\documentclass[12pt]{amsart}

\usepackage{verbatim}
\usepackage{fullpage}

\usepackage{amssymb,amsmath,amsthm}
\usepackage{amsfonts}
\usepackage[mathscr]{eucal}

\input{isomath}
\input{mathenv}

\renewcommand{\theequation}{\arabic{problem}.\arabic{equation}}
\renewcommand{\labelenumi}{(\alph{enumi})}
\def\ve{\varepsilon}

%--------------------------------------------------------------------
\begin{document}
\title{Assignment 1 $\cdot$ MAT 999 $\cdot$ Fall 1798}
\author{C. F. Gauss}
\date{\today}
\maketitle

%% ----------------------------------------------------------------

This is a 5-minute intro to \LaTeX for use in typing up homework.  That is, we
are not writing to formal formatting requirements, e.g.  for a thesis.  This
file isn't exhaustive (there already are plenty of good documents on \LaTeX)
--- instead, this is a quick sample of some frequently used symbols and
formattings, in a format hopefully less intimidating than a 100-page manual.
It's easy to use \LaTeX to make your homework nice, so don't fear!

As you read, please look at the output (e.g.
\texttt{.dvi} or
\texttt{.pdf}
file) as well as the
source (\texttt{.tex}) so you will know what I'm talking about.

John Kerl (\texttt{kerl at mathpost dot asu dot edu}, now at
\texttt{kerl.john.r@gmail.com}) 2003/09/18

%% ----------------------------------------------------------------
\begin{problem*}[1]
Here is what you are supposed to prove.

\begin{proof}
Here is where you prove it.
Notice that here I didn't supply an alternate name, so I got ``Proof''.
\end{proof}
\end{problem*}

%% ----------------------------------------------------------------
\begin{problem*}[2]
Here is what you are supposed to prove.

\begin{proof}[Here is where you specify an alternate name for your proof]
For example, I might use that to say \emph{Proof by contradiction}.
\end{proof}
\end{problem*}

%% ----------------------------------------------------------------
\begin{problem*}[3]
Here is what you are supposed to prove.

\begin{answer}
For text, just type it in as usual.  Three notable exceptions:
(1) use a triple dash --- here for example --- (2) use \texttt{ldots} instead
of just typing in three periods, i.e. \ldots instead of ..., and (3) rather
than using the double-quote key on your keyboard, use double backticks for open
double quote and double apostrophe for close double quote.  You get not "this"
but ``this'' which looks nicer.

A blank line starts a new paragraph.
Other
than
that,
though,
paragraphs are formatted
for you (in particular,
compare this sentence in the \texttt{.tex} file vs in the
\texttt{.dvi} or
\texttt{.pdf} output).  \emph{This is how you make italics.}

For mathematical content, if you want put a statement in-line like this: $G
\lhd H$ then just use single dollar signs.  If you want it on its own line like
this: $$ G \lhd H $$ then use double dollar signs.  If you have multi-line
statements, you might use \texttt{eqnarray*} as follows.  (Notice that the
double ampersands control vertical alignment (usually, but not necessarily, you
align on the equals signs), and you must use a double backslash at the end of
each line in the \texttt{eqnarray*}, except optionally the last.)

\begin{eqnarray*}
	(xy)^3 &=& xyxyxy \\
	&=& xxxyyy \textrm{ since we have commutativity here}\\
	&=& x^3y^3
\end{eqnarray*}

For superscripts and subscripts, just use caret and underscore, e.g.
$a_{i,j}^2, M_{\sigma(i)}^{p_{n_k}}$.  Remember that if you have a single
character, you can just put it after the caret, but if you have more than one,
use curly braces.  E.g. $x^12$ is not the same as $x^{12}$.  Since curly braces
are used for clumping, you have to escape them with a backslash if you want the
curly braces to be visible: $N = \{1,2,\ldots,n\}$.

It's considered polite to make cos, sin, log etc. upright instead of italic.
E.g. $\cos(x)$ or $\mathrm{cos}(x)$ rather than $cos(x)$.

Notice your whitespace isn't used in math expressions, e.g. this $xyz$ looks
the same as this $x  y  z$.  But sometimes you want to insert some space ---
e.g. in cycle notation for permutations, compare $(1 2 3)(5 7)$ to $(1\,2\,3)
(5\,7)$.  Here are some examples of how to insert extra spacing: $xy$, $x\,y$,
$x\:y$, $x\;y$, $x\quad y$, $x\qquad y$.

