\begin{frame} \frametitle{Not Forall and Not Exists} Let us reconsider two examples \begin{align*} \neg\myall{x}{P(x)} &\quad\quad \hint{Not everybody is green.}\\ \myex{x}{\neg P(x)} &\quad\quad \hint{There exists someone who is not green.} \end{align*} \pause \emph{Note that both statements are equivalent!} \pause \bigskip \begin{goal}{} In general we have the following equivalences: \begin{talign} \neg\myall{x}{\phi} \quad\iff\quad \myex{x}{\neg \phi} \\[1ex] \neg\myex{x}{\phi} \quad\iff\quad \myall{x}{\neg \phi} \\[1ex] \myall{x}{\phi} \quad\iff\quad \neg\myex{x}{\neg \phi} \\[1ex] \myex{x}{\phi} \quad\iff\quad \neg\myall{x}{\neg \phi} \end{talign} \end{goal} \pause Note that all these equivalences follow from each other. \end{frame}