% This is a TeX file
% It is essentially a programming language for mathematics documents
% You can ignore this first portion of the file, as it is essentially a bunch of "include" statements.
\documentclass[10pt]{article}

\usepackage{pictex, latexsym, graphicx,amsmath,amssymb,amsbsy,amsfonts,amsthm,verbatim}
\usepackage{graphics}
\usepackage{fullpage}
\usepackage{fancyhdr}

\usepackage{algorithm,algorithmic}
\usepackage{multirow}

\setlength{\voffset}{-0.25in}
\setlength{\headsep}{0.25in}
\setlength{\parskip}{1em}
\setlength{\parindent}{0em}

% The following denotes a counter for the problems.
\newcounter{problem}
\newcommand{\problem}{\textbf{\refstepcounter{problem}Problem \theproblem.} }

% These are macro shortcuts! Helpful!
\def\vu{\mathbf{u}}
\def\vx{\mathbf{x}}
\def\vb{\mathbf{b}}
\def\vv{\mathbf{v}}
\def\vw{\mathbf{w}}

\renewcommand{\implies}{\rightarrow}
\renewcommand{\lor}{\vee}
\renewcommand{\land}{\wedge}
\newcommand{\xor}{\oplus}
\renewcommand{\iff}{\leftrightarrow}
\newcommand{\TRUE}{\mathbf{T}}
\newcommand{\FALSE}{\mathbf{F}}
\newcommand{\universe}{\mathcal{U}}

\begin{document}
\pagestyle{fancyplain}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% NOW you can start typing your text that will appear in the PDF!!!

% Note that anything after a '%' is a comment!

% Anything immediately after a '\' is a command. 
% Some denote formatting, some are for special characters.
% The '$' symbol starts and ends "math mode". Use this only for mathematical statements
% In math mode, any sequence of letters is meant to be a list of variables, so use "\sin x" for the sine of x, which will look much different from "sin x"
\chead{COM S 330 --- Homework 03 --- INSERT NAME HERE}

\textit{Type your answers to the following questions and submit a PDF file to Blackboard. 
One page per problem.}


\problem [5pts] Let $P(x)$ be the statement ``$\frac{1}{x} = x$.'' If the domain consists of real numbers, then what are these truth values?

a. $P(2)$.

b. $P(1)$.

c. $\exists x P(x)$.

d. $\forall x \neq 0 (P(x))$.



% Be sure to use this \clearpage to make sure your problems are on separate pages!
%\clearpage
\problem [5pts] Let $P(x,y)$ be the statement ``$x > y$ and $x$ is divisible by $y$.'' Let the domain consist of positive integers and write English sentences describing the following propositions.

a. $\exists x \forall y P(x,y)$

b. $\exists y \forall x P(x,y)$

c. $\forall x \exists y P(x,y)$

d. $\forall y \exists x P(x,y)$

% Be sure to use this \clearpage to make sure your problems are on separate pages!
%\clearpage
\problem [10pts] Let $P(x,y)$, $Q(x,y)$, and $R(x)$ be propositional functions. Use logical equivalences to show that
	$\lnot \forall x ((\exists y (P(x,y) \implies Q(x,y))) \lor R(x))$ and $\exists x (\lnot R(x) \land \forall y ( \lnot Q(x,y) \land  P(x,y)))$ are equivalent.

% Be sure to use this \clearpage to make sure your problems are on separate pages!
%\clearpage
\problem [10pts] Use rules of inference to show that if $p \land q$, $r \lor s$, and $p \implies \lnot r$, then $s$ is true.

% Be sure to use this \clearpage to make sure your problems are on separate pages!
%\clearpage
\problem [10pts] Use rules of inference to show that if $(p\implies q) \land (q \implies p)$, $t \lor q$, $t \lor p$, and $(p \land q)\implies t$, then $t$ is true.

\end{document} 