\begin{frame}
  \frametitle{Minimality of Keys}

  \begin{exampleblock}{}
    \begin{minipage}{.4\textwidth}
    \centerline{%
      \ttfamily\small
      \colorbox{rellight}{%
        \begin{tabular}[t]{|r|r|r|}
          \multicolumn{3}{c}{Students}        \\ \hline
          \hd{\underline{sid}} & \hd{first} & \hd{last} \\ \hline
          103 & Lisa & Simpson \\ 
          104 & Bart & Simpson \\ 
          106 & Bart & Smit  \\ 
          \hline
        \end{tabular}%
      }      
    }
    \end{minipage}
    \begin{minipage}{.59\textwidth}
    What keys satisfy the key constraints?\vspace{-.75ex}%
    \begin{itemize}\setlength{\itemsep}{-.5ex}
    \pause
    \item \{\sql{sid}\} \onslide<9->{\alert{minimal}}
    \pause
    \item \{\sql{first}, \sql{last}\} \onslide<9->{\alert{minimal}}
    \pause
    \item \{\sql{sid}, \sql{first}\}
    \pause
    \item \{\sql{sid}, \sql{last}\}
    \pause
    \item \{\sql{sid}, \sql{first}, \sql{last}\}
    \end{itemize}
    \end{minipage}
  \end{exampleblock}
  \pause\medskip

  \begin{block}{Implication between key constraints}
    If $A$ is a key and $A \subsetneq B$, then $B$ is also a key. \\
    The key $B$ is \emph{weaker} (more database states are valid) than $A$.
  \end{block}
  Any superset of a key is itself a key.  
  \pause\medskip
  
  \begin{goal}{}
    A key $\{A_1, \dots, A_k\}$ is \textbf{minimal} if no proper subset is a key.\\
  \end{goal}
  \pause\pause
  
  In the literature, often keys are required to be minimal.
\end{frame}