Functional Dependency Closure Quiz

The closure of attribute set {A} includes all attributes that can be functionally determined from A. Starting with A, the dependency A → B adds B, then B → C adds C, and A → D adds D. Thus, the closure of {A} is {A, B, C, D}, encompassing all attributes in the relation.

In Armstrong's axioms, the reflexivity rule asserts that any set of attributes can determine any subset of itself. This means that if you have a complete set of attributes, it inherently contains the information needed to determine any smaller selection of those attributes.

Attribute closure refers to the set of attributes that can be functionally determined by a given set of attributes X based on a set of functional dependencies F. It includes not only the attributes in X but also any additional attributes that can be derived through the dependencies, thus accurately reflecting all attributes determined by X.

To compute the closure of {A, C}, we start with the attributes A and C. Using the dependency A → B, we can add B to the closure. Then, with B → C, C is already included, and using C → D, we can add D. Thus, the closure becomes {A, B, C, D}.

A candidate key must ensure that each value is unique, identifying a single record in a database table (uniqueness). Additionally, it must be minimal, meaning that no subset of the key can uniquely identify records, ensuring that the key is as concise as possible while still maintaining its uniqueness.

Transitivity is a fundamental principle in logic and mathematics that states if one element relates to a second, and that second element relates to a third, then the first element must also relate to the third. In this case, if X leads to Y and Y leads to Z, it follows that X leads to Z.

Augmentation is an Armstrong axiom that states if a functional dependency X → Y holds, then for any attribute set Z, the dependency XZ → YZ also holds. This allows the addition of attributes to both sides of the functional dependency while maintaining its validity, thus facilitating the expansion of dependencies in database design.

A relation is in Boyce-Codd Normal Form (BCNF) when every determinant, which is an attribute or set of attributes that can uniquely determine other attributes, must be a candidate key. This ensures that there are no partial or transitive dependencies, thus eliminating redundancy and maintaining data integrity in the database.

A is a candidate key because it uniquely identifies each tuple in the relation R. Given the functional dependencies A → B and B → C, A not only determines B but also indirectly determines C through B. Thus, A can uniquely identify all attributes in the relation, fulfilling the requirements of a candidate key.

The decomposition rule relies on the augmentation axiom, which allows us to add attributes to both sides of a functional dependency, and the reflexivity axiom, which states that if Y is a subset of X, then X → Y holds. Together, they support the breakdown of X → YZ into X → Y and X → Z.

Two sets of functional dependencies, F and G, are considered equivalent if they yield the same closure for any attribute set. This means that the implications of the dependencies in both sets lead to the same conclusions about attribute relationships, ensuring that they provide the same information regarding the database schema.

Finding all minimal sets whose closure equals all attributes directly identifies candidate keys by determining the smallest combinations of attributes that can uniquely identify all other attributes in the relation. This method ensures that only essential attributes are considered, streamlining the process of identifying candidate keys effectively.