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+\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}
+