Question

Consider a relation with attributes X, Y, Z, W where the only CK is X, and the FDs are X Y, X Z, X W and Y Z. Is there a transitive dependency

Short answer

Yes, there is a transitive dependency in the given relation. Specifically, Z is transitively dependent on X through Y.

Step-by-step solution

Define Transitive Dependency

A transitive dependency in a database is an indirect relationship that causes redundancy. It's a functional dependency which holds by virtue of transitivity among three sets of attributes. A dependency A -> C is transitive if there is an attribute set B such that, A -> B and B -> C hold and neither A -> C nor B -> C are trivial functional dependencies.

Examine Given Functional Dependencies

The functional dependencies given are X -> Y, X -> Z, X -> W, and Y -> Z. Since X is the only candidate key, it is used to determine all other attributes. Y -> Z is determined by an attribute other than X.

Identify Transitive Dependency

From the given dependencies, we can see that X -> Y and Y -> Z. This means Z is transitively dependent on X through Y. Therefore, a transitive dependency exists in the given relation.

Key concepts

Functional Dependency

In a database, functional dependency is a critical concept. It defines the relationship between different attributes. Specifically, if you have two attributes, A and B, we say that B is dependent on A (denoted as A → B) if each value of A is associated with exactly one value of B. This means that knowing the value of A gives us the value of B, creating a predictable pattern.

Functional dependencies are used to determine whether a database is well-structured. They help identify redundancies and potential anomalies in data, guiding the normalization process. For example, in the given relation involving attributes X, Y, Z, and W, several functional dependencies are present: X → Y, X → Z, X → W, and Y → Z. These dependencies suggest how changes in one attribute might affect others, acting as rules within the dataset.

Candidate Key

A candidate key is a minimal set of attributes that uniquely identify a tuple (record) in a relation. Among all possible options, it is the smallest subset that suffices for uniqueness. The existence of a candidate key ensures that no duplicate records exist within a relation. In database design, identifying the correct candidate key is crucial, as it helps in maintaining integrity and preventing redundancy.

In the problem given, X is defined as the only candidate key. This means that the attribute X alone is sufficient to distinguish each record uniquely in the relation. No other combinations of the attributes are necessary or smaller than X to ensure such uniqueness. The function X has over other attributes in the set, such as Y, Z, and W, underscores its importance in the database schema.

Transitive Dependency

Transitive dependency occurs in a relational database when there is an indirect relationship between attributes. It can lead to redundancy and hence, is an important consideration during normalization. To be precise, a transitive dependency is present when an attribute C is functionally dependent on attribute A, but only through another attribute B. So if A → B and B → C, then a transitive dependency A → C exists, provided neither A → C nor B → C is trivial.

In our exercise, examining the functional dependencies: X → Y and Y → Z reveal a transitive relationship. Specifically, Z depends transitively on X via Y; X determines Y, and Y in turn determines Z. Therefore, Z's dependency on X is not direct but occurs through Y, constituting a classic case of transitive dependency. Awareness of such dependencies is vital to minimize redundancy and optimize the database structure.