\begin{frame}
  \frametitle{Multiple Keys}
  
  A relation may have \textbf{more than one minimal key}.
  
  \begin{goal}{}
    In the relational model, one key is designated as
    \textbf{primary key.} A primary key \textbf{cannot be null.}
  \end{goal}
  All other keys are called \textbf{alternate} or \textbf{secondary keys.}
  \pause\medskip

  \begin{exampleblock}{}
    The primary key attributes are often marked by \textbf{underlining}: % them in the relation schema specifications:\vspace{-.5ex}
    \begin{talign}
    R(\underline{A_1}, \dots, \underline{A_k}, A_{k+1},\dots,A_n)
    \end{talign}
    Here $\{\, A_1,\ldots,A_k \,\}$ is the primary key of $R$.
  \end{exampleblock}
  \pause\medskip
  
  It is good design practice to define a primary key that
  \begin{itemize}
    \item 
      consists of a \textbf{single (simple) attribute} only,
    \item 
      is \textbf{never updated}.
%   \pause
%     \item better for indexing and retrieving items
  \end{itemize}
  \smallskip
  
  This is good for 
  \begin{itemize}
    \item consistency (applications might store the key), and
    \item indexing and retrieving items.
  \end{itemize}
  %   \begin{quiz}{0.9\textwidth}{Primary keys}
  %     What might be the rationale behind this design advice?
  %   \end{quiz}
  \medskip
  \vspace{10cm}
\end{frame}