\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}