diff options
author | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
---|---|---|
committer | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
commit | e0c6872cf40896c7be36b11dcc744620f10adf1d (patch) | |
tree | 60335e10d2f4354b0674ec22d7b53f0f8abee672 /usergrps/uktug/baskervi/4_5/bailey.tex |
Initial commit
Diffstat (limited to 'usergrps/uktug/baskervi/4_5/bailey.tex')
-rw-r--r-- | usergrps/uktug/baskervi/4_5/bailey.tex | 433 |
1 files changed, 433 insertions, 0 deletions
diff --git a/usergrps/uktug/baskervi/4_5/bailey.tex b/usergrps/uktug/baskervi/4_5/bailey.tex new file mode 100644 index 0000000000..33ab346155 --- /dev/null +++ b/usergrps/uktug/baskervi/4_5/bailey.tex @@ -0,0 +1,433 @@ +\title{Maths in \LaTeX: Part~2, Getting more Serious} +\author[R.~A.~Bailey]{R.~A.~Bailey\\ +School of Mathematical Sciences\\ +Queen Mary and Westfield College\\ +Mile End Road\\ +London E1 4NS} +\def\realrm{\rmfamily\upshape} +\newtheorem{preqn}{Exercise} +\newenvironment{qn}{\begin{preqn}\realrm}{\end{preqn}} +\newcommand{\lamport}{{\it \LaTeX: A Document Preparation System\/} by Leslie +Lamport} +\newcommand{\shortlamp}{\emph{The Manual}} +\let\torl\AllTeX +\begin{Article} +\section*{Recall} + +This is the second in a sequence of tutorials on typesetting +Mathematics in \LaTeX. It includes some things which can be found in +\lamport, but I am gradually working in more things which, while +straightforward and necessary for Mathematical work, are not in +\shortlamp. In case you missed the first tutorial, two warnings are +now repeated. + +I expect you, the reader, to do some work. Every so +often comes a group of exercises, which you are supposed to do. Use \LaTeX\ to +typeset everything in the exercise except sentences in italics, which are +instructions. If you are not satisfied that you can do the exercise, then tell +me. Either write +to me at Queen Mary and Westfield College (my full +address is given at the end of this article) +with hard copy of your input and output, or email me at +\mbox{\ttfamily r.a.bailey@qmw.ac.uk} with a copy of the smallest +possible piece of \LaTeX\ input file that contains your attempt at the +answer. In either case I will include a solution in the following +issue of \BV: you will remain anonymous. + +A word on the controversial issue of fonts. Fonts in Mathematics are +handled differently in \LaTeX\ 2.09, in NFSS, and in the new standard +\LaTeX, \LaTeXe. Rather than compare these systems every time that I +mention fonts, I limit myself to \LaTeX\ 2.09. When you upgrade to +\LaTeXe, all these commands will still work, so long as you use the +standard styles \latexword{article}, \latexword{report} and +\latexword{book}. + +\addtocounter{section}{1} +\section{What does it mean?} +\subsection{Arrows} +Arrows are relations. Four of them are +\[ +\begin{tabular}{cc@{\qquad}cc} +\verb!\leftarrow! & $\leftarrow$ & \verb!\longleftarrow! & $\longleftarrow$\\ +\verb!\Leftarrow! & $\Leftarrow$ & \verb!