If you want to make a table, you can use something like this:

\bigskip
\begin{tabular}{|c|c|c|c|}
\hline $x$ & $\sqrt{x}$ & $1/x$ & lah-dee-dah \\
\hline
\hline  1 & 1 & 1 & haw \\
\hline  4 & 2 & 1/4 & haw \\
\hline  9 & 3 & 1/9 & haw \\
\hline
\end{tabular}
\bigskip

Sometimes, though, I use \texttt{verbatim} for pre-formatted ASCII,
that is, if I have a text file that already looks the way I want it
and I don't want to fuss with the tabular environment:

\begin{verbatim}
mod 13:             mod 19:             mod 1f:           
02 = 0       [  0]  02 = 0       [  0]  02 = 0       [  0]
03 = 1       [  1]  03 = 1       [  1]  03 = 1       [  1]
07 = 6 7     [  3]  07 = a b     [  3]  07 = c d     [  3]
13 = 2 3 4 5 [ 15]  13 = 6 7 c d [ 15]  13 = 6 7 a b [ 15]
19 = 9 b d e [ 15]  19 = 2 4 9 e [ 15]  19 = 3 5 9 e [ 15]
1f = 8 a c f [  5]  1f = 3 5 8 f [  5]  1f = 2 4 8 f [  5]
\end{verbatim}

Here is a matrix example:
$$
\left(\begin{array}{rrr}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9 \\
\end{array}\right)
+
\left(\begin{array}{rrr}
1 & 1 & 1 \\
2 & 2 & 2 \\
3 & 3 & 3 \\
\end{array}\right)
=
\left(\begin{array}{rrr}
2 & 3 & 4 \\
6 & 7 & 8 \\
10 & 11 & 12 \\
\end{array}\right)
$$

Here is a bigger matrix:
$$
M =
\left(\begin{array}{rrrrr}
a_{1,1} & a_{1,2} & \cdots & a_{n,n} \\
a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\
\vdots  & \vdots  & \ddots & \vdots  \\
a_{n,1} & a_{n,2} & \cdots & a_{n,n} \\
\end{array}\right)
$$

For parentheses, square brackets, and curly braces wrapped around bigger stuff,
use \texttt{\textbackslash{}left(} and \texttt{\textbackslash{}right)}.  These
will cause the parentheses (or brackets or braces) to be sized to fit.  E.g.
not
$$
	(\frac{123}{456})
$$
but rather
$$
	\left(\frac{123}{456}\right).
$$

Sums, products and integrals use subscripts and superscripts just like
anything else:

$$
	F(k) = \int_{-\infty}^{\infty} e^{-i2 \pi k x} \dx , \quad
	e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} , \quad
	f(x) = \prod_{\sigma \in G} \sigma(x)^{f(\sigma)}
$$

Path integrals use $\oint$.

Here is itemize:

\begin{itemize}
\item[a.] Here is one item.
\item[b.] Here is another item.

	Note that I have indentation here.

\item[c.] The last one.
\end{itemize}

Here is itemize with default bullets:
\begin{itemize}
\item Here is one item.
\item Here is another item.
\end{itemize}

Here is enumerate:
\begin{enumerate}
\item Here is one item.
\item Here is one item.
\end{enumerate}

Greek letters are just a backslash followed by their usual name, e.g. $\alpha$,
$\beta$, $\gamma$.  Uppercase them by using an uppercase name in the \LaTeX
source, e.g. $\Gamma$.  Note, however, that there aren't codes for some
uppercase Greek letters, since they are identical to a Roman capital (e.g.
capital alpha is just a capital A).  Also, there's no lower-case omicron,
since that's just a Roman o.

$\alpha \beta \gamma \delta \epsilon (\varepsilon) \zeta \eta \theta
(\vartheta) \iota \kappa \lambda \mu \nu \xi o \pi \rho \sigma \tau \upsilon
\phi (\varphi) \chi \psi \omega$.

$A B \Gamma \Delta E Z H \Theta I K \Lambda M N \Xi O \Pi P \Sigma T \Upsilon
\Phi X \Psi \Omega$.

Some alternate typefaces:
$$\mathcal{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$
$$\mathscr{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$
$$\bbb{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$
$$\mathbf{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$
$$\mathbf{abcdefghijklmnopqrstuvwxyz}$$
$$\mathfrak{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$
$$\mathfrak{abcdefghijklmnopqrstuvwxyz}$$
The \texttt{isomath.tex} file which I
obtained from Dr. Kaliszewski (\texttt{isomath.tex} is included by this file,
for example) has abbreviations for frequently used symbols such as $\R$.  More
examples: $\C$ $\Q$ $\Z_p$ $\F_p^n$ $\mathcal{S}_n$ $\mathcal{A}_n$
$\mathcal{D}_n$.

