\begin{frame}
  \frametitle{Properties of Relations}

  \begin{goal}{Properties of a relation $R$}
    \vspace{-2ex}
    \begin{eqnarray*}
    \pause
    \text{\emph{Reflexive}}  && \myall{x}{R(x,x)}  \\
    \pause
    \text{\emph{Symmetric}}  && \myall{x}{\myall{y}{\big(\;R(x,y) \to R(y,x)\;\big)}}  \\
    \pause
    \text{\emph{Transitive}}  && \myall{x}{\myall{y}{\myall{z}{\big(\;(R(x,y) \wedge R(y,z)) \to R(x,z)\;\big)}}} \\
    \pause
    \text{\emph{Serial}}     && \myall{x}{\myex{y}{R(x,y)}} \\
    \pause
    \text{\emph{Functional}} && \myall{x}{\myex{y}{\big(\; R(x,y) \wedge \myall{z}{(R(x,z) \to z=y)}\;\big)}}
    \end{eqnarray*}
    \vspace{-2ex}
  \end{goal}
  \pause

  \begin{goal}{}
    An \emph{equivalence relation} is a relation that is 
    \begin{itemize}
      \item reflexive,
      \item symmetric and 
      \item transitive.
    \end{itemize} 
  \end{goal}
\end{frame}