7.6

Compute the closure of the following set $F$ of functional dependencies for relation schema $R=(A,B,C,D,E)$.

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

List the candidate keys for $R$.

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Note: It is not reasonable to expect students to enumerate all of $F^{+}$. Some shorthand representation of the result should be acceptable as long as the nontrivial members of $F^{+}$ are found.

Starting with $A\to BC$, we can conclude that: $A\to B$ and $A\to C$.

Since $A\to B$ and $B\to D$, $A\to D$ (decomposition, transitive)

Since $A\to CD$ and $CD\to E$, $A\to E$ (union, decomposition, transitive)

Since $A\to A$, we have $A\to ABCDE$ from the above steps (reflexive, union)

Since $E\to A$, $E\to ABCDE$ (transitive)

Since $CD\to E$, $CD\to ABCDE$ (transitive)

Since $B\to D$ and $BC\to CD$, $BC\to ABCDE$ (augmentative, transitive)

Also, $C\to C$, $D\to D$, $BD\to D$, etc.

Therefore, any functional dependency with $A,E,BC$ or $CD$ on the left-hand side of the arrow is in $F^{+}$, no matter which other attributes appear in the functional dependency. Allow $\ast$ to represent any set of attributes in $R$, the $F^{+}$ is $BD\to B$, $BD\to D$ , $C\to C$, $D\to D$, $BD\to BD$, $B\to D$, $B\to B$, $B\to BD$, and all functional dependencies of the form $A\ast \to \alpha$, $BC\ast \to \alpha$, $CD\ast \to \alpha$, $E\ast \to \alpha$ where $\alpha$ is any subset of $\{A,B,C,D,E\}$. The candidate keys are $A$, $BC$, $CD$, and $E$.