\begin{frame} \frametitle{How to Determine Keys: Examples} \begin{exampleblock}{} Find \emph{all} minimal keys the relation $\sql{R(A,B,C,D,E)}$ with \begin{talign} A, D &\to B, D & B, D &\to C & A &\to E \end{talign} We get \begin{enumerate} \pause \item $\textit{Candidates} = \{\; \{\, A \,\} \;\}$ since $A$ not in any right-hand side \pause \item $\{\, A \,\}^+ = \pause \{\, A,E \,\}$\pause, so we extend with $B,C,D$: $\textit{Candidates} = \{\; \{\, A,B \,\}, \{\, A,C \,\}, \{\, A,D \,\} \;\}$ \pause \item $\{\, A,D \,\}^+ = \pause \{\, A,B,C,D,E \,\}$. So $\{\, A,D \,\}$ is a \emph{key}. \pause \item $\{\, A,B \,\}^+ = \pause \{\, A,B,E \,\}$\pause, so we extend with $C,D$: $\textit{Candidates} = \{\; \{\, A,B,C \,\}, \{\, A,B,D \,\}, \{\, A,C \,\} \;\}$ \pause \item $\{\, A,C \,\}^+ = \pause \{\, A,C,E \,\}$\pause, so we extend with $B,D$: $\textit{Candidates} = \{\; \{\, A,B,C \,\}, \{\, A,B,D \,\}, \{\, A,C,D \,\} \;\}$ \pause \item Remove $\{\, A,B,D \,\}$ and $\{\, A,C,D \,\}$ since they contain a key. \pause \item $\{\, A,B,C \,\}^+ = \{\, A,B,C,E \,\}$ is not a key!\\ Extension with $D$ again contains a key. \end{enumerate} \end{exampleblock} \end{frame}