If a relational table has been normalized so that every determinant is a candidate key, then in which normal form is the table?

Introduction / Context:
Database normalization is a step by step process used to design relational tables that minimize redundancy and update anomalies. Each normal form adds extra rules on top of the previous one. Boyce Codd normal form, often written as BCNF, is a stronger version of third normal form that focuses on determinants. A determinant is any attribute or set of attributes on which some other attribute is fully functionally dependent. This question asks you to connect the definition of BCNF with the idea that every determinant in a table must be a candidate key.

Given Data / Assumptions:

We have a single relational table that has already been normalized to at least third normal form.
Every functional dependency in the table has been identified and analyzed.
Every determinant in the table is a candidate key, not just a superkey that contains extra attributes.
There are no multi valued dependencies or other advanced anomalies discussed beyond classical functional dependencies.
The terms first normal form, second normal form, third normal form, and Boyce Codd normal form use their standard textbook meanings.

Concept / Approach:
First normal form removes repeating groups and ensures atomic values. Second normal form removes partial dependencies of non key attributes on part of a composite primary key. Third normal form removes transitive dependencies, so that non key attributes do not depend on other non key attributes. Boyce Codd normal form goes further. A table is in BCNF if, for every nontrivial functional dependency X → Y that holds in the relation, X is a superkey. A candidate key is a minimal superkey, meaning it has no unnecessary attributes. The condition that all determinants are candidate keys is an even stronger statement that directly matches the essence of BCNF, and in many exam questions it is used as a shortcut definition.

Step-by-Step Solution:
Step 1: Recall that in a table that is in third normal form, certain dependencies with non key attributes on other non key attributes are not allowed. However, 3NF still allows some dependencies where a determinant is a superkey but not necessarily a candidate key. Step 2: Recall the formal definition of BCNF, which states that for every functional dependency X → Y in the relation, X must be a superkey. In practice, we usually think of X as a candidate key, since including extra attributes in X is unnecessary. Step 3: Match the statement in the question. If every determinant is a candidate key, then every X that determines other attributes is at least a candidate key, and therefore a superkey. This exactly matches the BCNF requirement that all determinants be keys. Step 4: Conclude that the table is in Boyce Codd normal form and choose the corresponding option.

Verification / Alternative check:
Consider an example table with attributes (StudentId, CourseId, Instructor). Suppose the functional dependency is (StudentId, CourseId) → Instructor, and no smaller set of attributes determines all other attributes. Then (StudentId, CourseId) is a candidate key and the only determinant. This table is in BCNF because the only determinant is a candidate key. If there were another functional dependency such as Instructor → CourseId, and Instructor was not a candidate key, then Instructor would be a determinant that is not a key, and the table would not be in BCNF. This example shows that the condition in the question is precisely the BCNF condition.

Why Other Options Are Wrong:
First normal form is too weak. It requires only that attribute domains be atomic and that there be no repeating groups. It says nothing about determinants or candidate keys.
Second normal form deals with partial dependencies on part of a composite primary key. It does not require that every determinant be a candidate key, so it is less restrictive than the condition in the question.
Third normal form removes transitive dependencies but still allows some determinants that are not candidate keys, as long as certain additional conditions hold, so it does not fully match the statement.

Common Pitfalls:
Learners often confuse the definitions of 3NF and BCNF and treat them as equivalent, but BCNF is stricter. Another common mistake is to think that as long as a table is in 3NF, all design problems are solved, even though some anomalies can remain if non key determinants exist. Students may also forget the difference between a candidate key and a superkey and may think that any determinant that contains a key is acceptable, which misses the minimality condition. Remembering that BCNF is characterized by every determinant being a key helps clarify these distinctions.

Final Answer:
If every determinant is a candidate key, the table is in Boyce Codd normal form (BCNF).