\begin{frame}
  \frametitle{Multivalued Dependencies}

  \begin{goal}{}
    If the FD $A_1, \dots A_n \to B_1, \dots B_m$ holds, the
    corresponding MVD 
    \begin{center}
      $A_1, \dots, A_n \mvd B_1, \dots, B_m$
    \end{center} 
    is trivially satisfied.
  \end{goal}
  \remark{The FD means that if tuples $t,u$ agree on the $A_i$ then 
      also  on the $B_j$.  Swapping thus has no effect (yields
      $t,u$ again).}
  \pause
  
  \begin{block}{\emph{Deduction rules} to derive \emph{all} implied FDs/MVDs}
    \begin{itemize}
    \item The three Armstrong Axioms for FDs.
    \item If $\alpha \mvd \beta$ then $\alpha \mvd \gamma$, where
      $\gamma$ are all remaining columns.
    \item If $\alpha_1 \mvd \beta_1$ and $\alpha_2 \supseteq \beta_2$
      then $\alpha_1 \cup \alpha_2 \mvd \beta_1 \cup \beta_2$.
      %% if two tuples agree on attributes alpha_2 they also agree on the
      %% beta_2 (which are a subset of alpha_2), thus swapping has no effect
    \item If $\alpha \mvd \beta$ and $\beta \mvd \gamma$ then $\alpha
      \mvd (\gamma - \beta)$.
    \item If $\alpha \to \beta$, then $\alpha \mvd \beta$.
    \item If $\alpha \mvd \beta$ and $\beta' \subseteq \beta$ and there
    is $\gamma$ with $\gamma \cap \beta = \emptyset$ and $\gamma \to
    \beta'$, then $\alpha \to \beta'$.
    \end{itemize}
  \end{block}
\end{frame}