Which one of the following statement is false?
2022
Which one of the following statement is false?
- A.
Any relation with two attributes is in BCNF.
- B.
A relation in which every key has only one attribute is in 2NF.
- C.
A prime attribute can be transitively dependent on a key in a 3NF relation.
- D.
A prime attribute can be transitively dependent on a key in a BCNF relation.
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Correct answer: D
(a) In a 2-attribute relation any nontrivial functional dependency A→BA\to BA→B (or B→AB\to AB→A) makes the left side a key (it determines the whole relation). Hence such a relation satisfies BCNF. → (a) true .
(b) If every key is a single attribute there are no composite keys, so no possibility of partial dependency; therefore the relation is in 2NF . → (b) true .
(c) 3NF allows dependencies X→AX\to AX→A when AAA is a prime attribute even if XXX is not a superkey. So a prime attribute may be transitively dependent in a 3NF relation. → (c) true .
(d) BCNF requires every nontrivial X→AX\to AX→A to have XXX as a superkey. A transitive dependency (key → B → primeAttr) implies an intermediate non-key determines the prime attribute, which would violate BCNF (unless that intermediate determinant is also a superkey). So in general (d) is false .