How to reconstruct functional dependencies from 3NF decomposition?
If AD is the only candidate key of R, then we know that in the original relation the dependency AD → BC holds.

But, since the relation is not in BCNF, we know that there are also other dependencies for which the determinant is not a superkey. Given the fact that the decomposition preserves the dependencies, we know that the other dependencies are constituted by attributes that are in the decomposed relations.

The simplest possible case is that there are dependencies A→B and B→C. So the original relation can be:

R&lt;(ABCD), {AD→BC, A→B, B→C}&gt;


In fact, if we apply both the classical BCNF decomposition algorithm as well as the 3NF synthesis algorithm we obtain the specified decomposition.