% Program: LatexExample.ltx % Author: Erin McNicholas % Date: 1/26/07 % % Description: Example latex program for modification by students. % I've tried to include some commonly used commands. % % Note: Comments made after a % sign are not read by the compiler. % are notational comments in the code. % ********************************************************************* % ********************************************************************* % Header Commands: These are commands that format the document and % define new command shortcuts. You can use the \newcommand % function to define shortcuts for commonly used commands % ********************************************************************* \documentclass{article} \usepackage[dvips]{graphicx,color} \usepackage{a4wide} \usepackage{amsmath} \usepackage{euscript} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amsopn} \theoremstyle{definition} \newtheorem*{definition}{Definition} \newtheorem{theorem}{Theorem} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} %***************************************************************** % Text of document: This is the code that will generate % the produced text %***************************************************************** \begin{document} \begin{center} \Large{Math 251: Foundations of Advanced Mathematics} \normalsize{Solutions to Chapter 1.3} \vspace{0.2cm} \hfill {\bf Name:} Erin McNicholas \vspace{0.1cm} \hline \end{center} Above I have have demonstrated a possible heading for your homework solutions. You should feel free to format your header however you like. You might want to use \fbox{textboxes}, {\it italic script}, {\bf bold face}, \underline{underline}, etc. In this document, I will try to demonstrate some of the more common commands you'll be using in \LaTeX. To begin with, lets cover some different text sizes: \Huge{This is Huge} \LARGE{This is LARGE} \Large{This is Large} \normalsize{This is normalsize} \small{This is small} \tiny{This is tiny} \vspace{0.25cm} \normalsize You have to use two returns in \LaTeX to start a new line. All new lines will be indented unless you start with a "noindent" command. \noindent The following list was created using itemize. \begin{itemize} \item Variables and symbols should all be in math mode, but regular text should not. \LaTeX will choose font types and spacing based on which mode you are in, so you want to make sure text is in text mode and math is in math mode. \item Now lets look at some examples using tables. The "begin\{array\}" command sets the number of columns. In the commands for tables and arrays, the c, l, and r refer to the alignment of the entries in each column: center, left, or right. Columns are separated by ampersands ($\&$) and rows are ended with double backslashes ($\backslash\backslash$). \end{itemize} \begin{center} $\begin{array}{|c|c|c|c|c|} \hline \N & \Z & \Q & \R & \C\\\hline \mbox{Natural Numbers} & \mbox{Integers} & \mbox{Rationals} & \mbox{Reals} & \mbox{Complex}\\\hline \end{array}$ \end{center} \vspace{.25cm} $\begin{array}{c|c|c|ccccccccccc}P&A&B&P&\rightarrow&(A&\vee&B)&\leftrightarrow&(P&\wedge&\neg A)&\rightarrow&B\\\hline T & T & T & T & T & T & T & T & T & T & F & F & T & T\\ T & T & F & T & T & T & T & F & T & T & F & F & T & F\\ T & F & T & T & T & F & T & T & T & T & T & T & T & T\\ T & F & F & T & F & F & F & F & T & T & T & T & F & F\\ F & T & T & F & T & T & T & T & T & F & F & F & T & T\\ F & T & F & F & T & T & T & F & T & F & F & F & T & F\\ F & F & T & F & T & F & T & T & T & F & F & T & T & T\\ F & F & F & F & T & F & F & F & T & F & F & T & T & F\\ & & & 1 & 8 & 2 & 7 & 3 & 11 & 4 & 9 & 5 & 10 & 6 \end{array}$ \vspace{.25cm} $\begin{array}{lr} P\rightarrow Q&\\ R\wedge\neg Q&\\\hline \neg Q& Simplification\\ \neg P& Contrapositive \end{array}$ \vspace{0.6cm} The "mbox" command produces normal text in the math setting, while \$ signs produce math mode text. You can also switch into math mode using the "begin\{equation\}" or "begin\{eqnarray\}" commands. \begin{equation} e^7\ge x_{i^2-2} \end{equation} \begin{eqnarray} \sin(\frac{x\pi}{6})&=&\ln(y^2)\\ 2y&=&5f(x)\\\nonumber 12z &\le&\sum_{n=0}^{\infty}f(x_n)\\ &\le& g(z) \end{eqnarray} ``$\backslash$begin\{eqnarray\}'' is primarily used to generate a list of aligned equations. To get the previous equation array I entered \begin{verbatim}\begin{eqnarray} \sin(\frac{x\pi}{6})&=&\ln(y^2)\\ 2y&=&5f(x)\\\nonumber 12z &\le&\sum_{n=0}^{\infty}f(x_n)\\ &\le& g(z) \end{eqnarray}\end{verbatim}. The \& symbol is used to distinguish where the equations are to be aligned, the $\backslash\backslash$ symbol is used to end a line in the equation array. Here are some other examples: \vspace{0.25cm} Let $K$ be the statement "you will kiss me," and let $J$ be the statement "I will dance a jig." The question is whether $(K\rightarrow J)\wedge J$ implies $K$, which it does not. You can verify it doesn't using a truth table. The failing case is when $K$ is false, and $J$ is true. Thus $(K\rightarrow J)\wedge J$ is true while $K$ is false. \vspace{0.5cm} \begin{enumerate} \item $e^5\le 0$ \item $3\ge 5$ and $7<8$ \item $\sin(\pi/2)\ge 0$ or $\tan(0)<0$ \item $y=3$ and $y^2\not= 7$ \item $w-3>0$ and $w^2+9\le 6w$ \item $a-b=c$ and $a\not= b+c$, or $a-b \not= c$ and $a=b+c$ \end{enumerate} \vspace{0.6cm} \begin{itemize} \item[] $\neg(P\rightarrow \neg Q)\Leftrightarrow P\wedge Q$ \item[] $A\rightarrow (A\wedge B)\Leftrightarrow A\rightarrow B$ \item[] Tautology $(X\wedge Y)\rightarrow X \Leftrightarrow X\rightarrow X$ \item[] Contradiction $\neg(M\vee L)\wedge L)\Leftrightarrow\neg L \wedge L$ \item[] $(P\rightarrow Q)\vee Q \Leftrightarrow P\rightarrow Q$ \item[] $\neg(X\rightarrow Y)\vee Y\Leftrightarrow (X\wedge \neg Y)\vee Y\Leftrightarrow X\vee Y$ \end{itemize} \vspace{0.6cm} Here's a sample proof. \vspace{0.3cm} \noindent \underline{proposition:} If $P$ is an invertible $n$x$n$ matrix and $C$ is any $k$x$n$ matrix, then $rank(CP)=rank(C)$ and $nullity(CP)=nullity(C)$. \vspace{0.4cm} \fbox{proof} For all matrices $rank(A)=rank(A^T)$. Thus, $rank(CP)=rank((CP)^T)=rank(P^TC^T)$. If $P$ is invertible, then so too must $P^T$ be. Thus, by our B.K-A.U. Theorem, $P^T$ is expressible as the product of elementary matrices. Multiplying $C^T$ by $P^T$ is equivalent to performing elementary row operations on $C^T$, which by theorem 7.3.7 does not change the row space or null space. Thus, $rank(CP)=rank((CP)^T)=rank(P^TC^T)=dim(row(P^TC^T))=dim(row(C^T))=rank(C^T)=rank(C)$. By the dimension theorem and our previous result, $nullity(CP)=k-rank(CP)=k-rank(C)=nullity(C)$. $\blacksquare$ \end{document}