Question

Prove that any relation schema with two attributes is in BCNF.

Short answer

The proof that any relation schema with two attributes is always in BCNF can be done by checking all possible non-trivial functional dependencies. In all cases (A -> B and B -> A), we find that the determinant of the dependency is a superkey, which complies with the requirements of the BCNF, hence the schema is in BCNF.

Step-by-step solution

Analyzing A -> B functional dependency

If A -> B, then A must be a superkey. In other words, using A we can uniquely identify all other attributes in our relation schema. Since there is only one other attribute, B, then A is a superkey and our schema is in BCNF.

Analyzing B -> A functional dependency

Similarly with the previous step, if B -> A, then B must be a superkey. In other words, using B we can uniquely identify all other attributes in our relation schema. Since there is only one other attribute, A, then B is a superkey and our schema is in BCNF.

Concluding proof

Considering all possible dependencies for a relation schema with two attributes, we can conclude that it is always in BCNF because all non-trivial functional dependencies comply with the requirements of the BCNF. The superkey condition is satisfied automatically due to the limited number of attributes.

Key concepts

Relation Schema

A relation schema can be thought of as a blueprint for a table in a database. It is a way to define the structure of data, which includes specifying the names of each attribute (or column) and the data type that each attribute can contain. In a sense, a relation schema is a set of rules that determines the shape and the contents of the actual data in the database.

For example, consider a simple relation schema for a table with only two attributes, say 'StudentID' and 'StudentName'. The schema doesn't contain the data itself but specifies that the 'StudentID' attribute is likely to be a unique identifier (such as an integer), and the 'StudentName' is a string representing the name of a student.

When discussing a relation schema, it's important to understand its properties in the context of various normalization forms, particularly BCNF, which is designed to eliminate redundancy and prevent anomalies in database operations.

Functional Dependency

A functional dependency is a constraint between two sets of attributes in a database. It is an expression of the relationship between attributes in which one attribute (or a group of attributes) uniquely determines another attribute. This relationship describes the property that, if two tuples (rows) of the relation have the same value for one set of attributes, they must also have the same value for another set of attributes.

For a functional dependency to hold, say A -> B, every time we encounter a particular value of A in the relation, it should be associated with the same value of B. This means that A functionally determines B. In databases, functional dependencies help us to understand which attribute can serve as a unique identifier for tuples, and they play a major role in normalization processes that help to optimize the database design to reduce redundancy and avoid update anomalies.

Superkey

A superkey is an attribute or a set of attributes that can be used to uniquely identify a tuple within a relation. It is a critical concept in the normalization of databases, particularly in defining what constitutes a candidate key or a primary key for a table. The uniqueness property means that no two distinct tuples (rows) can have the same value for a superkey.

However, a superkey might include additional attributes that are not necessary for uniqueness - this where the idea of a 'minimal' comes into play, leading to the concept of a candidate key. Being a more refined version of a superkey, a candidate key has no superfluous attributes; in other words, every attribute in a candidate key is necessary for uniqueness.

Understanding superkeys is essential when one considers the Boyce-Codd Normal Form (BCNF) because for a relation schema to be in BCNF, each of its functional dependencies X -> Y should have X as a superkey, which effectively ensures that every attribute in the schema is fully functionally dependent on a key.