichappearsntimes,cansuccessfullyjoinwithtupletinT,whichappearsotimes,weexpecttheresulttocontainnocopies.AlsogiventhattuplerinR,whichappearsmtimes,cansuccessfullyjoinwiththejoinedtuplesofsandt,weexpectthefinalresulttohavenomcopies.

Theorderinwhichweperformthenaturaljoindoesnotmatterforbags.

Forsets:

Thisisasimilarcasewhendealingwithbagsexceptthejoinedtuplescanonlyappearatmostonceineachresult.Iftherearetuplesr,s,tinrelationsR,S,Tthatcansuccessfullyjoin,thentheresultwillcontainatuplewiththeschemaoftheirjoinedattributes.

Exercise5.1.4d

Forbags:

SupposeatupletoccursnandmtimesinbagsRandSrespectively.IntheunionofthesetwobagsR⋃S,tupletwouldappearn+mtimes.Likewise,intheunionofthesetwobagsS⋃R,tupletwouldappearm+ntimes.Bothsidesoftherelationyieldthesameresult.

Forsets:

Atupletcanonlyappearatmostonetime.TupletmightappeareachinsetsRandSoneorzerotimes.ThecombinationsofnumberofoccurrencesfortupletinRandSrespectivelyare（0,0）,（0,1）,（1,0）,and（1,1）.OnlywhentupletappearsinbothsetsRandSwilltheunionR⋃Shavethetuplet.ThesamereasoningholdswhenwetaketheunionS⋃R.

Thereforethecommutativelawforunionholds.

Exercise5.1.4e

Forbags:

SupposeatupletoccursnandmtimesinbagsRandSrespectively.IntheintersectionofthesetwobagsR∩S,tupletwouldappearmin（n,m）times.LikewiseintheintersectionofthesetwobagsS∩R,tupletwouldappearmin（m,n）times.Bothsidesoftherelationyieldthesameresult.

Forsets:

Atupletcanonlyappearatmostonetime.TupletmightappeareachinsetsRandSoneorzerotimes.ThecombinationsofnumberofoccurrencesfortupletinRandSrespectivelyare（0,0）,（0,1）,（1,0）,and（1,1）.OnlywhentupletappearsinatleastoneofthesetsRandSwilltheintersectionR∩Shavethetuplet.ThesamereasoningholdswhenwetaketheintersectionS∩R.

Thereforethecommutativelawforintersectionholds.

Exercise5.1.4f

Forbags:

SupposeatupletoccursntimesinbagRandtupleuoccursmtimesinbagS.Supposealsothatthetwotuplest,ucansuccessfullyjoin.TheninthenaturaljoinofthesetwobagsR

S,thejoinedtuplewouldappearnmtimes.LikewiseinthenaturaljoinofthesetwobagsS

R,thejoinedtuplewouldappearmntimes.Bothsidesoftherelationyieldthesameresult.

Forsets:

Anarbitrarytupletcanonlyappearatmostonetimeinanyset.Tuplesu,vmightappearrespectivelyinsetsRandSoneorzerotimes.Thecombinationsofnumberofoccurrencesfortuplesu,vinRandSrespectivelyare（0,0）,（0,1）,（1,0）,and（1,1）.OnlywhentupleuexistsinRandtuplevexistsinSwillthenaturaljoinR

Shavethejoinedtuple.ThesamereasoningholdswhenwetakethenaturaljoinS

R.

Thereforethecommutativelawfornaturaljoinholds.

Exercise5.1.4g

Forbags:

SupposetupletappearsmtimesinRandntimesinS.IfwetaketheunionofRandSfirst,wewillgetarelationwheretupletappearsm+ntimes.TakingtheprojectionofalistofattributesLwillyieldaresultingrelationwheretheprojectedattributesfromtupletappearm+ntimes.IfwetaketheprojectionoftheattributesinlistLfirst,thentheprojectedattributesfromtupletwouldappearmtimesfromRandntimesfromS.Theunionoftheseresultingrelationswouldhavetheprojectedattributesoftupletappearm+ntimes.

