\begin{frame}
  \frametitle{Functional Dependencies}
  
  A functional dependency with $m$ attributes on the \emph{right}
  \begin{talign}
    A_1, \dots A_n \to B_1, \dots B_m
  \end{talign}
  is \emph{equivalent} to the $m$ FDs:
  $$
  \begin{array}{ccc}
    A_1, \dots, A_n & \to & B_1 \\[-.5ex]
    \vdots          &     & \vdots \\[-.5ex]
    A_1, \dots, A_n & \to & B_m 
  \end{array}\vspace{-1.3ex}%
  $$
  \pause

  \begin{exampleblock}{}
    \vspace{-1.5ex}
    \begin{talign}
      A,B \to C,D
    \end{talign}
    is equivalent to the combination of
    \begin{talign}
      A,B &\to C &
      A,B &\to D
    \end{talign}
    but \emph{not} equivalent to
    \begin{talign}
      A &\to C,D &
      B &\to C,D
    \end{talign}
  \end{exampleblock}
  \pause\vspace{-.75ex}
  
  \begin{goal}{}
    So, in the sequel it suffices to consider functional dependencies
    with a single column name on the right-hand side.
  \end{goal}
\end{frame}

\theme{\\Keys vs. Functional Dependencies}