\begin{frame} \frametitle{Exercises} \begin{minipage}{0.49\textwidth} \begin{center} \begin{tikzpicture}[ default, point/.style={circle, draw=blue, thick, inner sep=3pt, minimum size=9mm}, node distance=20mm] \node (1) [point] {\lab{p}}; \node [alo=1] {$w_1$}; \node (2) [point, below left of=1] {\lab{q}}; \node [alo=2] {$w_2$}; \node (3) [point, below right of =1] {\lab{p,\;r}}; \node [aro=3] {$w_3$}; \begin{scope}[shorten <= 1mm, shorten >= 1mm, very thick,>=stealth] \draw [->] (1) to[bend left=15] (2); \draw [->,red] (2) to[bend left=15] (1); \draw [->] (1) to (3); \draw [->] (3) to (2); \end{scope} \draw [rounded corners=2mm, dashed] (-25mm,-20mm) rectangle (25mm,8mm); \end{tikzpicture} \vspace{1ex} \end{center} \end{minipage} \begin{minipage}{0.50\textwidth} \begin{itemize} \item $\BLUE{W}= \{w_1,w_2,w_3\}$ \item $R = \{\langle w_1,w_2\rangle ,\langle w_1,w_3\rangle, $ \\ \;\hfill $ \langle w_2,w_1\rangle , \langle w_3,w_2\rangle \}$ \item $\lab{L}(w_1) = \{\lab{p}\}$ \\ $\lab{L}(w_2) =\{ \lab{q}\}$ \\ $\lab{L}(w_3) =\{ \lab{p,r}\}$ \end{itemize} \end{minipage} \medskip \begin{exampleblock}{Check for yourself:} \begin{malign} w_2 &\;\models\; \all r \wedge \all p & \mbox{\DARKRED{?}}\\ w_1 &\;\models\; \all p & \mbox{\DARKRED{?}}\\ w_1 &\;\models\; \some \all p & \mbox{\DARKRED{?}}\\ w_1 &\;\models\; \all \some p & \mbox{\DARKRED{?}} \end{malign} \end{exampleblock} \end{frame}