Problem
• Let R=ABCDEG
• Let F={C->D, E->C, EG->A, G->B}
• What are all the keys of R? EG
• What is the normal form of R? not in 3NF, not
BCNF
• Is the following decomposition lossless? Yes
– R1=AEG, R2=CE, R3=BG, R4=DEG
• Is the decomposition dependency preserving? No.
Does not preserve C->D
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Problem
• Given R = (A,B,C,D,E,G) and F = {A  E,
B  G, CD  B, EB  CD, G  ACE}
1. Find all keys of R. B, CD, GD
2. What is the normal form of R? not 3NF, not BCNF
3. Find a minimal cover for R A->E, B->G, CD->B, B->D,
G->A, G->C
4. Find a 3NF decomposition of R that preserves all functional
dependencies and is lossless AE, BG, CDB, GA, GC
5. What is the normal form of each relation in the result?
All in BCNF. For CDB, we have the minimal cover:
CD->B, B->CD
2
Problem
• Given R = (A,B,C,D,E) and F = {A  B,
AC  D, CE  D, ED  B, AE  C, B  A}
• We decompose R to R1=(A,B,D,E) and
R2=(C,D,E)
• Find a minimal cover of FR1. What is the normal
form of R1? A->B, B->A, AE->D, ED->A, 3NF
• Find a minimal cover of FR2. What is the normal
form of R2? CE->D, ED->C, BCNF
• Is the decomposition lossless? Dependency
preserving? Lossless, not dependency preserving
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Problem
• Prove or give a counter example
– (X+)+ = X+, True. Easy to see by considering the
algorithm for computing the closure.
– (X  Y)+ = X +  Y+ Wrong. F = {A->C, B->C}, X
= A, Y = B
– (X  Y)+ = X +  Y+ Wrong. F = {AB->C}, X = A,
Y=B
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Problem
• R=ABCDEGHI
• In each of the following we are given F and a subset of R.
For each sub-relation, (1) state the normal form (2) if it is
not in BCNF, decompose to a collection of BCNF relations
– R1=ABCD, F= {AE, EB, C D} Not in 3NF. Decomposition to
AB, BC, CD
– R2=ABG, F={ACE, BG} Not in 3NF: Decomposition to BG, AB
– R3=AD, F={DG, GH} In BCNF
– R4=DCHG, F={D I, I C} Not in 3NF: Decomposition to DC, DHG
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Problem
• Show that the algorithm for computing a
BCNF decomposition may return different
results depending on the order in which it
runs, or prove that this is incorrect.
• Counter example, R= ABC, F ={A->B, C->B}
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Problem
• Consider a relation R=ABCDE with F = {ABC,
BC  D, E  AD, D  B, CD  E}
– Find all the keys for R.
E, CD, BC, AB, AD
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