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\begin{document}
\begin{center}

{\Huge{Somemore Logic}}\\
with quantifiers!
\end{center}



Let $p$, $q$, and $r$ be propositions for the entirety of this worksheet (front \& back).


\begin{enumerate}
\item Below are several ways to combine (using textbook notation) $\neg$, $\wedge$, and $\vee$.  
\begin{enumerate}
\item Find the truth tables for each by hand or by using Sage.

\hspace{-0.8in}\begin{tabular*}{6.8in}{l@{\extracolsep{\fill}}l@{\extracolsep{\fill}}l@{\extracolsep{\fill}}l}
(i) $\neg (p\wedge q)$	&(ii) $(\neg p) \wedge q$  &(iii) $p \wedge (\neg q)$  &(iv) $(\neg p)\wedge (\neg q)$  \\
~\\
~\\
~\\
~\\
~\\
~\\
~\\
~\\
(v) $\neg (p\vee q)$	&(vi) $(\neg p) \vee q$  &(vii) $p \vee (\neg q)$  &(viii) $(\neg p)\vee (\neg q)$  \\
~\\
~\\
~\\
~\\
~\\
~\\
~\\
~\\
\end{tabular*}

\item Do any of the above truth tables look the same (i.e. are there any logical equivalences)?  Which ones?
\vspace{.7in}
\end{enumerate}
 

\item Perform the same investigation on the following (using textbook notation) and try to identify another logical equivalence.  
\vspace{.7in}


\hspace{-0.5in}\begin{tabular*}{6.8in}{l@{\extracolsep{\fill}}l@{\extracolsep{\fill}}l@{\extracolsep{\fill}}l}
(i) $ p\vee (q\wedge r)$	&(ii) $p \wedge (q\vee r)$  &(iii) $(p \vee q) \wedge (p\vee r)$  &(iv) $(p\wedge q) \vee (p \wedge r)$  \\
~\\
~\\
~\\
\end{tabular*}





\newpage
\noindent Let $x$ be an integer between -3 and 3.  

\item Translate the following symbolic propositions into English sentences and determine the truth value.
\begin{enumerate}
\item $\forall x, x+1>x$.
\vfill
\item $\exists x, x+1>x$.
\vfill
\item $\forall x<0, x^2<0$.
\vfill
\item $\forall x, (x<0 \rightarrow x^2<0)$
\vfill
\item $\exists x>0, x^2=2$.
\vfill
\item $\exists x, (x>0 \wedge x^2=2)$.
\vfill
\item $\neg \forall x, x^2>x$.
\vfill
\item $\exists x, \neg(x^2>x)$.
\vfill
\end{enumerate}

\item Do any of the above statements seem logically equivalent?  Which ones?



\end{enumerate}
\end{document}
