\documentclass[11pt]{article}

\usepackage{exscale}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{latexsym}
\usepackage{times,mathptm}
\usepackage{epsfig}
\usepackage{amsmath}
\usepackage{enumitem}
\usepackage{mathpazo}
\usepackage{amssymb}

\textwidth 6.5truein          
\textheight 9.5truein
\oddsidemargin 0.0in
\topmargin -0.5in
\thispagestyle{empty} 

\parindent 0pt          
\parskip 0pt

\begin{document}

{\bf Name: }
\hspace{13cm}
{\bf Section: }

\begin{large}
\centerline {\bf Computer Science 243}
\centerline {\bf Spring 2025}
\centerline {\bf Homework 3}

\vskip .7cm

\bf Due: 11:59 p.m., Friday, 2/14/25 \hfill Points: 50
\end{large}
\vskip 0.5cm
Answer the following questions and show your work.  Your final submission must be completely your own work.

\begin{enumerate}

\item {[3 points each]}  Rewrite the following statements such that no negations appear outside a quantifier or expression involving 
a logical operators (show each step):

\begin{enumerate}[label=\alph*.]
\itemsep-0mm
\item $\neg \forall y \forall x$ $P(x, y)$
\item $\neg \exists y \forall x$ $(\neg Q(y) \vee R(x,y))$
\item $\neg \forall x \exists y$ $(P(x,y) \rightarrow \neg Q(y))$
\end{enumerate}

\vskip 0.5cm
\item {[8 points]}  Prove the following equivalence.  Label each logical equivalence used in your proof. 

\itemsep-0mm
\hspace*{1cm}$\neg (\forall x$ $(\neg p(x) \wedge (\forall y$ $(q(y) \wedge \neg r (x, y))))) \equiv \exists x$ $(\neg p(x) \rightarrow$
$(\exists y$ ($q(y) \rightarrow r(x,y))))$


\vskip 0.5cm
\item {[10 points]}  Use rules of inference and/or logical equivalences to prove the following.  Number your statements and
label each rule of inference/logical equivalence used in your proof.

\itemsep-0mm
\hspace*{2cm} $(P \rightarrow Q) \wedge R$\\
\hspace*{2cm} $\neg (R \wedge Q)$\\
\hspace*{2cm} $S \rightarrow P$\\
\hspace*{2cm} ---------------------\\
\hspace*{2cm} $\therefore$ $\neg S$


\vskip 0.5cm
\item {[10 points]}  Use rules of inference and/or logical equivalences to prove the following.  Number your statements and
label each rule of inference/logical equivalence used in your proof.

\itemsep-0mm
\hspace*{2cm} $\exists x$ $(q(x) \wedge r(x))$\\
\hspace*{2cm} $\forall x$ $(q(x) \rightarrow p(x))$\\
\hspace*{2cm} ------------------------\\
\hspace*{2cm} $\therefore$ $\exists x$ $(p(x) \wedge r(x))$


\vskip 0.5cm
\item {[13 points]}  After converting the premises and conclusion below to predicate expressions, use the rules of inference and/or 
logical equivalences to prove the conclusion, where the domain of $x$ is all people.  Use $T(x)$, $D(x)$, and $C(x)$ as obvious predicates.

\itemsep-0mm
\hspace*{2cm} Premises:  Every teacher has a degree.\\
\hspace*{3.8cm} Anyone who did not go to college is not a teacher.\\
\hspace*{3.8cm} Someone is a teacher.\\ \\
\hspace*{2cm} Conclusion:  Someone went to college and has a degree.\\

Label each rule of inference used in your proof.

\end{enumerate}

\end{document}
