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\begin{frame}
  \frametitle{Implication of Functional Dependencies}
  
  \begin{goal}{}
      Whenever $A \to B$ and $B \to C$ hold, then $A \to C$ is automatically
      satisfied.    
  \end{goal}
  \begin{exampleblock}{}
    Note that $\sql{CRN} \to \sql{PHONE}$ is a consequence of
      \begin{center}
        $\sql{CRN} \to \sql{INAME}$ \quad and\quad $\sql{INAME} \to \sql{PHONE}$
      \end{center}
  \end{exampleblock}
    
  \bigskip\pause
  
  \begin{goal}{}
    FDs of the form $A \to A$ always hold.
  \end{goal}
  \begin{exampleblock}{}
    $\sql{PHONE} \to \sql{PHONE}$ holds, but is not interesting
  \end{exampleblock}
  \bigskip\pause
  
  \begin{block}{Implication of Functional Dependencies}
    A set of FDs $\{ \alpha_1 \to \beta_1, \dots, \alpha_n \to \beta_n\}$ 
    \emph{implies} an FD $\alpha \to \beta$ 
    if and only if every DB state which satisfies all $\alpha_i \to \beta_i, 1 \leqslant i \leqslant n$, 
    also satisfies $\alpha \to \beta$.
  \end{block}
\end{frame}