What is the difference between 3NF and BCNF?

Question

Can someone please explain the difference between 3NF and BCNF to me? It would be great if you could also provide some examples. Thanks.

Answer 1

The difference between 3NF and BCNF is subtle.

3NF

Definition

A relation is in 3NF if it is in 2NF and no non-prime attribute transitively depends on the primary key. In other words, a relation R is in 3NF if for each functional dependency X ⟶ A in R, at least one of the following conditions are met:

1. X is a key or superkey in R
2. A is a prime attribute in R

Example

Given the following relation:

EMP_DEPT(firstName, employeeNumber, dateOfBirth, address, departmentNumber, departmentName)

An employee can only work in one department and each department has many employees.

The candidate key is employeeNumber.

Consider the following functional dependencies:

1. employeeNumber ⟶ firstName, dateOfBirth, address, departmentNumber
2. departmentNumber ⟶ departmentName

Given the definition above, it is possible to conclude that the relation EMP_DEPT is not in 3NF because the second functional dependency does not meet any of the 2 conditions of the 3NF:

1. departmentNumber is not a key or superkey in EMP_DEPT
2. departmentName is not a prime attribute in EMP_DEPT

BCNF

Definition

A relation R is in BCNF if it is in 3NF and for each functional dependency X ⟶ A in R, X is a key or superkey in R. In other words, the only difference between 3NF and BCNF is that in BCNF it is not present the second condition of the 3NF. This makes BCNF stricter than 3NF as any relation that is in BCNF will be in 3NF but not necessarily every relation that is in 3NF will be in BCNF.

Example

Given the following relation:

STUDENT_COURSE(studentNumber, socialSecurityNumber, courseNumber)

A student can assist to many courses and in a course there can be many students.

The candidate keys are:

1. socialSecurityNumber, courseNumber
2. studentNumber, courseNumber

Consider the following functional dependencies:

1. studentNumber ⟶ socialSecurityNumber
2. socialSecurityNumber ⟶ studentNumber

Given the definition above, it is possible to conclude that STUDENT_COURSE is not in BCNF as at least studentNumber is not a key or superkey in STUDENT_COURSE.