% If you are working from this template to just write up your own
% solutions, I'm trying to keep it simple for you and there are only a
% few places you need to edit. On line 43 there's a place where the
% phrase "YOUR NAME HERE" appears; change it to your actual
% name. Several places below there are places where the phrase "YOUR
% ANSWER HERE" appears, and directly below (or replacing) those lines
% are good places to put your answers.
%
% One important thing you need to do, however (due to the complicated
% way I use a single form to produce both the question sheets and
% solution writeups) is to make sure that you keep the word 'solution'
% in your filename, or else your answers won't be printed at all.
\documentclass[12pt]{article}
 
\usepackage[margin=0.7in,dvips]{geometry}
\usepackage{fancyhdr,lastpage}
\usepackage{amsmath,amsfonts,amsthm,amssymb}
\usepackage{tikz}
\usepackage{substr}

\IfSubStringInString{\detokenize{solution}}{\jobname}{%
  % Solutions
  \newcommand\qfill{}
  \newcommand\qpage{}
  \newcommand\qonly[1]{}
  \newcommand\answer[1]{#1}
  \newcommand\question[1]{{\em #1}}
  \newcommand\rubric[1]{}
}
{%
  % Question sheets
  \newcommand\qfill{\vfill}
  \newcommand\qpage{\newpage}
  \newcommand\qonly[1]{#1}
  \newcommand\answer[1]{}
  \newcommand\question[1]{#1}
  \newcommand\rubric[1]{}
}

\pagestyle{fancy}
\lhead{MATH 311-01}
\chead{Problem Set \#6\answer{ solutions}}
\rhead{\answer{}}
\lfoot{}
\cfoot{Page \thepage\ of \pageref{LastPage}}
\rfoot{{\bf due March 2, 2017}}

\newtheorem{prop}{Proposition}

% The command below redefines \neg (the logical negation character)
% to be a big tilde, in line with the usage in the book. Comment it
% out if you prefer the more traditional negation symbol.
\renewcommand*{\neg}{\mathord{\sim}}

\begin{document}
\begin{enumerate}
\item \question{Prove that for a real number $x$, if $0<x<1$, then $\frac1{x(1-x)}\geq 4$.}

\answer{
  \begin{proof}

  \end{proof}
}

\item \question{Prove (if true) or provide a counterexample (if false)
    each of the following statements:}
  \begin{enumerate}
  \item \question{For sets $A$, $B$, and $C$,
      $\bigl(A-(B-C)\bigr)=(A-B)\cup(A\cap C)$.}

    \answer{

    }

  \item \question{For sets $A$, $B$, and $C$, if $A\subseteq B$, then $A-C\subseteq C$.}

    \answer{

    }

  \item \question{For sets $A$, $B$, and $C$, if $A-C\subseteq B-C$, then $A\subseteq B$.}

    \answer{

    }

  \item \question{\textbf{For bonus credit:} for any false statements
      above, find an additional premise which will make them true, and
      then prove your new, improved propositions.}

    \answer{

    }
  \end{enumerate}

\item \question{Prove that for sets $A$, $B$, $C$, and $D$, it is the
    case that $(A\times B)\cup (C\times D)\subseteq (A\cup C)\times
    (B\cup D)$.}

  \answer{
    \begin{proof}
      
    \end{proof}
  }
\end{enumerate}

\end{document}
