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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}