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Database Relations | Heighest normal form

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A relation R(A,B,C,D) has only trivial functional dependencies of the form ( A→A, AB→AB,ABC→A, etc)

Then consider the following options:

  1. The relation is surely in BCNF
  2. The relation is surely in 3NF
  3. The relation is surely in 2NF
  4. None of the above

 

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If no valid FDs holds (a part from the trivial dependencies), then the only candidate key is ABCD, and the relation is all in 1NF, 2NF, 3NF and BCNF.
Hence A,B and C are all correct.

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Option 1 is correct. A relation is in BCNF (Boyce-Codd Normal Form) if and only if every determinant in the relation is a candidate key. In other words, if a functional dependency X → Y holds in a relation and X is not a superkey, then the relation is not in BCNF.

In this case, the functional dependencies given in the question are all trivial, meaning that they are of the form X → X or X → XY, where X is a subset of the attributes in the relation. These functional dependencies do not violate the BCNF condition because the determinant (X) is always a superkey. Therefore, the relation is surely in BCNF.

Option 2 (The relation is surely in 3NF) is not necessarily true because a relation can be in 3NF without being in BCNF. 3NF (Third Normal Form) is a less strict normal form than BCNF and requires that all non-prime attributes in the relation are fully functionally dependent on the relation's primary key.

Option 3 (The relation is surely in 2NF) is also not necessarily true because a relation can be in 2NF without being in BCNF. 2NF (Second Normal Form) is a less strict normal form than BCNF and requires that all non-prime attributes in the relation are fully functionally dependent on the relation's primary key.

Option 4 (None of the above) is not correct because one of the options (Option 1) is correct.
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