7.1

Suppose that we decompose the schema $R=(A,B,C,D,E)$ into

$$\begin{gathered} (A,B,C) \\ (A,D,E) \end{gathered}$$

Show that this decomposition is a lossless decomposition if the following set $F$ of functional dependencies holds:

$$\begin{gathered} A\to BC \\ CD\to E \\ B\to D \\ E\to A \end{gathered}$$

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A decomposition $\{R_1,R_2\}$ is a lossless decomposition if $R_1\cap R_2\to R_1$ or $R_1\cap R_2\to R_2$. 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), $R_1\cap R_2\to R_1$.