\documentclass[12pt,english]{amsart}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{geometry}
\geometry{verbose,tmargin=.8in,bmargin=1in,lmargin=1in,rmargin=1in}
\usepackage[all]{xy}
\usepackage{bussproofs}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{plain}
\newtheorem{lemma}[theorem]{Lemma}
\theoremstyle{plain}
\newtheorem{example}[theorem]{Example}
\theoremstyle{remark}
\newtheorem{proposition}[theorem]{Proposition}

\newcounter{problems}
\theoremstyle{definition}
\newtheorem{problem}[problems]{Problem}

\theoremstyle{remark}
\newtheorem*{solution}{Solution}



\title{Math 301: Homework 1}

\author{insert your name here}

\begin{document}
\maketitle

\begin{problem}
	Let $A$ and $B$ be propositional variables denoting "Alice is in the room" and "Bob is in the room" respectively. Write the propositional formulas corresponding to the following statements.
	\begin{enumerate}
		\item Alice and Bob are not both in the room.
	
	\begin{solution}
		
	\end{solution}	 
		 
		\item Alice and Bob are both not in the room.
		
	\begin{solution}
		
	\end{solution}		
		
		\item Either Alice or Bob is not in the room.
		
	\begin{solution}
		
	\end{solution}	
	
		\item Neither Alice nor Bob is in the room.
		
		\begin{solution}
			
		\end{solution}	
	\end{enumerate}
\end{problem}

\begin{problem}
	Analyze the logical form of the following statements by rewriting them as propositional formulas.
	\begin{enumerate}
		\item It will rain tomorrow or it will snow tomorrow, but it will not both rain and snow tomorrow.
		
		\begin{solution}
			
		\end{solution}
		
		
		\item If it rains tomorrow, I will either drive to work or call a cab.
		
		\begin{solution}
			
		\end{solution}
		
		\item If it does not rain tomorrow and it does not snow tomorrow, I will neither drive nor call a cab.  
		
		\begin{solution}
			
		\end{solution}
		
	\end{enumerate}
\end{problem}

\begin{problem}
	On an island inhabited by knights and knaves, where the former always tell the truth and the latter always lie, you meet three individuals: Alice, Bob, and Eve. Alice says that Bob is a knight. Bob say that Alice is a knight but Eve is a knave. Eve says that both Alice and Bob are knights. Determine who is a knight and who is a knave by constructing a truth table.
\end{problem}

\begin{solution}
	% It might be easier to draw the truth table by hand. Otherwise, you can use the following template for a table: 
	
%	\begin{center}
%		\begin{tabular}{ |c|c|c|c|c|c|c|c| } 
%			\hline
%			$p$ & $q$ & r &&&&& \\ \hline
%			-&-&-&-&-&-&-&-\\
%			-&-&-&-&-&-&-&-\\
%			-&-&-&-&-&-&-&-\\
%			-&-&-&-&-&-&-&-\\ \hline
%
%		\end{tabular}
%	\end{center}

% To add vertical space, use the following command 
% \vspace*{3in}


\end{solution}

\begin{problem}
	Let $p,q,r$ be propositional variables. Find two distinct proof trees whose conclusions are $(p \wedge q) \vee r$.
\end{problem}

\begin{solution}
% This will be much easier to add by hand, otherwise you can use the bussproofs pachage. The userguide is at http://mirrors.ibiblio.org/CTAN/macros/latex/contrib/bussproofs/BussGuide2.pdf 
% Here is a sample of a proof tree using bussproofs

%\begin{prooftree}
%	\AxiomC{$p$}
%	\AxiomC{$q$}
%	\BinaryInfC{$p\wedge q$}
%	\AxiomC{$r$}
%	\BinaryInfC{$(p\wedge q) \wedge r$}
%\end{prooftree}



% To add vertical space, use the following command 
% \vspace*{3in}


\end{solution}

\begin{problem}
	Let $n$ be an integer. Prove that $n^3$ leaves the same remainder as $n$ when divided by $3$.
\end{problem}

\begin{solution}
	
\end{solution}






\end{document}
