%If you start a line with a "percent" symbol (like %), then that line is a "comment" and won't show up in your actual document.  

%Every document starts with a documentclass. 
\documentclass[11pt]{article}

%After that, it's useful to list an author and give a title.
\author{Davis Lazowski}
\title{An example math pset}

%Graphicx is used to include pictures in LaTeX files. 
\usepackage{graphicx}

%The AMS packages. They contain a lot of useful math-related goodies.
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{amsfonts}

%The below command makes sure that every section starts on a new page. That way if you have a new section for every CA, they'll all print out on separate pieces of paper. 
\usepackage{titlesec}
\newcommand{\sectionbreak}{\clearpage}

%The amsthm package lets you format different types of mathematical ideas nicely. You use it by defining "\newtheorem"s as below:
\newtheorem{problem}{Problem}
\newtheorem{theorem}{Theorem}
\newtheorem*{proposition}{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{defn}[theorem]{Definition}

%The "\newcommand" command lets you specify a custom command. This should be used wisely to add semantic meaning to otherwise confusing sequences of commands - not just speed up typing. (If you want suggestions for shortcuts you can ask Thayer).
%Here is an example definition of a bra and ket from Quantum Mechanics.
\newcommand{\bra}[1]{\langle #1 |}
\newcommand{\ket}[1]{| #1 \rangle}

%Adding your name here lets you make sure every page has your name, so that your psets don't get mixed up.
\usepackage{fancyhdr}
\pagestyle{fancy}
\lhead{Davis Lazowski}
\rhead{Example Pset 1}

%Everything above here is just commands which don't create any main-document text directly.
%You have to put all your writing within \begin{document} and \end{document} clauses.
\begin{document}

% You could format a pretty title page with the command below but 3/4 of the CAs would miss out so I've commented it out.
%\maketitle

%You might want to divide your document into sections. Good ones might be for each CA. 
\section{For Thayer}

\subsection{Problem 1}

%If you want to write normal text, just write it down!
%Don't care about my random math: I just tried to write things that used a lot of symbols you might want to use. 
I would like to show that the integers are not dense in the reals over any compact interval.
%If you want to add math into a LaTeX file, enclose it in dollar signs. If you want fancy math letters, use \mathbb{}.
Let $[a,b] \subset \mathbb{R}$.  Then $[a,b] \cap \mathbb{Z}$ is a finite set. Let $r \in \mathbb{R} \setminus \mathbb{Z}$. Then ... blablabla

\subsection{Problem 2} 
I would like to show that $1 + 2 + 3 + 4 + 5 + \frac{1}{2} = 15.5$.

%Use \begin{equation} \end{equation} for equations, or math you want alone.
\begin{equation}
(1 + 2) + 3 + 4 + 5 + \frac{1}{2} = (3+3)+4+5+ \frac{1}{2} = (6 + 4)+5+ \frac{1}{2} = (10 + 5)+ \frac{1}{2} = 15+ \frac{1}{2} = 15.5 \text{this is text}
\end{equation}

But that was really hard to read so let me do it again.
%If you want to split your equation over multiple lines, use \begin{split} \end{split}. Then use "\\" to split. One "&" symbol can be placed on each line to align the lines horizontally at the &.

\begin{equation}\label{eq:equationlabel}
\begin{split}
(1 + 2) + 3 + 4 + 5     &= (3 + 3) + 4 + 5 + \frac{1}{2}\\
          &= (6 + 4) + 5 + \frac{1}{2}\\
                    &= (10 + 5) + \frac{1}{2} \\
                        &= 15+ \frac{1}{2} \\
                        &= 15.5
\end{split}
\end{equation}

You can refer back to equation \eqref{eq:equationlabel} without typing its label directly. 
Or another way (this time with the align environment):

%The align environment is useful for showing the process of calculations when the calculations do not represent a specific result or single thought that will be referenced later.
\begin{align*}
  (1 + 2) + 3 + 4 + 5     &= (3 + 3) + 4 + 5 + \frac{1}{2}\\
  &= (6 + 4) + 5 + \frac{1}{2}\\
  &= (10 + 5) + \frac{1}{2} \\
  &= 15+ \frac{1}{2} \\
  &= 15.5
\end{align*}

If we aren't careful, our parentheses can be ugly:
\begin{equation*}
  (\frac{\frac{1}{x}}{y})
\end{equation*}
but we can fix that with the appropriate delimiters.
\begin{equation*}
  \left(\frac{\frac{1}{x}}{y}\right)
\end{equation*}

\section{For Davis}
\subsection{Problem 1}
We would like to show that transitivity and symmetry do not imply reflexivity, by way of an example which is transitive and symmetric but not reflexive.