Sizes:
\begin{itemize}
\item \begin{tiny}tiny: Here is some text\end{tiny}
\item \begin{scriptsize}scriptsize: Here is some text\end{scriptsize}
\item \begin{footnotesize}footnotesize: Here is some text\end{footnotesize}
\item \begin{small}small: Here is some text\end{small}
\item \begin{normalsize}normalsize: Here is some text\end{normalsize}
\item \begin{large}large: Here is some text\end{large}
\item \begin{Large}Large: Here is some text\end{Large}
\item \begin{LARGE}LARGE: Here is some text\end{LARGE}
\item \begin{huge}huge: Here is some text\end{huge}
\item \begin{Huge}Huge: Here is some text\end{Huge}
\end{itemize}

The \texttt{mathenv.tex} file, which I also obtained from Dr. Kaliszewski, has
nice environments such as problem, proof, answer, etc. as used above.

\begin{lem*}
Here is an example of the lemma environment.
\begin{proof}
Here is your proof of your lemma.  Notice the nice little white box which lets
people know you're done.
\end{proof}
\end{lem*}

Here are several symbols: $a \cdot b$, $a \circ b$, $a \pm b$, $a < b$, $a \not
< b$, $a \le b$, $a > b$, $a \not > b$, $a \ge b$, $a = b$, $a \ne b$, $a \gg
b$, $a \ll b$, $a/b$, $\frac{a}{b}$, $a \choose b$, $\left(\frac{a}{b}\right)$,
$(\frac{a}{b})$.  Notice that you can put \texttt{not} in front of many
operators to negate them.

$y=\sqrt{x}$, $y=\sqrt[3]{x}$, $y=\sqrt[4]{x}$, $f:A \to B$, $x \mapsto y$, $A
\leftarrow B$, $A \hookleftarrow B$, $A \hookrightarrow B$, $A \leftrightarrow
B$, $\lfloor x \rfloor$, $\lceil x \rceil$.

$|A|$, $a \mid b$, $a \nmid b$, $A\overline{BCD}E$,
$\underbrace{A \times \ldots \times A}_{r \;\textrm{times}}$,
$\overbrace{A \times \ldots \times A}^{r \;\textrm{times}}$,
$A \cap B$, $A \cup B$,
$A \setminus B$, $A \wedge B$, $A \subset B$, $A \subseteq B$, $\langle x
\rangle$, $A \times B$, $A \otimes B$, $A \oplus B$,
$A \lhd B$,
$A \rhd B$,
$A \unlhd B$,
$A \unrhd B$,
$A \ltimes B$,
$A \rtimes B$,
$x \in A$, $x \not \in A$,
$A \not \subseteq B$, $(\Z_p, \oplus)$, $(\Z_p, \odot)$, $A \sim B$, $A \simeq
B$, $A \cong B$, $A \approx B$, $a \equiv b \pmod p$, $a \not \equiv b \pmod
p$.

Math mode: $\acute{a}$, $\bar{a}$, $\breve{a}$, $\check{a}$, $\ddot{a}$,
$\dot{a}$, $\grave{a}$, $\hat{a}$, $\vec{a}$, $\tilde{a}$, $X / \sim$,
$\tau_\sim$, $\Delta$, $\nabla$.

Text mode: \"{a}, \'{a}, \.{a}, \={a}, \^{a}, \`{a}, \`{a}, \H{a}, \r{a},
\u{a}, \v{a}, \~{a}.

Pr\"{u}fer, Erd\H{o}s, \v{C}ech, Kl{\o}ve, Osterg\r{a}rd, polyn\^{o}me,
th\'{e}or\`{e}me, na\"{\i}ve, lima\c{c}on, Gro{\ss}encharacter, \textdollar
2.56.

$$
(a)
\big(b\big)
\Big(c\Big)
\bigg(d\bigg)
\Bigg(e\Bigg)
$$

$$
x_i = \left\{
\begin{array}{ll}
m_{d_i,k}, & \textrm{if } d_i > 0 \\
1, & \textrm{if } i = k \\
0, & \textrm{otherwise}
\end{array}
\right.
$$

\bigskip

Here is an example of using the picture environment.  Note (see the {\LaTeX}
source) that the run and rise arguments to the line and vector commands (in
parentheses) must be integers between -6 and 6.  The length argument (in curly
braces) may be any floating-point number.  However, it signifies the length of
the horizontal projection, not the length of the hypotenuse.  (For vertical
lines, i.e. (0,1) or (0,-1), it is the line length.)

