\begin{frame}
  \frametitle{Covers}
  
  \begin{goal}{}
  Simpler way to \emph{check if $\alpha \to \beta$ is implied by a set $\mathcal{F}$ of FDs}:%
  \begin{itemize}
    \item compute the \emph{cover} $\alpha_{\mathcal{F}}^+$ of $\alpha$, and
    \item then check if $\beta \subseteq \alpha_{\mathcal{F}}^+$.
  \end{itemize}
  \end{goal}
  \pause
  
  \begin{block}{Cover}
    The \emph{cover} $\alpha_{\mathcal{F}}^+$ of attributes $\alpha$ 
    with respect to a set $\mathcal{F}$ of FDs is
    \begin{talign}
      \alpha_{\mathcal{F}}^+ := \{ \;A \mid \mathcal{F} \text{ implies } \alpha \to A \;\} \;,
    \end{talign}
    the set of all attributes $A$ that are uniquely determined by $\alpha$.
  \end{block}
  \pause
  \begin{goal}{}
    The cover $\gamma_{\mathcal{F}}^+$ can be \emph{computed} as follows:
    \begin{itemize}
      \item Let $x = \gamma$, and repeat the next step until $x$ is stable.
      \item If $\alpha \subseteq x$ for some $(\alpha \to \beta) \in \mathcal{F}$, then let $x = x \cup \beta$.
    \end{itemize}
    Finally $x$ is the cover $\gamma_{\mathcal{F}}^+$ of $\gamma$ with respect to the set $\mathcal{F}$ of FDs.
  \end{goal}
  \pause
  
  \begin{block}{}
    A set of FDs $\mathcal{F}$ implies an FD $\alpha \to \beta$ 
    if and only if $\beta \subseteq \alpha^+_{\mathcal{F}}$.
  \end{block}
  \vspace{10cm}
\end{frame}