\begin{frame}
  \frametitle{Multivalued Dependencies}

  \begin{block}{Multivalued Dependency}
    A \emph{multivalued dependency (MVD)}
    \begin{talign}
      A_1, \dots, A_n \mvd B_1, \dots, B_m
    \end{talign}
    is satisfied in a DB state $I$ $\iff$ for all tuples $t,u \in I(R)$ with
    \begin{tcenter}
      $t.A_i = u.A_i, 1 \leqslant i \leqslant n$
    \end{tcenter}
    there are two tuples $t',u' \in I(R)$ such that
      \begin{enumerate}
      \item $t'$ agrees with $t$ except that $t'.B_i = u.B_i, 1 \leqslant i \leqslant m$, and
      \item $u'$ agrees with $u$ except that $u'.B_i = t.B_i, 1 \leqslant i \leqslant m$.
      \end{enumerate}
  \end{block}
  \medskip
  The condition means that the values of the $B_i$ are swapped:
  \begin{center}\vspace{-2ex}
    \scalebox{.85}{
    \begin{tikzpicture}[thick,r/.style={rectangle,draw,fill=yellow!20}]
      \node [r] (t) {$a_1,\ldots,a_n,\;b_1,\ldots,b_m,\;c_1,\ldots,c_k$};
      \node [r,yshift=-8mm] (u) {$a_1,\ldots,a_n,\;b'_1,\ldots,b'_m,\;c'_1,\ldots,c'_k$};
      \node [lo=t] {$t$};
      \node [lo=u] {$u$};
      \node [r,xshift=65mm] (t) {$a_1,\ldots,a_n,\;b'_1,\ldots,b'_m,\;c_1,\ldots,c_k$};
      \node [r,xshift=65mm,yshift=-8mm] (u) {$a_1,\ldots,a_n,\;b_1,\ldots,b_m,\;c'_1,\ldots,c'_k$};
      \node [lo=t] {$t'$};
      \node [lo=u] {$u'$};
    \end{tikzpicture}
    }
  \end{center}
\end{frame}