\begin{frame} \frametitle{How to Determine Keys} \begin{block}{} Given a set of FDs and the set $\mathcal{A}$ of all attributes of a relation $R$: \begin{talign} \alpha \subseteq \mathcal{A} \text{ is key of $R$ } \quad\iff\quad \alpha^+ = \mathcal{A} \end{talign} \end{block} That is $\alpha$ is a key if the cover $\alpha^+$ contains all attributes. \pause\bigskip \begin{goal}{} We can use FDs to determine all possible keys of $R$. \end{goal} \bigskip\pause Normally, we are interested in \emph{minimal keys} only. \begin{block}{} A key $\alpha$ is \emph{minimal} if every $A \in \alpha$ is \emph{vital}, that is \begin{talign} (\alpha - \{ A \})^+ \neq \mathcal{A} \end{talign} \end{block} \end{frame}