85/296
\begin{frame}
\frametitle{How to Determine Keys}

\begin{block}{}
Given a set of FDs and the set of all attributes $\mathcal{A}$ of a relation $R$:
\begin{talign}
\alpha \subseteq \mathcal{A} \text{ is key of $R$ } \quad\iff\quad \alpha^+ = \mathcal{A}
\end{talign}
That is $\alpha$ is a key if the cover $\alpha^+$ contains all attributes.
\end{block}
\pause
\begin{goal}{}
We can use FDs to determine all possible keys of $R$.
\end{goal}
\bigskip\pause

Remember: normally, we are interested in \emph{minimal keys} only.
\pause
\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}