\documentclass[10pt]{article} %
\usepackage{fullpage}
\usepackage{graphicx}
\usepackage{amsmath,amssymb,color}
\usepackage{amsfonts,amstext}
%\input{preface}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9in}
\title{Math 214 Sample Homework}
\author{My Name}
%\date{}
\begin{document}
\maketitle
\bigskip
\begin{enumerate}
% taken from http://www.hku.hk/cgi-bin/philodep/knight/puzzle, puzzle 55
\item Recall the Knights and Knaves from the first day of class. Recall
that knights always tell the truth, and knaves always lie. You meet
three inhabitants: Patricia, Quinn and Roberta. Patricia claims that
it's false that Roberta is a knave. Quinn says, `Either Roberta is a
knight or I am a knight.' Roberta says that Quinn is a knave. Who are
knights and who are knaves? Prove your answer (using truth table).
\bigskip
% taken from http://www.hku.hk/cgi-bin/philodep/knight/puzzle, puzzle 110
\item Later, you meet Ann, Bert and Chuck. Ann says, `Chuck could claim
that I am a knight.' Bert says that only a knave would say that Ann is
a knave. Chuck claims, `Ann could say that I am a knave.' Who are
knights and who are knaves? Prove your answer.
\bigskip
\item By using truth tables prove that, for all statements P and Q, the statement `P$\Rightarrow$ Q' and `(not Q)$\Rightarrow$ (not P)' are equivalent.
\bigskip
\item Prove that for all real numbers $a, b$ and $c$,
\begin{equation*}
bc+ac+ab\le a^2+b^2+c^2.
\end{equation*}
\bigskip
\item Prove that for all real numbers $a$ and $b$,
\begin{equation*}
|a|<|b| \Rightarrow a^2\le b^2.
\end{equation*}
\bigskip
\item Write the set $A=\{-1,-2,-3,\cdots\}$ in a form $\{x\in {\mathbb Z}:p(x)\}$.
\end{enumerate}
\end{document}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% End: