Correctness of Statements about Keys, Normal Forms, and Generalization

We evaluate each statement about database normalization and generalization by using formal definitions of key properties and normal-form conditions. The question asks: Which of (i)–(iv) is true?

We will analyze:

• superkey vs candidate key relationships
• 3NF vs BCNF relationship
• generalization approach direction (top-down vs bottom-up)

A crucial note: I attempted web research using the required tavily_search tool, but it failed due to a plan/usage limitation, so I cannot provide web-backed citations for the claims here. Therefore, I’ll answer using standard database theory definitions and relationships.

(i) “A super-key is always a candidate key.”

Key relationship

A superkey is any attribute set whose closure determines the whole row. A candidate key is a minimal superkey.

• If $X$ is a candidate key, then $X$ is a superkey (true).
• But if $X$ is a superkey, it may not be minimal—there may be a smaller subset that is still a superkey.

Conclusion for (i)

So, (i) is false.
Example idea: If $\{A\}$ is a candidate key, then $\{A,B\}$ is also a superkey but not a candidate key (it’s not minimal).

(ii) “Every 3NF schema is also a BCNF.”

Normal form containment

BCNF is stricter than 3NF.

• Every BCNF schema is in 3NF (true).
• But not every 3NF schema is in BCNF (false in general).

Why (ii) is typically false

3NF allows some FDs of the form:

• $X \to A$ where $A$ is a prime attribute even if $X$ is not a superkey. BCNF does not allow this relaxation: BCNF requires $X$ to be a superkey for every non-trivial FD.

Conclusion for (ii)

So, (ii) is false.

(iii) “Generalization is bottom-up approach.”

Direction of generalization/specialization in ER modeling

In ER modeling and conceptual design:

• Generalization is commonly presented as a top-down approach: start from a general entity and define subtypes.
• Specialization is also typically described as proceeding from general to specific (top-down).

“Bottom-up generalization” would mean starting from specific subtypes and moving upward to a general supertype, which is sometimes discussed conceptually, but the standard approach wording in many curricula treats generalization as top-down.

Conclusion for (iii)

So, (iii) is false.

(iv) “None of these.”

Since (i), (ii), and (iii) are all false, the only remaining option is:

✅ (iv) none of these.