\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}