\hat{a} \acute{a} \bar{a} \dot{a}

˜ ˇ `

a a a a

\breve{a} \check{a} \grave{a} \vec{a}

¨ ˜

a a

\ddot{a} \tilde{a}

Table VIII.12: Some other Constructions

abc abc

\widetilde{abc} \widehat{abc}

←’ ’’

abc abc

\overleftarrow{abc} \overrightarrow{abc}

abc abc

\overline{abc} \underline{abc}

abc abc

\overbrace{abc} \underbrace{abc}

√ √

n

abc abc

\sqrt{abc} \sqrt[n]{abc}

abc

f f™ \frac{abc}{xyz}

xyz

Table VIII.13: AMS Delimiters

\ulcorner \urcorner \llcorner \lrcorner

Table VIII.14: AMS Arrows

\dashrightarrow \dashleftarrow

\leftleftarrows \leftrightarrows

\Lleftarrow \twoheadleftarrow

\leftarrowtail \looparrowleft

\leftrightharpoons \curvearrowleft

\circlearrowleft \Lsh

\upuparrows \upharpoonleft

\downharpoonleft \multimap

\leftrightsquigarrow \rightrightarrows

\rightleftarrows \rightrightarrows

\rightleftarrows \twoheadrightarrow

\rightarrowtail \looparrowright

\rightleftharpoons \curvearrowright

\circlearrowright \Rsh

\downdownarrows \upharpoonright

\downharpoonright \rightsquigarrow

Table VIII.15: AMS Negated Arrows

\nleftarrow \nrightarrow \nLeftarrow

\nRightarrow \nleftrightarrow \nLeftrightarrow

Table VIII.16: AMS Greek

107

SYMBOLS

VIII.8.

\digamma \varkappa

κ

Table VIII.17: AMS Hebrew

\beth \daleth \gimel

Table VIII.18: AMS Miscellaneous

\hbar \hslash

™¦

\square \lozenge

\measuredangle \nexists

\Game \Bbbk

k

\blacktriangle \blacktriangledown

\bigstar \sphericalangle

\diagup \diagdown

\vartriangle \triangledown

∠

\circledS \angle

\mho \Finv

\backprime \varnothing

…

\blacksquare \blacklozenge

\complement \eth

°

Table VIII.19: AMS Binary Operators

\dotplus \smallsetminus \Cap

\barwedge \veebar \doublebarwedge

\boxtimes \boxdot \boxplus

\ltimes \rtimes \leftthreetimes

\curlywedge \curlyvee \circleddash

\circledcirc \centerdot \intercal

\Cup \boxminus \divideontimes

\rightthreetimes \circledast

Table VIII.20: AMS Binary Relations

\leqq \leqslant \eqslantless

\lessapprox \approxeq \lessdot

\lessgtr \lesseqgtr \lesseqqgtr

\risingdotseq \fallingdotseq \backsim

\subseteqq \Subset \sqsubset

\curlyeqprec \precsim \precapprox

\trianglelefteq \vDash \Vvdash

\smallfrown \bumpeq \Bumpeq

\geqslant \eqslantgtr \gtrsim

\gtrdot \ggg \gtrless

\gtreqqless \eqcirc \circeq

\thicksim \thickapprox \supseteqq

∼ ≈

108 TYPESETTING MATHEMATICS

VIII.

\sqsupset \succcurlyeq \curlyeqsucc

\succapprox \vartriangleright \trianglerighteq

\shortmid \shortparallel \between

∝ ∴

\varpropto \blacktriangleleft \therefore

\blacktriangleright \because \lesssim

\lll \doteqdot \backsimeq

\preccurlyeq \vartriangleleft \smallsmile

\geqq \gtrapprox \gtreqless

\triangleq \Supset \succsim

\Vdash \pitchfork \backepsilon

Table VIII.21: AMS Negated Binary Relations

\nless \nleq \nleqslant

\lneq \lneqq \lvertneqq

\lnapprox \nprec \npreceq

\precnapprox \nsim \nshortmid

\nvdash \nvDash \ntriangleleft

\nsubseteq \subsetneq \varsubsetneq

\varsubsetneqq \ngtr \ngeq

\ngeqq \gneq \gneqq

\gnsim \gnapprox \nsucc

\nsucceq \succnsim \succnapprox

\nshortparallel \nparallel \nvDash

\ntriangleright \ntrianglerighteq \nsupseteq

\supsetneq \varsupsetneq \supsetneqq

\nleqq \lnsim \precnsim

\nmid \ntrianglelefteq \subsetneqq

\ngeqslant \gvertneqq \nsucceq

\ncong \nVDash \nsupseteqq

\varsupsetneqq

Table VIII.22: Math Alphabets

Required package

ABCdef \mathrm{ABCdef}

ABCdef \mathitABCdef

ABCde f \mathnormal{ABCdef}

ABC \mathcal{ABC}

ABC euscript with option: mathcal

\mathcal{ABC}

euscript with option: mathcr

\mathscr{ABC}

eufrak

ABCdef \mathfrak{ABCdef}

amsfonts or amssymb

\mathbb{ABC}

ABC

A BC mathrsfs

\mathscr{ABC}

TUTORIAL IX

TYPESETTING THEOREMS

A

THEOREMS IN L TEX

IX.1.

