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\begin{frame}
  \frametitle{Implication of Functional Dependencies}

  \vspace{-.5ex}
  \begin{goal}{}
    The DB designer is normally not interested in all FDs, 
    but only in a \emph{representative FD set} that implies all other FDs.
  \end{goal}
  \vspace{-.5ex}
  \pause
  
  \begin{block}{Armstrong Axioms}
  \pause
    \begin{itemize}
    \item \emph{Reflexivity:} \\
      If $\beta \subseteq \alpha$, then $\alpha \to \beta$.
      
  \pause
    \item \emph{Augmentation:}\\
      If $\alpha \to \beta$, then $\alpha \cup \gamma \to \beta \cup \gamma$.
  
  \pause
    \item \emph{Transitivity:}\\
      If $\alpha \to \beta$ and $\beta \to \gamma$, then $\alpha \to \gamma$.
    \end{itemize}
  \end{block}
  \pause
  
  \begin{quiz}{\textwidth}{}
  Use the Amstrong axioms to show that
  \begin{talign}
    \sql{ISBN}           & \to \sql{TITLE}, \sql{PUBLISHER} \\
    \sql{ISBN}, \sql{NO} & \to \sql{AUTHOR} \\
    \sql{PUBLISHER}      & \to \sql{PUB\_URL} 
  \end{talign}
  implies $\sql{ISBN} \to \sql{PUB\_URL}$.
  \end{quiz}
\end{frame}