Consider a relational schema \(π‘πππ(ππππ, πππ‘π¦, ππ€πππ)\),β¦
2025
Consider a relational schema \(π‘πππ(ππππ, πππ‘π¦, ππ€πππ)\), with functional dependencies \(\{ππππ β πππ‘π¦, ππππ β ππ€πππ\}\).
The relationΒ \(π‘πππ\) is decomposed into two relations,Β \(π‘1(ππππ, πππ‘π¦)\) and \(π‘2(ππππ, ππ€πππ)\). Which of the following statement(s) is/are TRUE?
- A.
The relationΒ
\(π‘πππ\)is NOT in BCNF. - B.
The relationsΒ
\(t1\)andΒ\(t2\)are in BCNF. - C.
The decomposition constitutes a lossless join.
- D.
The relationΒ
\(π‘πππ \)is NOT in 3NF.
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Correct answer: B, C
Identify the candidate key: name determines both city and owner, so name is a candidate key for the relation.
BCNF for the original relation: Every nonβtrivial functional dependency (name -> city, name -> owner) has the left side equal to a key, so the original relation is in BCNF.
t1 (name, city): name -> city, so name is the key for t1; therefore t1 is in BCNF.
t2 (name, owner): name -> owner, so name is the key for t2; therefore t2 is in BCNF.
Lossless join: The common attribute between the decomposed relations is name, which is a key for the relations. Since the common attribute is a key in at least one relation (here in both), the decomposition is lossless.
3NF: Because the relation is in BCNF, it also satisfies 3NF.
Conclusion: The statements that the decomposed relations are in BCNF and that the decomposition is lossless are true. The statements claiming the original relation is not in BCNF or not in 3NF are false.
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