GATE CS 2012 Q.2 (CS Q.2): Functional Dependencies and Normalization

Which of the following is TRUE?

Every relation in 3NF is also in BCNF

A relation R is in 3NF if every non-prime attribute of R is fully functionally dependent on every key of R

Every relation in BCNF is also in 3NF

No relation can be in both BCNF and 3NF

Official answer and solution

Official answer: C
From: Official GATE 2012 answer key

Solution

Recall the normalization hierarchy and definitions of 3NF and BCNF.

In relational database design:

A relation $R$ is in 3NF if for every non-trivial functional dependency $X \rightarrow Y$ , either $X$ is a superkey or $Y$ is a prime attribute.
A relation $R$ is in BCNF if for every non-trivial functional dependency $X \rightarrow Y$ , $X$ is a superkey.

Because the definition of BCNF requires $X$ to be a superkey without the alternative prime-attribute exemption, any relation that satisfies BCNF automatically satisfies 3NF. Thus:

\text{BCNF} \subset \text{3NF} \subset \text{2NF} \subset \text{1NF}

Evaluating the options:

(A) Every relation in 3NF is also in BCNF: FALSE (3NF allows $Y$ to be a prime attribute even if $X$ is not a superkey).
(B) A relation $R$ is in 3NF if every non-prime attribute is fully functionally dependent on every key: FALSE (this defines 2NF, not 3NF).
(C) Every relation in BCNF is also in 3NF: TRUE.
(D) No relation can be in both BCNF and 3NF: FALSE (every BCNF relation is in 3NF).

Therefore, option (C) is correct.

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2012 Computer Science and Information Technology (CS)
Question: 2 of 65, printed as CS Q.2
Section: Computer Science and Information Technology

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