\begin{frame} \frametitle{Examples: Semantics Intuitive} \begin{alertgoal}{} Typical exam question. \end{alertgoal} \begin{exampleblock}{} Which of the following semantic implications are true? \begin{talign} \text{\hint{(a)}}\quad\quad \myall{x}{\myex{y}{R(x,y)}} \;\;&\models\;\; \myex{x}{\myall{y}{R(x,y)}} \mpause[2]{\quad\quad \alert{NO}} \\ \text{\hint{(b)}}\quad\quad \myall{x}{\myex{y}{R(x,y)}} \;\;&\models\;\; \myex{y}{\myall{x}{R(x,y)}} \mpause{\quad\quad \alert{NO}} \\ \text{\hint{(c)}}\quad\quad \myex{y}{\myall{x}{R(x,y)}} \;\;&\models\;\; \myall{x}{\myex{y}{R(x,y)}} \mpause{\quad\quad \alert{YES}} \\ \text{\hint{(d)}}\quad\quad \myex{y}{\myall{x}{R(x,y)}} \;\;&\models\;\; \myall{y}{\myex{x}{R(x,y)}} \mpause{\quad\quad \alert{NO}} \end{talign} \pause For the non-valid implications give counter-models.\\ (Models that make the premise true, the conclusion false.) \begin{center} \begin{tikzpicture}[node distance=15mm, dot/.style={minimum size=4mm, circle, draw=none, fill=black, inner sep=0, outer sep=1mm, text=white}] \mpause[1]{ \begin{scope} \node at (-1.5,0) {\hint{(a)\mpause{, (b)}}}; \node [dot] (1) {1}; \node [dot,below right of=1] (2) {2}; \node [dot,above right of=2] (3) {3}; \begin{scope}[->,thick] \draw (1) -- (2); \draw (2) -- (3); \draw (3) -- (1); \end{scope} \end{scope} } \mpause{} \mpause{ \begin{scope}[xshift=50mm] \node at (-1,0) {\hint{(d)}}; \node [dot] (1) {1}; \node [dot,below right of=1] (2) {2}; \node [dot,above right of=2] (3) {3}; \begin{scope}[->,thick] \draw (1) -- (2); \draw (3) -- (2); \draw (2) to[out=-45,in=-45-90,looseness=5] (2); \end{scope} \end{scope} } \end{tikzpicture} \end{center}\vspace{-2ex} \pause\pause\pause\pause\pause Thus only (c) is a valid semantic implication! \end{exampleblock} \end{frame}