% We can add comments which you don't appear in the paper like this -- using the "percentage" symbol in the beginning
%
% Through this document, you'll notice that LaTeX commands begin with a backslash.  The first one is here; it tells
% LaTeX that we're writing an article, as opposed to a book or some other document type.  Your writing project
% doesn't need to be a full blown article, but this is the most appropriate choice.
\documentclass{article}

\usepackage{fullpage} % For standard smaller margins; You may comment this line to change margins
%\usepackage[margin=1in]{geometry}  % uncomment and play with this line to change margins in your document

\usepackage{amsthm, amsmath, amssymb, mathrsfs} %These are standard packages which you will need in almost every writing involving a lot of math.

\usepackage{hyperref} %For displaying urls

% Title and author information
\title{MATH 3283W \\ \TeX\ project template}
\author{Author's Name}
\date{Fall 2016}

% Here's where the document actually starts.
\begin{document}

% This creates the title and copies in your name and date, as defined above.
\maketitle

\section{Section Title}
The odds are that you won't need sections in your writing project, but this template includes a few section headers, so that you can see how \LaTeX automatically numbers them.

Perhaps you need to write things in a list:

\begin{itemize}
 \item like
\item this
\item here
\end{itemize}

Or maybe you need a list with numbers.
\begin{enumerate}
 \item like
\item that
\begin{enumerate}
\item you can even
\item have lists within lists!
\end{enumerate}

\end{enumerate}

\section{Second Section}
You can do math mode in the middle of the line, like this $x + y = 5$, or you can do 
``displaymath'' mode to get a separate line of math, like this: \[x + y = 5. \]

\section{Third Section}
Here are some more useful mathmode symbols.  Some of the code below uses the command \texttt{\textbackslash{quad}}, which inserts a bit of extra space in math mode.
\begin{itemize}
 \item $\mathbb{Z}, \; \mathbb{N}, \; \mathbb{R}$
\item $x_{n+1}, \quad x^{5+y}$
\item $\frac{4}{z-2}, \dfrac{4}{z-2}$
\item $\epsilon, \varepsilon, \delta, \Delta$
\item $\in, \; \notin$
\item $\subset, \; \subseteq, \; \subsetneq$ 
\item $\neq, \; \leq, \; \geq $
\item $\to, \; \rightarrow, \; \leftarrow$
\end{itemize}
Here are some more... but notice that sometimes things don't look as nice as you want them to... so use \texttt{\textbackslash{displaystyle}} in math mode to make the symbols larger.
\begin{itemize}
\item $\int_1^\infty{x^2}  \, dx \quad$  vs. $\quad \displaystyle{\int_1^\infty{x^2}  \, dx}$
\item $\sum_{i=1}^{\infty} e^{i^2} \quad$ vs. $\quad \displaystyle{\sum_{i=1}^\infty e^{i^2}}$
\item $\lim_{n\to\infty} \frac{a_n}{n} = 0 \quad$ vs. $\quad \displaystyle{\lim_{n\to\infty} \frac{a_n}{n} = 0}$

\end{itemize}

In contrast to Google Docs, LaTeX can make just about anything look beautiful and professional.
For instance, multi-line equations:
\begin{align*} %remove the * if you want each equation to be numbered for reference.
|x-y| &\le |x|+|y| \\
&< \frac{\epsilon}{2} + \frac{\epsilon}{2}\\
& = \epsilon
\end{align*}


\noindent There are lots of other things you can do within Latex!
Many of them you can figure out by googling your question and the word \texttt{latex}. 
Try googling ``piecewise function latex" with no quotes.\\


\section{Useful Tool}
You might find the website \url{http://detexify.kirelabs.org/classify.html} useful. You can draw symbols and it will 
tell you the \LaTeX code for them.  Another useful online reference is \url{http://en.wikibooks.org/wiki/LaTeX/}.


\section{Professor Mosher's List of Symbols}

Professor Mosher wrote a long template document with many mathematical symbols.  There's some overlap with what has already been written above, but because it is so comprehensive it is included for your benefit here.

\subsection{How to state a problem}

Here is a way to format the statement of your problem:
\vspace{1em}

\noindent \textbf{2.1 \#20.} Prove that $A \cap B$ and $A - B$ are disjoint and $$A=(A \cap B) \cup (A-B).$$

\noindent \textit{Proof:} Here is where your solution will go. \qed

\bigskip

What follows are symbols that might be relevant to this course.  It's definitely not complete in any sense. Delete all of this in your project submission, of course.  You can also search the internet -- google ``latex if and only if'' and you'll quickly see the \TeX\ command \begin{verbatim} \Leftrightarrow \end{verbatim} which produces $\Leftrightarrow$.

