Question-19

In a relational data model, which one of the following statements is TRUE?

• A relation with only two attributes is always in BCNF.
• If all attributes of a relation are prime attributes, then the relation is in BCNF.
• Every relation has at least one non-prime attribute.
• BCNF decompositions preserve functional dependencies.

Answer

• A relation with only two attributes is always in BCNF.
• If all attributes of a relation are prime attributes, then the relation is in BCNF.
• Every relation has at least one non-prime attribute.
• BCNF decompositions preserve functional dependencies.

Solution

1. A relation with two attributes always in BCNF: True. In BCNF, every non-trivial functional dependency \(x \rightarrow y\) implies \(x\) is a super key. For relations with two attributes, all non-trivial functional dependencies would indeed have the left side (\(x\)) as a super key.

2. If all attributes of a relation are prime attributes, then the relation is in BCNF: False. It’s possible for prime attributes to determine each other, which is allowed in 3NF but not in BCNF.

3. Every relation has at least one non-prime attribute: False. It’s not mandatory for a relation to have at least one non-prime attribute. However, every relation has at least one prime attribute.

4. BCNF decompositions preserve functional dependencies: False. BCNF decompositions might lose some functional dependencies, as exemplified by the case where \(D \rightarrow A\) in a relation \([AB \rightarrow CD, D \rightarrow A]\) where \(D\) is not a super key, and it’s not in BCNF.

Therefore, the only statement that holds true is (a) - a relation with only two attributes is always in BCNF.