\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}