\Longleftarrow! & $\Longleftarrow$ +\end{tabular} +\] +and eight others can be obtained by replacing {\tt left} by {\tt right} (all +versions) or by +{\tt up} or {\tt down} (not the long ones). A full list is given on page~45 of +\shortlamp. + +Because of its frequent use in defining functions, \verb+\rightarrow+ has the +short alternative name \verb+\to+. It should not be confused with +\verb+\mapsto+, which is also used in defining functions. +\begin{quote} +\verb+x\mapsto g^{-1}xg+ \qquad +$x\mapsto g^{-1}xg$ +\end{quote} + +The relation \verb+\iff+, which does indeed stand for the relation `iff', +is not quite the same as \verb+\Longleftrightarrow+: it has a little more +space on either side. +\[ +\begin{tabular}{c} +\verb!gh^{-1} \in K \iff g \in Kh! \\ +$gh^{-1} \in K \iff g \in Kh$ +\end{tabular} +\] + +\subsection{Fences} +Mathematical typesetters use the word \emph{fence} for anything like a bracket +that comes with a mate to enclose part of a formula. \TeX\ calls them \emph{ +delimiters}. Obvious examples are parentheses, brackets and braces, produced +with \verb+(+, \verb+)+, \verb+[+, \verb+]+, \verb+\{+ and \verb+\}+ +respectively. Angle brackets are produced with \verb+\langle+ and +\verb+\rangle+: +\begin{quote} +\verb+\langle u,v \rangle+ \qquad $\langle u,v \rangle$ +\end{quote} +Don't try to use the keyboard symbols \verb+<+ and \verb+>+ for this: +apart from the fact that they do not look right in typeset Maths, +\torl\ +does not think that they are fences and may well split the line between the $<$ +and the~$u$. + +Use \verb+\lfloor+, \verb+\rfloor+, \verb+\lceil+ and \verb+\rceil+ to obtain +`floor' and `ceiling': +\begin{quote} +\verb+\lceil 3.75 \rceil = 4+ \qquad $\lceil 3.75 \rceil =4$ +\end{quote} +Other fences are shown on page~48 of \shortlamp. + +Fences need to grow to enclose large formulas. They will automatically grow +to the correct size if you preface the opening fence with \verb+\left+ and the +closing fence with \verb+\right+. +The \verb+\left+s and \verb+\right+s must come in +properly matching pairs, but the fences which they qualify need not obviously +match. Compare +\begin{quote} +$\frac{1}{n} \in (0,1]$ \quad with \quad $\frac{1}{3} \in \left( 0, \frac{1}{2} +\right]$: +\end{quote} +the second half-open interval is produced with +\begin{verbatim} + \left( 0, \frac{1}{2} \right] +\end{verbatim} + +The solidus \verb+/+ is really a binary operator, but it is treated by \TeX\ as +a fence, partly because traditional typesetters do not put the same amount of +space around it as they do around other binary operators, partly because it +needs to grow when it is between two tall formulas. But it can't grow +unless it is matched with another fence, and it doesn't need one. So \TeX\ +makes the full stop into an invisible fence +(called the \emph{null delimiter}) when it is preceded by +\verb!\left! or \verb!\right!. +%provides the invisible fences \verb+\left.+ and \verb+\right.+\nobreak\,: +\[ +\begin{tabular}{c} +\verb!a(b+c) \left / \frac{1}{xy} \right.! \\[2\jot] +$\displaystyle a(b+c) \left / \frac{1}{xy} \right.$ +\end{tabular} +\] +How would the above expression be different if you typed +\verb!\left. a(b+c) \right /! \ldots? + +Sometimes you need to use \verb+\left+ and \verb+\right+ just to tell \TeX\ +that you are using fences, even if you do not need them to change size. A good +example is $\left|\phantom{A}\right|$ used for the modulus or cardinality +functions. If you type \verb!|+3|!, \TeX\ will typeset the first~$|$ as if +it is being added to the~$3$. +\[ +\begin{tabular}{c@{\qquad}c} +\verb!| +3 |! & $| +3 |$\\ +\verb!\left| +3 \right|! & $\left| +3 \right|$ +\end{tabular} +\] + + +\subsection{Standard functions with English names} +Some standard functions have written names with two or more letters, based on +their full English name. An example is $\cos$ for `cosine'. It is no good +simply typing \verb+cos+, for then the output will look like $c$~multiplied by +$o$ multiplied by~$s$. So there are standard commands such as \verb+\cos+, +\verb+\sin+, \verb+\log+, \verb+\exp+ and \verb+\dim+. +\begin{quote} +\verb!