GATE CS 2002 Q2.25 — Normal Forms

To determine which options are correct, we must understand a fundamental property of functional dependencies (FDs):

• A functional dependency $X \rightarrow Y$ is a semantic constraint that must hold for all possible instances of a relation schema.
• A single relation instance can disprove (falsify) a functional dependency if we find a violation.
• However, a single relation instance can never prove that a functional dependency holds for the schema, as future valid instances might violate it.

Let's analyze the given instance of relation $R(A, B, C)$:

1. Testing $B \rightarrow C$:

Look at the first two tuples:

• Tuple 1: $(1, 1, 1)$
• Tuple 2: $(1, 1, 0)$

Here, the $B$ values are identical ($1$), but the corresponding $C$ values are different ($1$ and $0$). This is a direct violation of the dependency $B \rightarrow C$.

Therefore, we can conclusively state that $B$ does not functionally determine $C$ ($B \not\rightarrow C$).

2. Testing $A \rightarrow B$:

In this specific instance, for every identical value of $A$, the values of $B$ are also identical:

• For $A = 1$, $B = \{1, 1\}$
• For $A = 2$, $B = \{3, 3\}$

While the dependency $A \rightarrow B$ is satisfied by this specific instance, we cannot conclude that $A \rightarrow B$ holds as a schema-level constraint, because a future instance might violate it.

Conclusion:

• We can definitely conclude that $B$ does not functionally determine $C$ (Option C).
• We cannot conclude that $A$ functionally determines $B$, which makes Option B incorrect.

Thus, the correct option is C.