\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}