\documentclass[11pt]{amsart}
\usepackage[margin=1in]{geometry}
\newcommand{\abs}[1]{\left\lvert#1\right\rvert}
\newcommand{\e}{\varepsilon}

\begin{document}
\title{COS341 Problem Set 01 Question 02}
%\author{Rory Gilmore} % anonymous grading!
\date{\today}
\maketitle

What did $\Sigma=\{0,1\}$ say to $\e$?
You're absolutely nothing: $\abs{\e}$.

\bigskip

Recall the definition of addition in Peano arithmetic:
\[
   a + b =
   \begin{cases}
      a         & \text{if $b = 0$}\\
      S(a + b') & \text{if $b = S(b')$}
   \end{cases}
\]
where $S$ denotes the successor.

I will now prove $1+1=2$, showing my steps in multiple lines and explaining them:
\begin{align*}
   1 + 1 &= 1 + S(0) && \text{by definition of $S$}\\
   & = S(1 + 0) && \text{by definition of $+$, case two}\\
   &= S(1) && \text{by definition of $+$, case one}\\
   &= 2 && \text{by definition of $S$},
\end{align*}
as desired.
\qed
\end{document}
