\begin{frame}
  \frametitle{Canonical Set of Functional Dependencies}

  \begin{exampleblock}{}
    Consider the relation $R(A,B,C,D,E)$ with FDs 
    \begin{talign}
      A \to D,E && B \to C && B,C \to D && D \to E
    \end{talign}\vspace{-2ex}
    \begin{enumerate}
    \pause
    \item
      Make the right-hand sides singular\pause
      \begin{talign}
        A \to D && A \to E && B \to C && B,C \to D && D \to E
      \end{talign}\vspace{-3ex}
    \pause
    \item
      Minimise left-hand sides\pause
      \begin{talign}
        A \to D && A \to E && B \to C && B \to D && D \to E
      \end{talign}
      We drop $C$ from $B,C \to D$ since $D \in \{\,B\,\}^+$ due to $B \to C$.
    \pause
    \item Remove implied FDs\pause
      \begin{talign}
        A \to D && B \to C && B \to D && D \to E
      \end{talign}
      $A \to E$ can still be derived from $A \to D$ and $D \to E$.
    \end{enumerate}
    \pause\smallskip
    
    Thus we have obtained the following \emph{canonical} set of FDs:
      \begin{talign}
        A \to D && B \to C && B \to D && D \to E
      \end{talign}
  \end{exampleblock}
\end{frame}