\cos\pi = -1! \qquad $\cos\pi = -1$ +\end{quote} +The following ten standard functions +\[ +\begin{tabular}{ccccc} +\verb+\max+ & \verb+\sup+& \verb+\limsup+ & \verb+\lim+ & \verb+\det+ \\ +\verb+\min+ & \verb+\inf+ & \verb+\liminf+ & \verb+\gcd+ & \verb+\Pr+ +\end{tabular} +\] +can have expressions above and/or below them to show what range of variables +they apply to: these are typed in as if they were super- or subscripts. +\[ +\begin{tabular}{c} +\verb!\lim_{x \to 0} \frac{\sin x}{x} = 1! \\[2\jot] +$\displaystyle\lim_{x \to 0} \frac{\sin x}{x} = 1$\\[4\jot] +\verb!\max_{i=1}^{n} \theta_i! \qquad $\displaystyle\max_{i=1}^{n} \theta_i$ +\end{tabular} +\] +In displayed Maths the range expressions come above and below the name of the +standard function; in text they come somewhat to the right: +$\max_{i=1}^{n} \theta_i$. + +The standard function `modulo' has two forms: +\[ +\begin{tabular}{ll} +\verb!7 = 10 \bmod 3! & $7 = 10 \bmod 3$\\ +\verb!7 = 10 \pmod 3! & $7 = 10 \pmod 3$ +\end{tabular} +\] + + +The remaining standard functions are listed on page~46 of \shortlamp. + + +\subsection{Large repeated binary operators} +When a binary operator is commutative and associative it often has a a special +symbol to show its repeated application: $\sum$ for repeated~$+$, $\bigcap$ for +repeated~$\cap$, and so on. Some of these have special commands in \TeX: +\[ +\begin{array}{cc@{\qquad}cc@{\qquad}cc} +\verb!\sum! & \sum & \verb!\bigcap! & \bigcap & \verb!\bigoplus! & \bigoplus\\ +\verb!\prod! & \prod & \verb!\bigcup! & \bigcup & \verb!\bigotimes! & +\bigotimes +\end{array} +\] +(Note that the operator \verb+\sum+ is \emph{not} the same as the Greek letter +\verb+\Sigma+.) +More are shown on page~45 of \shortlamp. Each of these has two sizes: the big +one is used in displayed Maths (except inside arrays and fractions) and the +small one is used in text. Each can also take sub- and superscripts to show the +range of operation: as with the ten listed standard functions, these sub- and +superscripts appear above and below the operator in displayed Maths (except +inside arrays and fractions) and a little to the right otherwise. +\[ +\begin{tabular}{c} +\verb!x_1 + \cdots + x_n = \sum_{i=1}^{n} x_i! \\[2\jot] +$\displaystyle x_1 + \cdots + x_n = \sum_{i=1}^{n} x_i$ +\end{tabular} +\] +\[ +\begin{tabular}{c} +\verb!M_1 \vee\cdots\vee M_t = \bigvee_1^t M_i! \\[2\jot] +$\displaystyle M_1 \vee \cdots \vee M_t = \bigvee_1^t M_i$ +\end{tabular} +\] +But in text: $\alpha_1 \alpha_2 \ldots \alpha_m = \prod_{j=1}^{m} \alpha_j$. + +\subsection{Integrals} +There are two integral signs: +\[ +\begin{tabular}{cc@{\qquad}cc} +\verb!\int! & $\displaystyle \int$ & +\verb!\oint! & $\displaystyle \oint$ +\end{tabular} +\] +These behave somewhat like the large repeated operators in that they have a +larger size in displayed Maths and their limits are typed in as sub- and +superscripts. However, these limits stay in the same position even in displayed +Maths. +%Alan, this is a deliberate lie, OK? + +\subsection{More symbols} +There are some miscellaneous Mathematical symbols that we have not covered +elsewhere. Important ones include: +\[ +\begin{tabular}{cc@{\qquad}cc@{\qquad}cc} +\verb!\emptyset! & $\emptyset$ & \verb!\forall! & $\forall$ & \verb!\Box! & +$\Box$\\ +\verb!