% 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}
\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 \#1\answer{ solutions}}
\rhead{\answer{YOUR NAME HERE}}
\lfoot{}
\cfoot{Page \thepage\ of \pageref{LastPage}}
\rfoot{{\bf due January 19, 2017}}

\begin{document}
For each question, explain your reasoning in full sentences.
\begin{enumerate}
\item \question{For two sets $A$ and $B$, suppose that all you know
    about the two of them is that $|A|=6$ and $|B|=4$.}

  \begin{enumerate}
  \item \question{What is the largest possible value of $|A\cup B|$?
      What is its smallest possible value?}

    \answer{
      % YOUR ANSWER HERE

    }

  \item \question{What is the largest possible value of $|A\cap B|$?
      What is its smallest possible value?}

    \answer{
      % YOUR ANSWER HERE

    }

  \item \question{What is the largest possible value of $|A-B|$?
      What is its smallest possible value?}

    \answer{
      % YOUR ANSWER HERE

    }

  \item \question{What is the largest possible value of $|B-A|$?
      What is its smallest possible value?}

    \answer{
      % YOUR ANSWER HERE

    }
  \end{enumerate}

\item \question{Describe two sets $S$ and $T$ such that $S$ is both an
    element \emph{and} a subset of $T$.}

  \answer{
    % YOUR ANSWER HERE

  }

\item \question{If $X$ is an infinite set and $X\in Y$, can you be
    certain that $Y$ is also infinite?}

  \answer{
    % YOUR ANSWER HERE

  }

\item \question{If $X$ is an infinite set and $X\subseteq Y$, can you be
    certain that $Y$ is also infinite?}

  \answer{
    % YOUR ANSWER HERE

  }

\item \question{Does it appear to be possible or meaningful to you for
    there to be a set $S$ which is an element of itself, that is, a
    set $S$ such that $S\in S$? Does this appear to be either a useful
    or problematic concept? Your response to this need not be
    ``mathematically justified'', but should express a personal
    consideration of the ramifications of such an idea.}

  \answer{
    % YOUR ANSWER HERE

  }
\end{enumerate}

\end{document}
