\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}