8.1SupposethatwedecomposetheschemaR=(A,B,C,D,E)into
(A,B,C)
(A,D,E).
Showthatthisdecompositionisalossless-joindecompositionifthe
followingsetFoffunctionaldependenciesholds:
A→BC
CD→E
B→D
E→A
Answer:Adecomposition{R1,R2}isalossless-joindecompositionif
R1∩R2→R1orR1∩R2→R2.LetR1=(A,B,C),R2=
(A,D,E),andR1∩R2=A.SinceAisacandidatekey(seePractice
Exercise8.6),ThereforeR1∩R2→R1.
8.2ListallfunctionaldependenciessatisedbytherelationofFigure8.17.
Answer:Thenontrivialfunctionaldependenciesare:A→Band
C→B,andadependencytheylogicallyimply:AC→B.Thereare
19trivialfunctionaldependenciesoftheform→,where.C
doesnotfunctionallydetermineAbecausetherstandthirdtupleshave
thesameCbutdifferentAvalues.ThesametuplesalsoshowBdoesnot
functionallydetermineA.Likewise,Adoesnotfunctionallydetermine
CbecausethersttwotupleshavethesameAvalueanddifferentC
values.ThesametuplesalsoshowBdoesnotfunctionallydetermineC.
8.3Explainhowfunctionaldependenciescanbeusedtoindicatethefol-
lowing:
9
10Chapter8RelationalDatabaseDesign
Aone-to-onerelationshipsetexistsbetweenentitysetsstudentand
instructor.
Amany-to-onerelationshipsetexistsbetweenentitysetsstudent
andinstructor.
Answer:LetPk(r)denotetheprimarykeyattributeofrelationr.
ThefunctionaldependenciesPk(student)→Pk(instructor)and
Pk(instructor)→Pk(student)indicateaone-to-onerelationship
becauseanytwotupleswiththesamevalueforstudentmusthave
thesamevalueforinstructor,andanytwotuplesagreeingon
instructormusthavethesamevalueforstudent.
ThefunctionaldependencyPk(student)→Pk(instructor)indicatesa
many-to-onerelationshipsinceanystudentvaluewhichisrepeated
willhavethesameinstructorvalue,butmanystudentvaluesmay
havethesameinstructorvalue.
8.4UseArmstrong’saxiomstoprovethesoundnessoftheunionrule.(Hint:
Usetheaugmentationruletoshowthat,if→,then→.Applythe
augmentationruleagain,using→,andthenapplythetransitivity
rule.)
Answer:Toprovethat:
if→and→then→
Followingthehint,wederive:
→given
→augmentationrule
→unionofidenticalsets
→transitivityruleandsetunioncommutativity
8.5UseArmstrong’saxiomstoprovethesoundnessofthepseudotransitiv-
ityrule.
Answer:ProofusingArmstrong’saxiomsofthePseudotransitivityRule:
if→and→,then→.
→augmentationruleandsetunioncommutativity
→transitivityrule
8.6ComputetheclosureofthefollowingsetFoffunctionaldependencies