7.35

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.

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Take the schema \(R = (A,B,C,D,E)\) given on Practice Exercise 1. Assume the following set \(F\) of functional dependencies holds:

\[ F := \{A \rightarrow BC, CD \rightarrow E, B \rightarrow D, E \rightarrow A\} \]

Take the decomposition of \(R\) given on Exercise 7.29.

\[ (A,B,C) \\ (C,D,E) \\ \]

The above two decompsitions are clearly in BCNF.

They are also lossy decompositions (We have shown this in Exercise 7.29).