\begin{frame}
  \frametitle{Splitting Relations: Lossless Splits}

  \begin{goal}{}
    When is a split lossless?
  \end{goal}
  \pause
  \begin{block}{Decomposition Theorem}
    The split of relations is \emph{guaranteed to be lossless} if 
    the set of shared attributes of the new tables is a key of at least one.
  \end{block}
  \remark{%
    The join connects tuples depending on the shared attributes.  
    If these values uniquely identify tuples in one relation we do not lose information.
  }
  \pause
  
  \begin{exampleblock}{``Lossy'' decomposition}
    \small
    \centering
    \renewcommand{\arraystretch}{0.8}
    \begin{tabular}{@{}cccc@{}}
      Original table & \multicolumn{2}{c@{}}{Decomposition} &
      ``Reconstruction'' \\
      (key $A,B,C$)  &    $R_1$  & $R_2$ & $R_1 \join R_2$ \\[-1ex]
      \colorbox{rellight}{%
        $\begin{array}[t]{|@{}c@{}|@{}c@{}|@{}c@{}|}
          \hline
          \hd{$A$} & \hd{$B$} & \hd{$C$}  
          \\
          \hline
          a_{11} & b_{11} & c_{11} \\
          a_{11} & b_{11} & c_{12} \\
          a_{11} & b_{12} & c_{11} \\
          \hline
        \end{array}$%
      }
      &
      \colorbox{rellight}{%
        $\begin{array}[t]{|@{}c@{}|@{}c@{}|}
          \hline
          \hd{$A$} & \hd{$B$}  
          \\
          \hline
          a_{11} & b_{11} \\
          a_{11} & b_{12} \\
          \hline
        \end{array}$%
      }
      &
      \colorbox{rellight}{%
        $\begin{array}[t]{|@{}c@{}|@{}c@{}|}
          \hline
          \hd{$A$} & \hd{$C$}  
          \\
          \hline
          a_{11} & c_{11} \\
          a_{11} & c_{12} \\
          \hline
        \end{array}$%
      }
      &
      \colorbox{rellight}{%
        $\begin{array}[t]{|@{}c@{}|@{}c@{}|@{}c@{}|}
          \hline
          \hd{$A$} & \hd{$B$} & \hd{$C$}  
          \\
          \hline
          a_{11} & b_{11} & c_{11} \\
          a_{11} & b_{11} & c_{12} \\
          a_{11} & b_{12} & c_{11} \\
          a_{11} & b_{12} & c_{12} \\
          \hline
        \end{array}$%
      }
    \end{tabular}
  \end{exampleblock}
\end{frame}