\documentclass[12pt]{article}

\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{graphicx}
\usepackage{csquotes}

\usepackage{tikz}
\usetikzlibrary{positioning, scopes} 			%always use these
\usetikzlibrary{backgrounds}

%table spacing
\newcommand{\T}{\rule{0pt}{2.6ex}}
\newcommand{\B}{\rule[-1.2ex]{0pt}{0pt}}


\newcommand{\RR}{\mathbb{R}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\lcm}{\operatorname{lcm}}
\newcommand{\xor}{\operatorname{xor}}


\newtheorem{theorem}{Theorem}
\renewcommand*{\thetheorem}{I.\arabic{theorem}}
\newtheorem{exercise}[theorem]{Exercise}
\theoremstyle{definition}
\newtheorem*{definition}{Definition}

\setlength{\oddsidemargin}{0in} \setlength{\evensidemargin}{0in}
\setlength{\marginparwidth}{0in} \setlength{\marginparsep}{0in}
\setlength{\topmargin}{0in} \setlength{\headheight}{0pt}
\setlength{\headsep}{0in} \setlength{\footskip}{.3in}
\setlength{\textwidth}{6.5in} \setlength{\textheight}{8.2in}%\setlength{\textheight}{9.2in}
\setlength{\parskip}{4pt}

%\pagestyle{fancyplain} \chead{Math 3345, Autumn 2021, Kutler}


\begin{document}

\begin{center}
{\large{\textsc{Homework 4}}} \\
\textsc{Math 3345 -- Autumn 2022 -- Kutler}
\end{center}

\thispagestyle{empty}

Please complete the following problems on your own paper. Solutions should be written clearly, legibly, and with appropriate style.

\begin{enumerate}
\item \textbf{{[Falkner Section 2 Exercise 14]}} Show by means of an explanation in words that the sentence $(P \wedge Q) \Rightarrow (P \vee Q)$ is a tautology. (As usual, you should use the method of conditional proof.)

\item \textbf{{[Falkner Section 2 Exercise 15]}} Use the method of conditional proof to explain in words why the sentence
\[ \big\{ (P \vee Q) \, \wedge \, [ (P \Rightarrow R) \wedge (Q \Rightarrow S) ] \big\} \ \Rightarrow \ (R \vee S) \]
is a tautology. \textbf{You do NOT need to use the book's method of ``discharging assumptions.'' However, please be clear about which assumptions you are making, and why you are justified in making these assumptions.}
\end{enumerate}

\vspace{.25 in}

\begin{center}
\textbf{Practice Problems}
\end{center}

It is strongly recommended that you complete the following problems. There is no need to write up polished, final versions of your solutions (although you may find this a useful exercise). Please do not submit any work for these problems.

\begin{enumerate}
\item \textbf{{[Falkner Section 2 Exercise 13]}} Show by means of an explanation in words that the sentence $P \Rightarrow (P \vee Q)$ is a tautology.
\end{enumerate}




\end{document}