\infty! & $\infty$ & \verb!\exists! & $\exists$ & \verb!\partial! & +$\partial$ +\end{tabular} +\] +More are shown on page~45 of \shortlamp. + +Unfortunately, some of these, such as \verb+\Box+, are not standard in +the new standard \LaTeX. To continue using such symbols after you have +upgraded, you must +include the package \latexword{amsfonts}, if available, +or the package \latexword{latexsym}. + Some operators and relations have +been similarly banished, and can be rescued in just the same way. + +\subsection{Punctuation} +In Maths mode, \torl\ treats a full stop as an ordinary symbol, so that decimal +points look right. However, the comma +and semi-colon are treated as punctuation, and get a little space after them, +but not so much as the space they usually get in text. To suppress the space, +put the comma or semi-colon in a pair of braces. + +The colon is treated as a relation, because of its frequent use in defining +sets. To obtain a colon as a piece of punctuation, type \verb+\colon+. Many +people think that the punctuation form is more correct for defining functions. + +\subsection{Braces for grouping} +Grouping has two extra properties in Maths mode. In the first place, it can +prevent line-breaking. \torl\ is usually very good at knowing where to break a +line within a Mathematical expression, but it doesn't always do it exactly to +your liking. For example, in Exercise~7 in the first tutorial, one equation +was broken after a $+$~sign, leaving the single character~$\varepsilon$ on the +next line. This could have been prevented by enclosing the whole right-hand +side of the equation in braces: \TeX\ never breaks a line between grouping +braces in Maths mode. Use this force sparingly: the more line breaks that you +ban the harder is it for \TeX\ to build beautiful paragraphs. + +The second property is more subtle: injudiciously placed grouping braces can +destroy \TeX's fine knowledge of what is a relation, an operator etc., and +thus prevent it from applying the proper spacing. \verb!2{+3}! is not the same +as \verb!2+3! and neither is \verb!2{+}3!; +while \verb!{}-4! is different from \verb!-4!. +Likewise, \verb!{\cos} \theta! is not the same as +\verb!\cos \theta!: the former +turns \verb!\cos! from an operator +into a symbol. + +Some \TeX ies recommend always putting \verb!{}! after a command with no +argument. That advice can be disastrous in Maths mode. Look at +\begin{quote} +\verb!\sum{}_{1}^{3} y_j! \qquad $\displaystyle\sum{}_{1}^{3} y_j$ +\end{quote} +and compare +\[ +\begin{tabular}{ll} +\verb!k \geq{} -b! & $k \geq{} -b$\\ +\verb!k \geq -b! & $k \geq -b$ +\end{tabular} +\] +In the first example the sub-and superscripts have been placed on the +\verb!{}!, not on the \verb!\sum!: in particular, they do not move to the +correct +place in displayed Maths. In the second, the minus sign has become a binary +operator between \verb!{}! and~$b$ instead of a unary operator on~$b$. + +This apparently awkward property of braces can be turned to advantage when +\TeX's first interpretation is not the correct one, as we have already seen in +some examples. + +\subsection{Ties} +The tie~\verb!~!, which prevents line breaks both within and without Maths +mode, can be used to make your Mathematical +text easier to read. A piece of notation +consisting of a single letter or symbol should almost always be tied to one of +its neighbours. If the notation names a noun, tie it to the noun: +\verb!group~$G$! and \verb!element~$h$!. If it is the subject or single +direct object of a verb, tie it to the verb: +\begin{verbatim} + If $v$~annihilates~$W$ then ... +\end{verbatim} +If it follows a preposition, tie it to the preposition: +\verb!of~$\Lambda$!, \verb!in~$T$!, \verb!onto~$Y$!. +If an adjective qualifies it, tie it to the adjective: +\verb!positive~$\delta$!. +A notational +adjective, particularly a number, should be tied to its following noun: +\verb!$n$~points!, \verb!$21$~lines!. +Be careful about the beginnings and ends of lists: +\begin{verbatim} + Only one of $A$, $B$ and~$C$ occurs ... + For $i=1$, $2$, \ldots,~$n$, put ... +\end{verbatim} +Finally, use a tie if a small piece of notation ends a sentence, to prevent a +line beginning something like +\begin{quote} +$B$. Therefore \ldots +\end{quote} +which makes it look like the second item in a list. + +Try to get in the habit of inserting these ties when you first type the text: +don't leave them until you see bad line-breaks. + +Of course, like all rules in typography, some of the above rules conflict, so +you have to use common sense about which ones to follow. Also, there will +always be times when the paragraph simply cannot be broken into lines nicely if +all the ties are observed: wait until the final run and then relax the least +important ones. + +\section*{Exercises} +\addtocounter{preqn}{10} +\begin{qn} +The function $f\colon Z\to Z$ defined by $f\colon n\mapsto n^2$ is neither +injective nor surjective. However, the function $g\colon R^+ \to R^+$ given by +$g(x) = \sqrt[7]{x}$ is both. +\end{qn} + +\begin{qn} +If $\{x_1, \ldots, x_n\}$ and $\{y_1, \ldots, y_m\}$ are bases for $\cal X$ and +$\cal Y$ respectively then +\[ +\left\{x_i\otimes y_j: \mbox{$i=1$, \ldots, $n$, $j=1$, \ldots, $m$}\right\} +\] +is a basis for ${\cal X}\otimes {\cal Y}$. +\end{qn} + +\begin{qn} +If $A = \{n\in N: \mbox{$n$ is prime}\}$ and if +$B = {\{n\in N: \mbox{$n$ is even}\}}$ +then $\left|A\cap B\right|=1 $. +\end{qn} + +\begin{qn} +If $z$ is any real number then $\left|+z\right| = \left|-z\right|$. +\end{qn} + +\begin{qn} +{\it Express the following use large binary operators instead of dots. +Try them both +in text and in display.} +\[ +\begin{array}{ll} +x_1 + \cdots + x_{153} & 1 + 2 + \cdots +r \\ + 1 + 1/2 + 1/4 + \cdots & p \times (p-1) \times \cdots \times 1 \\ +{\cal U}_1 \otimes \cdots \otimes {\cal U}_m +& T_4 \cup T_6 \cup \cdots \cup T_{10}\\ +\mbox{the sum of $\gamma$ for $\gamma\in\Gamma$} & P_1 \wedge \cdots \wedge P_r +\end{array} +\] +\end{qn} + +\begin{qn} +\[ +\prod_{k\ge 0} \frac{1}{(1 - q^kz)} = \sum_{n\ge 0} z^n \left / \prod_{1\le +k\le n} (1 - q^k) \right. +\] +\label{dek} +\end{qn} + +\begin{qn} +{\it +Redo Exercise~3 from the first tutorial without using built-up fractions. You +may want to change the way the square root is shown.} +\end{qn} + +\begin{qn} +\[ +\sum_{n=1}^{N} n^3 = \left(\frac{N(N+1)}{2}\right)^2 +\] +\end{qn} + +\begin{qn} +$\int_1^2 \frac{1}{x}dx = [\log x ]_1^2 = \log 2$. +\end{qn} + +\begin{qn} +$\lim_{n\to \infty} (1 + \frac{x}{n})^n = \exp (x)$. +\end{qn} + +\begin{qn} +If $Z$ is a normal random variable with mean~$\mu$ and variance $\sigma^2$ +then +\[ +\Pr (Z < x) = \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}\sigma} \exp +\left(-\frac{(x-\mu)^2}{\sigma^2}\right). +\] +\end{qn} + +\begin{qn} +\[\frac{\partial {\rm e}^{-xt}}{\partial t} = -x{\rm e}^{-xt}.\] +\end{qn} + +\end{Article} + |