72/296
\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}