Although the BCNF algorithm ensures that the resulting decomposition is lossless, it is possible to have a schema and a decompsition that was not generated by the algorithm, that is in BCNF, and is not lossless. Give an example of such a schema and its decomposition.
Take the schema given on Practice Exercise 1. Assume the following set of functional dependencies holds:
Take the decomposition of given on Exercise 7.29.
The above two decompsitions are clearly in BCNF.
They are also lossy decompositions (We have shown this in Exercise 7.29).