GATE CS 2007 Q62 — Normal Forms

To determine which statement is FALSE, let us analyze each option systematically:

• Option A: Any relation with two attributes is in BCNF.

  Let $R(A, B)$ be a relation with two attributes. The only possible non-trivial functional dependencies (FDs) are $A \to B$ or $B \to A$.

  • If $A \to B$ holds, then $A$ is a candidate key (and thus a superkey).
  • If $B \to A$ holds, then $B$ is a candidate key (and thus a superkey).
  • If both hold, both are superkeys.
  • If neither holds, there are no non-trivial FDs, which vacuously satisfies BCNF.

  In all cases, for every non-trivial FD $X \to Y$, the left-hand side $X$ is a superkey. Thus, any binary relation is always in BCNF. This statement is TRUE.

• Option B: A relation in which every key has only one attribute is in 2NF.

  A relation is in 2NF if there is no partial dependency (i.e., no proper subset of a candidate key functionally determines a non-prime attribute). If every candidate key consists of only a single attribute, then no candidate key has any proper non-empty subset. Consequently, a partial dependency cannot exist. Thus, the relation is guaranteed to be in 2NF. This statement is TRUE.

• Option C: A prime attribute can be transitively dependent on a key in a 3NF relation.

  By definition, a relation is in 3NF if for every non-trivial FD $X \to Y$, either $X$ is a superkey or $Y$ is a prime attribute.

  Consider the relation $R(A, B, C)$ with FDs: $$A \to B, \quad B \to C, \quad C \to A$$ The candidate keys are $A$, $B$, and $C$. All attributes are prime attributes. Here, $C$ is transitively dependent on key $A$ via $B$ ($A \to B$ and $B \to C$, where $B \not\to A$). Since $C$ is a prime attribute, the FD $B \to C$ satisfies the 3NF condition. Thus, a prime attribute can be transitively dependent on a key in 3NF. This statement is TRUE.

• Option D: A prime attribute can be transitively dependent on a key in a BCNF relation.

  In BCNF, for every non-trivial FD $X \to Y$, $X$ must be a superkey.

  For a transitive dependency $K \to Y \to Z$ to exist (where $K$ is a candidate key), we must have $Y \to Z$ such that $Y \not\to K$. However, in BCNF, since $Y \to Z$ is a non-trivial FD, $Y$ must be a superkey. If $Y$ is a superkey, then $Y \to K$ must hold, which contradicts the condition for transitive dependency ($Y \not\to K$).

  Therefore, no attribute (prime or non-prime) can be transitively dependent on a key in a BCNF relation. This statement is FALSE.