GATE CS 2002 Q2.24 — Normal Forms

To identify which decomposition is in $\text{BCNF}$ and which is in $\text{3NF}$, we must analyze the relationship between these two normal forms:

• Every relation in $\text{BCNF}$ is also in $\text{3NF}$ (i.e., $\text{BCNF} \subset \text{3NF}$).
• A relation in $\text{3NF}$ is not necessarily in $\text{BCNF}$.

Let us evaluate the tests:

1. Using the $\text{3NF}$ definition test (Option D): Since both $\text{BCNF}$ and $\text{3NF}$ decompositions satisfy the definition of $\text{3NF}$, both decompositions will pass this test. Thus, we cannot distinguish between them.
2. Using the $\text{BCNF}$ definition test (Option C):
   • The decomposition that is in $\text{BCNF}$ will pass the test.
   • The decomposition that is only in $\text{3NF}$ (and not in $\text{BCNF}$) will fail the test.
   This provides a guaranteed identification of both decompositions.
3. Dependency-preservation (Option A) & Lossless-join (Option B): Both $\text{3NF}$ and $\text{BCNF}$ decompositions can be lossless-join. Furthermore, while a $\text{3NF}$ decomposition is always dependency-preserving, a $\text{BCNF}$ decomposition may or may not be dependency-preserving. Hence, these tests cannot guarantee identification.

Therefore, the $\text{BCNF}$ definition test must be used.