Define some relation $\sim$ on $\{1,2,3,4\}$ as the subset : 
\begin{equation}
\{ (2,2), (2,3) , (3,2) , (2,4), (4,2), (4,3), (3,4), (3,3), (4,4)\} \subset \{1,2,3,4\} \times \{1,2,3,4\}
\end{equation}
(Or, alternatively, by the equivalent below truth table for clarity)

%Use tabular to make tables. To start, you list the number of columns as an argument. Add another |c| per column you want, remove one per column you want to get rid of.
% Add "\\" to start a new line. Add "&" to go to the next row.
\begin{tabular}{|c|c|c|c|c|}
\hline 
Is $a$ related to $b$? & 1 & 2 & 3 & 4 \\ \hline
1 & False & False & False & False\\ \hline 
2 & False & True & True & True \\ \hline 
3 & False & True & True & True \\ \hline 
4 & False & True & True & True \\ \hline 
\end{tabular}


This is not reflexive, as $1 \not \sim 1$. But it is transitive, as $2 \sim 3 \text{ and } 3 \sim 4 \implies 2 \sim 4$ (alongside other transitive relations). And it is symmetric, as $ 2 \sim 3 \iff 3 \sim 2, 4 \sim 2 \iff 2 \sim 4$, etc. 

\section{For Handong}
\subsection{Problem 1}
Define a matrix : 
%This is how you define a matrix!
\begin{equation}
A = \begin{pmatrix}
     3 & -1 & 2 & -y_1\\
     2 & 1 & 1 & -y_2 \\
     1 & -3 & 0 & -y_3 \\
    \end{pmatrix}
\end{equation}
After row-reduction, one acquires:
\begin{equation}
A' = \begin{pmatrix}
     1& 0 & 0 & \frac{y_1 - 2y_2 - y_3}{2}\\[0.3em]
     0 & 1 & 0 & \frac{y_1 - 2y_2 + y_3}{6} \\[0.3em]
     0 & 0 & 1 & \frac{-7y_1 + 8y_2 + 5y_3}{6} \\
    \end{pmatrix}
\end{equation}

\section{For Rohil}
\subsection{Problem 1}
%Greek letters start with \ and then their english name. Capitalise for capital greek letter.

%Use _{ } for subscript, ^{ } for superscript.
Let $\mu$ be the average of $\alpha, \beta, \gamma$.  Then $\mu < \delta$, and $\mu \geq \sum_{j \in \mathbb{N}} 2^{-j} \epsilon_j$.

\subsection{Problem 2}
Let $A :=[ B \cup (C \cap D)] \cup U$, $\tilde{A} := \bigcup_{q \in \mathbb{Q}} \Gamma_q$.

\subsection{Problem 3}
%You should use the \langle and \rangle command for inner products as opposed to < and >:
Let $v, w$ be vectors in an inner product space. We denote their inner product as $\langle v, w \rangle$.
%Here we use the \bra and \ket command defined in the header of the file:
Or in physics notation, let $\ket{v}$ and $\ket{w}$ be states. We denote their inner product as $\langle v | w \rangle$.

%Problem numbering with amsthm is automatic based on the number of problems that have appread already.
\begin{problem}
  Here is an example of a problem statement formatted with amsthm: Let $K$ be a Galois extension of $F$ whose Galois group is $S_4$. What numbers may occur as degrees of elements of $K$ over $F$. (Recall that the degree of $\alpha \in K$ over $F$ is $[F(\alpha) : F]$).
\end{problem}

%The amsthm proof environment gives a semantic singal that a proof is beginning and provides some nice formatting sugar.
\begin{proof}
  We have $K \supset F(\alpha) \supset F$ and thus we have $[K:F(\alpha)][F(\alpha):F]$. We use the main theorem and see that the quantity $[F(\alpha):F]$ must correspond to the index of a subgroup in $S_4$. There are subgroups of every order dividing 24 in $S_4$ and it follows that elements may have any order dividing 24.
\end{proof}


\end{document}