It is well known from relational database theory , that the schema R(A, B, C) is then in 3NE but is not in BCNE As a result, instances of R(A, B, C) have redundancies, but decomposition the schema into BCNF leads to two schemas RI(C, A) and R2(C, B), which are free of redundancy but do not preserve dependency A, B [right arrow] C.
The main task in relational schema normalization is producing such a set of schemas that posses the required form, usually 3NF or BCNF. The normalization process consists in decomposition of a given input schema.
A schema is in BCNF if only the first condition of the two above is allowed.
* A relation in 3NF + there is only one primary key within the relation.
As a corollary of this result we are able to show that when F satisfies the intersection property, then it also satisfies the split-freeness property, i.e., is monodependent, if and only if every lossless join decomposition of R, which is in BCNF, is also dependency preserving.
A lossless join decomposition of R, which is in BCNF, is said to be optimum if it has the smallest possible size.
In Section 9, we show that, when a set of FDs F over R satisfies the intersection property, it also satisfies the split-freeness property (i.e., is monodependent), if and only if every lossless join decomposition of R is also dependency preserving, and show that a unique, optimum, lossless join decomposition of R, which is in BCNF and is also dependency preserving, can be obtained in polynomial time in the size of F.
Later, Markowitz and Shoshani  proposed a precise design approach that produces BCNF relations for an entity-relationship schema.
In this way, our approach does not produce objects that are equivalent to the 3NF, BCNF, or 4NF (if an object is seen as a relation), but produces an object that derives the user's GD-constraints.
In addition to being free of the redundancy-related problems cited, the decomposition has several very desirable properties: (1) it is lossless in the sense that the original relation can be reconstructed by taking the natural join of the two projections and, thus, no information is lost; (2) the decomposition preserves the FDs in the sense that the FDs associated with the schemes in the decomposition are equivalent (identical in our example) to those associated with the original scheme; and (3) the two schemes are in Boyce-Codd normal form (BCNF), the strongest normal form in terms of FDs as the only dependencies.
To remove data redundancy, if we use the FDs corresponding to constraints 1 and 2, we obtain the following BCNF decomposition for COURSES: (Course#, Credits) and (Course#, Lab-Assist, Lab-Assist-Pay, #Students-in-Class, Day).
The central part of the article gives the definitions of temporal BCNF (TBCNF) and temporal 3NF (T3NF) and algorithms for achieving TBCNF and T3NF decompositions.