\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}