\setlength{\unitlength}{1.0in}
\begin{picture}(6.0,1.25)
%\put(0.0,0.0){\framebox(6.0,1.25)}
\put(0.2,0.1){$A$}
\put(0.4,0.15){\vector(1,0){0.7}}
\put(0.4,0.25){\vector(1,1){0.7}}
\put(0.4,0.5){$\alpha$}
\put(0.8,0.25){$\phi$}
\put(1.2,0.1){$B$}
\put(1.25,0.25){\vector(0,1){0.7}}
\put(1.2,1.0){$C$}
\put(1.4,0.5){$\pi$}
\end{picture}

Here is another example:

\setlength{\unitlength}{1.0in}
\begin{picture}(6.0,1.75)
%\put(0.0,0.0){\framebox(6.0,1.75)}
\put(0.5,0.2){\line(1,0){1.0}}
\put(0.5,0.2){\line(0,1){1.0}}
\put(0.5,1.2){\line(1,0){1.0}}
\put(1.5,0.2){\line(0,1){1.0}}
%
\put(0.75,0.45){\line(1,0){1.0}}
\put(0.75,0.45){\line(0,1){1.0}}
\put(0.75,1.45){\line(1,0){1.0}}
\put(1.75,0.45){\line(0,1){1.0}}
%
\put(0.5,0.2){\line(1,1){0.25}}
\put(1.5,0.2){\line(1,1){0.25}}
\put(0.5,1.2){\line(1,1){0.25}}
\put(1.5,1.2){\line(1,1){0.25}}
%
\put(0.0,0.15){\small{000}}
\put(1.8,0.15){\small{100}}
\put(0.0,1.15){\small{001}}
\put(1.8,1.15){\small{101}}
\put(0.25,0.40){\small{010}}
\put(2.05,0.40){\small{110}}
\put(0.25,1.40){\small{011}}
\put(2.05,1.40){\small{111}}
%
\thicklines
\put(0.5,0.2){\line(1,1){1.25}}
\thinlines

%%%

\put(3.5,0.2){\line(1,0){1.0}}
\put(3.5,0.2){\line(0,1){1.0}}
\put(3.5,1.2){\line(1,0){1.0}}
\put(4.5,0.2){\line(0,1){1.0}}
%
\put(3.75,0.45){\line(1,0){1.0}}
\put(3.75,0.45){\line(0,1){1.0}}
\put(3.75,1.45){\line(1,0){1.0}}
\put(4.75,0.45){\line(0,1){1.0}}
%
\put(3.5,0.2){\line(1,1){0.25}}
\put(4.5,0.2){\line(1,1){0.25}}
\put(3.5,1.2){\line(1,1){0.25}}
\put(4.5,1.2){\line(1,1){0.25}}
%
\put(3.0,0.15){\small{000}}
\put(4.8,0.15){\small{100}}
\put(3.0,1.15){\small{001}}
\put(4.8,1.15){\small{101}}
\put(3.25,0.40){\small{010}}
\put(5.05,0.40){\small{110}}
\put(3.25,1.40){\small{011}}
\put(5.05,1.40){\small{111}}
%
\thicklines
\put(3.5,0.2){\line(1,1){1.00}}
\put(3.5,0.2){\line(1,5){0.25}}
\put(3.5,0.2){\line(5,1){1.25}}
\put(3.75,1.45){\line(3,-1){0.75}}
\put(4.75,0.45){\line(-1,3){0.25}}
\put(4.75,0.45){\line(-1,1){1.00}}
\thinlines
\end{picture}

For more symbols, Google for \texttt{lshort.pdf}, and/or see the Computing
link at \texttt{math.asu.edu}.

%$$
%\mathrm{erfc}(x) = \frac{e^{-x^2}}{x \sqrt{\pi}} \left(
%1 - \frac{1/2}{
%2 + X - \frac{1 \cdot 3/2}{
%4 + X - \frac{2 \cdot 5/2}{
%6 + X - \ddots
%}
%}
%}
%\right), \quad X = x^2 - 1/2
%$$

\end{answer}
\end{problem*}
\end{document}
