\begin{frame}{Proving and Disproving Semantical Entailment} \begin{exampleblock}{} Which of the following semantic implications are true? \begin{talign} \text{\hint{(a)}}\quad\quad \formula{\forallst{x}{\existsst{y}{\,\binpred{R}{x}{y}}}} \;\;&\models\;\; \formula{\existsst{x}{\forallst{y}{\,\binpred{R}{x}{y}}}} \quad\quad \alert{NO} \\ \text{\hint{(b)}}\quad\quad \formula{\forallst{x}{\existsst{y}{\,\binpred{R}{x}{y}}}} \;\;&\models\;\; \formula{\existsst{y}{\forallst{x}{\,\binpred{R}{x}{y}}}} \quad\quad \alert{NO} \\ \text{\hint{(c)}}\quad\quad \formula{\existsst{y}{\forallst{x}{\,\binpred{R}{x}{y}}}} \;\;&\models\;\; \formula{\forallst{x}{\existsst{y}{\,\binpred{R}{x}{y}}}} \quad\quad \alert{YES} \\ \text{\hint{(d)}}\quad\quad \formula{\existsst{y}{\forallst{x}{\,\binpred{R}{x}{y}}}} \;\;&\models\;\; \formula{\forallst{y}{\existsst{x}{\,\binpred{R}{x}{y}}}} \quad\quad \alert{NO} \end{talign} \end{exampleblock} \end{frame}