These formal definitions are taken from the book by Chris J. Date: An Introduction to Database Systems Volume 1 4th edition, © 1996, Addison-Wesley Publishing Co., Inc., Reading, Massachusetts.

First Normal Form

‘A relation R is in first normal form (1NF) if and only if all underlying domains contain atomic values only.’

Second Normal Form

‘A relation R is in second normal form (2NF) if and only if it is in 1NF and every nonkey attribute is fully dependent on the primary key.’

Third Normal Form

‘A relation R is in third normal form (3NF) if and only if it is in 2NF and every nonkey attribute is nontransitively dependent on the primary key.’

Boyce/Codd Normal Form

‘A relation R is in Boyce/Codd normal form (BCNF) if and only if every determinant is a candidate key.’

Fourth Normal Form

‘A relation R is in fourth normal form (4NF) if and only if, wherever there exists an MVD in R, say A -> -> B, then all attributes of R are also functionally dependent on A. In other words, the only dependencies (FDs or MVDs) in R are of the form K -> X (i.e. a functional dependency from a candidate key K to some other attribute X). Equivalently: R is in 4NF if it is in BCNF and all MVD’s in R are in fact FDs.’

Fifth Normal Form

‘A relation R is in fifth normal form (5NF) – also called projection-join normal form (PJ/NF) if and only if every join dependency in R is a consequence of the candidate keys of R.’

For every normal form it is assumed that every occurrence of R can be uniquely identified by a primary key using one or more attributes in R.

FD = Functional Dependency
MVD = Multi-Valued Dependency