Consider the following implications relating to functional and multivalued…
2007
Consider the following implications relating to functional and multivalued dependencies given below, which may or may not be correct.
i. If A ↠ B and A ↠ C then A → BC
ii. If A → B and A → C then A ↠ BC
iii. If A ↠ BC and A → B then A → C
iv. If A → BC and A → B then A ↠ C
Exactly how many of the above implications are valid?
- A.
0
- B.
1
- C.
2
- D.
3
Attempted by 180 students.
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Correct answer: C
Answer: Exactly 2 of the implications are valid (implications ii and iv).
i. A ↠ B and A ↠ C ⇒ A → BC — Not valid. Reason: Multivalued dependencies do not imply functional dependencies. You can have for a fixed A that B and C vary independently (Cartesian product), so A ↠ B and A ↠ C hold while BC is not functionally determined by A.
ii. A → B and A → C ⇒ A ↠ BC — Valid. Reason: Any functional dependency is also a multivalued dependency. From A → B and A → C we get A → BC, and therefore A ↠ BC holds.
iii. A ↠ BC and A → B ⇒ A → C — Not valid. Reason: Even if B is functionally determined by A, A ↠ BC allows C to vary independently for the same A. Example: for A=1 let B be fixed (by A) and let C take multiple values; this satisfies A ↠ BC but not A → C.
iv. A → BC and A → B ⇒ A ↠ C — Valid. Reason: From A → BC we get A → C (decomposition of functional dependency). Any functional dependency implies the corresponding multivalued dependency, so A ↠ C holds.
Conclusion: Exactly 2 implications are valid (ii and iv).