\begin{frame} \small \begin{definition}[Derived Relations of $\to$] \begin{itemize}[<+->] \item \makebox[2cm][l]{\alert<1>{$\from$} or \alert<1>{$\to^{-1}$}} inverse of $\to$ \item \makebox[2cm][l]{\alert<2>{$\to^=$}} reflexive closure of $\to$ \item \makebox[2cm][l]{\alert<3>{$\to^+$}} transitive closure of $\to$ \item \makebox[2cm][l]{\alert<4>{$\to^*$} or \alert<4>{$\twoheadrightarrow$}} transitive and reflexive closure of $\to$ \item \makebox[2cm][l]{\alert<5>{$\FromP{*}$} or \alert<5>{$\twoheadleftarrow$}} \makebox[22mm][l]{inverse of $\to^*$} (transitive and reflexive closure of $\from$) \item \makebox[2cm][l]{\alert<6>{$\fromto$}} symmetric closure of $\to$, that is, ${\fromto} = {\to} \cup {\from}$ \item \makebox[2cm][l]{\alert<7>{$\conv$}} \makebox[22mm][l]{\alert<7>{conversion}} (equivalence relation generated by $\to$) \item \makebox[2cm][l]{\alert<8>{$\join$}} \makebox[22mm][l]{\alert<8>{joinability}} ${\join} = {\to^* \cdot \FromP{*}}$ \item \makebox[2cm][l]{\alert<9>{$\uparrow$}} \makebox[22mm][l]{\alert<9>{meetability}} ${\uparrow} = {\FromP{*} \cdot \to^*}$ \end{itemize} \end{definition} \medskip \begin{minipage}{.2\textwidth} \only<2->{a relation $R$ is}\ \\\ \\\ \\ \end{minipage} \begin{minipage}{.69\textwidth} \begin{itemize} \item<2-> \alert<2,7>{reflexive} if $a \mathrel{R} a$ for all $a \in A$, \item<3-> \alert<3,7>{transitive} if $a \mathrel{R} c$ whenenver $a \mathrel{R} b$ and $b \mathrel{R} c$, \item<6-> \alert<6,7>{symmetric} if $a \mathrel{R} b$ whenenver $b \mathrel{R} a$. \end{itemize} \end{minipage} \bigskip \end{frame}