Third normal form

The third normal form (3NF) is a normal form used in database normalization. 3NF was originally defined by E.F. Codd in 1971. Codd's definition states that a table is in 3NF if and only if both of the following conditions hold:


 * The relation R (table) is in second normal form (2NF)
 * Every non-prime attribute of R is non-transitively dependent (i.e. directly dependent) on every key of R.

A non-prime attribute of R is an attribute that does not belong to any candidate key of R. A transitive dependency is a functional dependency in which X → Z (X determines Z) indirectly, by virtue of X → Y and Y → Z (where it is not the case that Y → X).

A 3NF definition that is equivalent to Codd's, but expressed differently, was given by Carlo Zaniolo in 1982. This definition states that a table is in 3NF if and only if, for each of its functional dependencies X → A, at least one of the following conditions holds:


 * X contains A (that is, X → A is trivial functional dependency), or
 * X is a superkey, or
 * A is a prime attribute (i.e., A is contained within a candidate key)

Zaniolo's definition gives a clear sense of the difference between 3NF and the more stringent Boyce-Codd normal form (BCNF). BCNF simply eliminates the third alternative ("A is a prime attribute").

"Nothing but the key"
A memorable summary of Codd's definition of 3NF, paralleling the traditional pledge to give true evidence in a court of law, was given by Bill Kent: every non-key attribute "must provide a fact about the key, the whole key, and nothing but the key." A common variation supplements this definition with the oath: "so help me Codd".

Requiring that non-key attributes be dependent on "the whole key" ensures that a table is in 2NF; further requiring that non-key attributes be dependent on "nothing but the key" ensures that the table is in 3NF.

Chris Date refers to Kent's summary as "an intuitively attractive characterization" of 3NF, and notes that with slight adaptation it may serve as a definition of the slightly-stronger Boyce-Codd normal form: "Each attribute must represent a fact about the key, the whole key, and nothing but the key." The 3NF version of the definition is weaker than Date's BCNF variation, as the former is concerned only with ensuring that non-key attributes are dependent on keys.

Example
An example of a 2NF table that fails to meet the requirements of 3NF is:

Because each row in the table needs to tell us who won a particular Tournament in a particular Year, the composite key {Tournament, Year} is a minimal set of attributes guaranteed to uniquely identify a row. That is, {Tournament, Year} is a candidate key for the table.

The breach of 3NF occurs because the non-prime attribute Winner Date of Birth is transitively dependent on the candidate key {Tournament, Year} via the non-prime attribute Winner. The fact that Winner Date of Birth is functionally dependent on Winner makes the table vulnerable to logical inconsistencies, as there is nothing to stop the same person from being shown with different dates of birth on different records.

In order to express the same facts without violating 3NF, it is necessary to split the table into two:

Update anomalies cannot occur in these tables, which are both in 3NF.

Derivation of Zaniolo's conditions
A lemma proved by Zaniolo states that a table is in 3NF if and only if, for each of its functional dependencies X → A, at least one of the following conditions holds:


 * X contains A, or
 * X is a superkey, or
 * A is a prime attribute (i.e., A is contained within a candidate key)

The lemma is proved in the following way: Let X → A be a nontrivial FD (i.e. one where X does not contain A) and let A be a non-key attribute. Also let Y be a key of R. Then Y → X. Therefore A is not transitively dependent on Y if and only if X → Y, that is, if and only if X is a superkey.

Normalization beyond 3NF
Most 3NF tables are free of update, insertion, and deletion anomalies. Certain types of 3NF tables, rarely met with in practice, are affected by such anomalies; these are tables which either fall short of Boyce-Codd normal form (BCNF) or, if they meet BCNF, fall short of the higher normal forms 4NF or 5NF.