4NF

Fourth Normal Form (4NF) is a database normalization form that addresses redundancy caused by multivalued dependencies. It extends the ideas of BCNF by focusing on the independent occurrence of multiple values within a single relation.

Formally, a relation R is in 4NF if for every nontrivial multivalued dependency X ->-> Y that holds in R, X is a superkey of R. A multivalued dependency X ->-> Y means that, for each value of X, the set of Y-values is independent of the other attributes, so the appearance of Y values is not constrained by the remaining attributes. Nontrivial means that Y is not empty and Y does not lie entirely within X.

In practice, 4NF eliminates certain types of redundancy that arise when multiple independent relationships share the same record. If R contains a nontrivial MVD X ->-> Y with X not a superkey, R can be decomposed into two relations: R1(X ∪ Y) and R2(X ∪ (R \ (X ∪ Y))). This decomposition preserves data while removing the redundancy caused by the MVD.

Example: Consider a relation R(Student, Course, Hobby) where a student can enroll in many courses and have many hobbies independently. If Student ->-> Course and Student ->-> Hobby hold and Student is not a key for R, the relation is not in 4NF. Decomposing into R1(Student, Course) and R2(Student, Hobby) brings the design into 4NF.

Relation to other normal forms: 4NF is stricter than 3NF/BCNF in that it requires the absence of nontrivial multivalued dependencies not based on a key. In practice, many schemas are designed to meet 4NF only when independent attribute values warrant the stricter separation.