\section{Logic and proof}

Here are some logical connectives: $\sim p$, $p \wedge q$, $p \vee q$, $p \Rightarrow q$, $p \Leftrightarrow q$.

\bigskip

Here is how you make a table:

\bigskip

\begin{tabular}{| c | c | c |} 
% this means: vertical line, centered entry, vertical line, etc. Use r, l in place of c if needed.
\hline
$p$ & $q$ & $p \Rightarrow q$ \\
\hline
T & T & T \\
T & F & F \\
F & T & T \\
F & F & T \\
\hline
\end{tabular}

\bigskip

Here are some quantifier symbols:

$$\forall x, \exists y \ni x < y.$$

\section{Sets and functions}

Here is some set notation: $x \in A$, $x \not\in A$, $A = B$, $A \not= B$, $A \subset B$, $A \subseteq B$, $A \not\subseteq B$, $A = \{ 1,2,3,4 \}$, ${\bf R}$, ${\mathbb R}$, ${\bf Q}$, ${\mathbb Q}$, ${\bf Z}$, ${\mathbb Z}$

% Sometimes I use bold and sometimes I use "blackboard bold" for the standard sets of numbers.

$[0,\infty)$, $\emptyset$, $\varnothing$

% I like \varnothing better than \emptyset

$\mathscr{A} = \{ A_i: i\in \mathbb{N} \}$

$$\bigcup_{i=1}^\infty A_i$$

$$\bigcap_{i\in \mathbb{N}} A_i$$

$A_1 \cup A_2 \cup \cdots A_3 \cup \cdots$

$$A \times B =\{ (a,b): a\in A \mbox{ and } b\in B \}$$

$\left< 2,4 \right> \in (1,3) \times (3,5)$

$f:A \to B$, $g:B \to C$, $g \circ f: A \to C$

$f^{-1}:B \to A$.

$g(x) = \sqrt{x-1}$

$h(x) = \sqrt[3]{x-1}$

$f(x) = \sin x$

$f(x) = 1/x$

$f(x) = \frac{1}{x}$

$$f(x) = \frac{1}{x}$$

$f(x) = 2x^{12}  + 3x^3 - \pi$

$$b_n = \begin{cases} 
2, &\mbox{if } a_{nn} \not= 2, \\
3, &\mbox{if } a_{nn} = 2.
\end{cases}$$

$|S| \leq |T|$

$|S| < \aleph_0$

$\mathscr{P}(A)$

$\mathcal{P}(B)$

\section{Natural numbers}

$$\frac{1}{3} + \frac{1}{15} + \cdots + \frac{1}{n(n+1)} = \frac{n}{n+1}, \mbox{ for all } n\in \mathbb{N}.$$

$$\binom{n}{r} = \frac{n!}{r!(n-r)!}$$

$\sup f(D) \leq \inf g(C)$

$$\left\{ \frac{n}{n+1}: n \in \mathbb{N}  \right\}$$

$\left\{ \frac{n}{n+1}: n \in \mathbb{N}  \right\}$

\bigskip

$\{ \frac{n}{n+1}: n \in \mathbb{N}  \}$

Let $\epsilon > 0$.

Let $\varepsilon > 0$. % I like this one better.

For all $\varepsilon > 0$, there exists $N\in \mathbb{N}$ such that if $n>N$, then \dots

For all $\varepsilon > 0$, there exists $\delta > 0$ such that \dots

$A = \partial B$

$A = \overline{B}$

$A = \mathring{B}$

\section{Sequences}

$$\lim_{n\to \infty} a_n = 1.$$

$$\left| \frac{1}{t_n} - \frac{1}{t} \right| < \varepsilon.$$

$$s_n = \sqrt{1+ \sqrt{1 + \sqrt{1 + \cdots}}}$$

$$s_m - s_n > \underbrace{
\frac{1}{m} + \frac{1}{m} + \cdots + \frac{1}{m}
}_{m-n \mbox{ \small{times}}}
= \frac{m-n}{m}$$

\section{Series}

$$s_n = \sum_{k=0}^n r^k$$

$\sum a_n = +\infty$.

$$\int_1^n \frac{1}{x^p} \, dx$$

$\int_1^n \frac{1}{x^p} \, dx$


\end{document}
