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McMaster University
SFWRENG 3DB3, Fall 2023
Assignment 3
Due: Dec. 1, 2023 at 10:00pm
November 23, 2023
I. Database Design (65 marks)
Question 1 (12 marks) Keys
Consider a relation schema
R
(
A, B, C, D, E
)
, and the set of functional dependencies
F
=
{
A
→
BC
CD
→
E
B
→
D
E
→
A
}
Find all candidate keys (i.e., minimal keys) of relation
R
. Show all the steps you took to derive
each key, and clearly state which of Armstrong’s axioms are used in each step.
Question 2 (8 marks) Decomposition
Consider the relation R(A,B,C,D,E), and the decomposition of R into R1(ABC) and R2(ADE).
(a) (4 marks) Give a set of functional dependencies (FDs) such that the decomposition into R1 and
R2 is lossless join and dependency preserving. Show your work and explain why your FDs
satisfy the criteria.
(b) (4 marks) Give a set of functional dependencies such that the decomposition into R1 and R2
is not lossless join, but dependency preserving. Show your work and explain why your FDs
satisfy the criteria.
Question 3 (35 marks) Functional Dependencies, BCNF
The Department has a database containing information about all lectures during a term. The
Schedule
relation has the following schema:
1
Schedule
(
C, D, T, R, P, A
)
where
C
represents a course,
D
for day (weekday),
T
for time,
R
for room,
P
for professor, and
A
for head TA. The set of functional dependencies
F
defined over
Schedule
are:
F
=
{
RDT
→
P, RDT
→
C, C
→
A, PDT
→
R, PDT
→
C, CDT
→
P, CDT
→
R
}
Answer the following questions:
(a)
(4 marks) Is the FD
RDT
→
AP
entailed by
F
? Explain and show your work with references
to Armstrong’s axioms.
(b)
(6 marks) Find all the key(s) of relation
Schedule
. Show your work (i.e., how each key is
derived).
(c)
(6 marks) Is
Schedule
in BCNF? If not, decompose it into smaller relations that are each in
BCNF. Show your work at each step.
(d)
(6 marks) We can create a new relation
ProfsSchedule
(
D, T, P
) by projecting some at-
tributes of
Schedule
. Are there new functional dependencies that hold over
ProfsSchedule
?
If so, state these FDs. If not, state why not. In both cases, show your work at each step to justify
your answer.
(e)
(13 marks) Find a minimal cover
F
min
for
F
. Show all the steps in your derivation of
F
min
.
Question 4 (10 marks) Armstrong’s Axioms
Prove the following using Armstrong’s axioms (using only the axioms presented in class). Show all
the steps of your proof, and indicate which of Armstrong’s axioms is applied in each step.
a) (5 marks) Consider the schema R(A,B,C,D,E,F), and the following functional dependencies:
A
→
BCD, BC
→
DE, B
→
D, D
→
A
. Show that
AF
is a superkey.
b) (5 marks) Given the relational schema
R
(
A, B, C, D, E, F
)
and the FDs
F
1
:
{
AB
→
C
,
A
→
D
,
CD
→
EF
}
. Show that
AB
→
F
.
II. Transactions and Concurrency (30 marks)
Question 5 (6 marks) Schedules I
Consider schedules
S
1
,
S
2
below.
State which of the following properties hold (or not) for each
schedule: strict, avoids cascading aborts, recoverability. Provide a brief justification for each answer.
(a) (3 marks)
S
1
:
R
1
(
X
);
R
2
(
Z
);
R
1
(
Z
);
R
3
(
X
);
R
3
(
Y
);
W
1
(
X
);
C
1
;
W
3
(
Y
);
C
3
;
R
2
(
Y
);
W
2
(
Z
);
W
2
(
Y
);
C
2
(b) (3 marks)
S
2
:
R
1
(
X
);
R
2
(
Z
);
R
3
(
X
);
R
1
(
Z
);
R
2
(
Y
);
R
3
(
Y
);
W
1
(
X
);
W
2
(
Z
);
W
3
(
Y
);
W
2
(
Y
);
C
3
;
C
1
;
C
2
2
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