51. Let r be a relation instance with schema R = (A, B, C, D). We define r1 = ΠA, B, C (r) and r2 = ΠA.D (r). Let s = r1 * r2 where * denotes natural join. Given that the decomposition of r into r1 and r2 is lossy, which one of the following is TRUE?
a. s ⊂ r
b. r ∪ s
c. r ⊂ s
d. r * s = s
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Answer: (c).r ⊂ s

52. Consider a relation scheme R = (A, B, C, D, E, H) on which the following functional dependencies hold: {A–>B, BC–>D, E–>C, D–>A}. What are the candidate keys of R?
a. AE, BE
b. AE, BE, DE
c. AEH, BEH, BCH
d. AEH, BEH, DEH
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Answer: (d).AEH, BEH, DEH

53. The relation scheme Student Performance (name, courseNo, rollNo, grade) has the following functional dependencies:
name, courseNo → grade
rollNo, courseNo → grade
name → rollNo
rollNo → name
The highest normal form of this relation scheme is
a. 2 NF
b. 3 NF
c. BCNF
d. 4NF
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Answer: (b).3 NF

54. Consider the following functional dependencies in a database:

Data_of_Birth → Age
Age → Eligibility
Name → Roll_number
Roll_number → Name
Course_number → Course_name
Course_number → Instructor
(Roll_number, Course_number) → Grade

The relation (Roll_number, Name, Date_of_birth, Age) is:
a. In second normal form but not in third normal form
b. In third normal form but not in BCNF
c. In BCNF
d. None of the above
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Answer: (d).None of the above

55. Relation R with an associated set of functional dependencies, F is decomposed into BCNF. The redundancy (arising out of functional dependencies) in the resulting set relations is.
a. Zero
b. More than zero but less than that of an equivalent 3NF decomposition
c. Proportional to the size of F+
d. Indeterminate
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Answer: (a).Zero

56. With regard to the expressive power of the formal relational query languages, which of the following statements is true?
a. Relational algebra is more powerful than relational calculus
b. Relational algebra has the same power as relational calculus
c. Relational algebra has the same power as safe relational calculus
d. None of the above
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Answer: (c).Relational algebra has the same power as safe relational calculus

57. Relation R is decomposed using a set of functional dependencies, F and relation S is decomposed using another set of functional dependencies G. One decomposition is definitely BCNF, the other is definitely 3NF, but it is not known which is which. To make a guaranteed identification, which one of the following tests should be used on the decompositions? (Assume that the closures of F and G are available).
a. Dependency-preservation
b. Lossless-join
c. BCNF definition
d. 3NF definition
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Answer: (c).BCNF definition

58. From the following instance of a relation scheme R (A, B, C), we can conclude that :
A B C
1 1 1
1 1 0
2 3 2
2 3 2
a. A functionally determines B and B functionally determines C
b. A functionally determines B and B does not functionally determine C
c. B does not functionally determine C
d. A does not functionally determine B and B does not functionally determine C
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Answer: (c).B does not functionally determine C

59. Consider a schema R(A,B,C,D) and functional dependencies A->B and C->D. Then the decomposition of R into R1(AB) and R2(CD) is
a. dependency preserving and lossless join
b. lossless join but not dependency preserving
c. dependency preserving but not lossless join
d. not dependency preserving and not lossless join
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Answer: (c).dependency preserving but not lossless join

60. Suppose the adjacency relation of vertices in a graph is represented in a table Adj(X,Y). Which of the following queries cannot be expressed by a relational algebra expression of constant length?
a. List of all vertices adjacent to a given vertex
b. List all vertices which have self loops
c. List all vertices which belong to cycles of less than three vertices
d. List all vertices reachable from a given vertex
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Answer: (d).List all vertices reachable from a given vertex

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