Forsets:

Anarbitrarytupletcanonlyappearatmostonetimeinanyset.TupletmightappearinsetsRandSoneorzerotimes.ThecombinationsofnumberofoccurrencesfortupletinRandSrespectivelyare（0,0）,（0,1）,（1,0）,and（1,1）.OnlywhentupletexistsinRorS（orbothRandS）willtheprojectedattributesoftupletappearintheresult.

Thereforethelawholds.

Exercise5.1.4h

Forbags:

SupposetupletappearsutimesinR,vtimesinSandwtimesinT.Onthelefthandside,theintersectionofSandTwouldproducearesultwheretupletwouldappearmin（v,w）times.WiththeadditionoftheunionofR,theoverallresultwouldhaveu+min（v,w）copiesoftuplet.Ontherighthandside,wewouldgetaresultofmin（u+v,u+w）copiesoftuplet.Theexpressionsonboththeleftandrightsidesareequivalent.

Forsets:

Anarbitrarytupletcanonlyappearatmostonetimeinanyset.TupletmightappearinsetsR,SandToneorzerotimes.ThecombinationsofnumberofoccurrencesfortupletinR,SandTrespectivelyare（0,0,0）,（0,0,1）,（0,1,0）,（0,1,1）,（1,0,0）,（1,0,1）,（1,1,0）and（1,1,1）.OnlywhentupletappearsinRorinbothSandTwilltheresulthavetuplet.

Thereforethedistributivelawofunionoverintersectionholds.

Exercise5.1.4i

SupposethatinrelationR,utuplessatisfyconditionCandvtuplessatisfyconditionD.SupposealsothatwtuplessatisfybothconditionsCandDwherew≤min（v,w）.Thenthelefthandsidewillreturnthosewtuples.Ontherighthandside,σC（R）producesutuplesandσD（R）producesvtuples.However,weknowtheintersectionwillproducethesamewtuplesintheresult.

Whenconsideringbagsandsets,theonlydifferenceisbagsallowduplicatetupleswhilesetsonlyallowonecopyofthetuple.Theexampleaboveappliestobothcases.

Thereforethelawholds.

Exercise5.1.5a

Forsets,anarbitrarytupletappearsonthelefthandsideifitappearsinbothR,SandnotinT.Thesameistruefortherighthandside.

Asanexampleforbags,supposethattupletappearsonetimeeachinbothR,TandtwotimesinS.Theresultofthelefthandsidewouldhavezerocopiesoftupletwhiletherighthandsidewouldhaveonecopyoftuplet.

Thereforethelawholdsforsetsbutnotforbags.

Exercise5.1.5b

Forsets,anarbitrarytupletappearsonthelefthandsideifitappearsinRandeitherSorT.ThisisequivalenttosayingtupletonlyappearswhenitisinatleastRandSorinRandT.Theequivalenceisexactlytherightside’sexpression.

Asanexampleforbags,supposethattupletappearsonetimeinRandtwotimeseachinSandT.Thenthelefthandsidewouldhaveonecopyoftupletintheresultwhiletherighthandsidewouldhavetwocopiesoftuplet.

Thereforethelawholdsforsetsbutnotforbags.

Exercise5.1.5c

Forsets,anarbitrarytupletappearsonthelefthandsideifitsatisfiesconditionC,conditionDorbothconditionCandD.Ontherighthandside,σC（R）selectsthosetuplesthatsatisfyconditionCwhileσD（R）selectsthosetuplesthatsatisfyconditionD.However,theunionoperatorwilleliminateduplicatetuples,namelythosetuplesthatsatisfybothconditionCandD.Thusweareensuredthatbothsidesareequivalent.

Asanexampleforbags,weonlyneedtolookattheunionoperator.IfthereareindeedtuplesthatsatisfybothconditionsCandD,