7.1

Suppose that we decompose the schema \(R = (A, B, C, D, E)\) into \[ (A, B, C) \\ (A, D, E) \] Show that this decomposition is a lossless decomposition if the following set \(F\) of functional dependencies holds: \[ A \rightarrow BC \\ CD \rightarrow E \\ B \rightarrow D \\ E \rightarrow A \\ \]

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A decomposition $ { R_1, R_2 }$ is a lossless decomposition if \(R_1 \cap R_2 \rightarrow R_1\) or \(R_1 \cap R_2 \rightarrow R_2\). Let \(R_1 = (A, B, C)\), \(R_2 = (A, D, E)\), and \(R_1 \cap R_2 = A\). Since \(A\) is a candidate key (see Practice Exercise 7.6), \(R_1 \cap R_2 \rightarrow R_1\).