GATE CS 2000 Q2.24 — Normal Forms

A functional dependency $A \rightarrow B$ is satisfied by a relation instance if and only if for any two tuples $t_1$ and $t_2$ in the relation, $t_1[A] = t_2[A]$ implies $t_1[B] = t_2[B]$. If all values of the determinant attribute $A$ are unique in the instance, then $A \rightarrow B$ is trivially satisfied.

Let's evaluate the given relation instance:

Let's check the functional dependencies in Option B:

1. Checking $YZ \rightarrow X$:

   The values of the composite attribute $YZ$ for each row are:

   • Row 1: $(4, 2) \rightarrow 1$
   • Row 2: $(5, 3) \rightarrow 1$
   • Row 3: $(6, 3) \rightarrow 1$
   • Row 4: $(2, 2) \rightarrow 3$

   Since all $YZ$ pairs are unique, $YZ \rightarrow X$ is satisfied.

2. Checking $Y \rightarrow Z$:

   The values of $Y$ are $4, 5, 6,$ and $2$. Since all values of $Y$ are unique, $Y \rightarrow Z$ is trivially satisfied.

Thus, both $YZ \rightarrow X$ and $Y \rightarrow Z$ are satisfied by the relation instance.

Why other options are incorrect:

• Option A: $Z \rightarrow Y$ is violated because $Z = 2$ maps to both $Y = 4$ (Row 1) and $Y = 2$ (Row 4).
• Option C: $X \rightarrow Z$ is violated because $X = 1$ maps to both $Z = 2$ (Row 1) and $Z = 3$ (Row 2).
• Option D: $XZ \rightarrow Y$ is violated because $XZ = (1, 3)$ maps to both $Y = 5$ (Row 2) and $Y = 6$ (Row 3).