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

\begin{goal}{}
Simpler way to \emph{check whether $a \to \beta$ is implied by an FD set}:%
\begin{itemize}
\item compute the \emph{cover} $\alpha^+$ of $\alpha$, and
\item then check if $\beta \subseteq \alpha^+$.
\end{itemize}
\end{goal}
\pause

\begin{block}{Cover}
The \emph{cover} $\alpha_{\mathcal{F}}^+$ of
\begin{itemize}
\item a set of attributes $\alpha$
\item with respect to an FD set $\mathcal{F}$
\end{itemize}
is the set of all attributes $B$ that are uniquely determined by $\alpha$:
\begin{talign}
\alpha_{\mathcal{F}}^+ := \{ \;B \mid \mathcal{F} \text{ implies } \alpha \to B \;\}
\end{talign}
\end{block}
\pause

\begin{block}{Implication Check}
A set of FDs $\mathcal{F}$ implies an FD $\alpha \to \beta$
if and only if $\beta \subseteq \alpha^+_{\mathcal{F}}$.
\end{block}
\end{frame}