In Mathematical documents we often have special statements such as axioms (which are

nothing but the assumptions made) and theorems (which are the conclusions obtained,

sometimes known by other names like propositions or lemmas). These are often typeset

in different font to distinguish them from surrounding text and given a name and a num-

ber for subsequent reference. Such distinguished statements are now increasingly seen in

other subjects also. We use the term theorem-like statements for all such statements.

LTEX provides the declaration \newtheorem to de¬ne the theorem-like statements

A

needed in a document. This command has two arguments, the ¬rst for the name we

assign to the environment and the second, the name to be printed with the statement.

Thus if you want

Theorem 1. The sum of the angles of a triangle is 180—¦ .

you ¬rst specify

\newtheorem{thm}{Theorem}

and then type

\begin{thm}

The sum of the angles of a triangle is $180ˆ\circ$.

\end{thm}

Note that in the command \newtheorem the ¬rst argument can be any name you fancy, in-

stead of the thm given here. Also, it is a good idea to keep all your \newtheorem commands

together in the preamble.

The \newtheorem command has a couple of optional arguments which control the

way the corresponding statement is numbered. For example if you want the above theo-

rem to be numbered 1.1 (the ¬rst theorem of the ¬rst section) rather than a plain 1, then

you must specify

\newtheorem{thm}{Theorem}[section]

in the \newtheorem command. Then the same input as above for the theorem produces

The sum of the angles of a triangle is 180—¦ .

Theorem IX.1.1.

The next Theorem will be numbered 1.2, the third Theorem in the fourth section

will be numbered 4.3 and so on.

109

110 TYPESETTING THEOREMS

IX.

The other optional argument of the \newtheorem command is useful when you have

several different types of theorem-like statements (such as lemmas and corollaries) and

you want some of them to share the same numbering sequence. For example if you want

The sum of the angles of a triangle is 180—¦ .

Theorem IX.1.2.

An immediate consequence of the result is the following

The sum of the angles of a quadrilateral is 360—¦ .

Corollary IX.1.3.

Then you must specify

\newtheorem{cor}[thm]{Corollary}

after the speci¬cation \newtheorem{thm}[section] and then type

\begin{thm}

The sum of the angles of a triangle is $180ˆ\circ$.

\end{thm}

An immediate consequence of the result is the following

The sum of the angles of a quadrilateral is 360—¦ .

Corollary IX.1.4.

The optional argument thm in the de¬nition of the cor environment speci¬es that

“Corollaries” and “Theorems” are to be numbered in the same sequence.

A theorem-like environment de¬ned using the \newtheorem command has also an

optional argument which is used to give a note about the theorem such as the name of its

discoverer or its own common name. For example, to get

(Euclid). The sum of the angles of a triangle is 180—¦ .

Theorem IX.1.5

you must type

\begin{thm}[Euclid]

The sum of the angles of a triangle is $180ˆ\circ$.

\end{thm}

Note the optional argument Euclid after the \begin{thm}. This use of [...] for optional

notes sometimes lead to unintended results. For example, to get

Theorem [0, 1] is a compact subset of R.

IX.1.6.

if you type

\begin{thm}

[0,1] is a compact subset of $\mathbb{R}$.

\end{thm}

then you get

Theorem (0,1). is a compact subset of R.

IX.1.7

Do you see what happened? The string 0,1 within [ ] at the beginning of the theorem is

considered an optional note by LTEX ! The correct way is to type

A

THEOREMS”THE AMSTHM PACKAGE 111

DESIGNER

IX.2.

\begin{thm}

$[0,1]$ is a compact subset of $\mathbb{R}$.

\end{thm}

Now all the theorem-like statements produced above have the same typographical form”

name and number in boldface and the body of the statement in italics. What if you need

something like

(EUCLID). The sum of the angles of a triangle is 180—¦ .

THEOREM IX.1.1

Such customization is necessitated not only by the aesthetics of the author but often by

the whims of the designers in publishing houses also.

THEOREMS”THE AMSTHM PACKAGE

DESIGNER

IX.2.

The package amsthm affords a high level of customization in formatting theorem-like

statements. Let us ¬rst look at the prede¬ned styles available in this package.

Ready made styles

IX.2.1.

The default style (this is what you get if you do not say anything about the style) is termed

plain and it is what we have seen so far”name and number in boldface and body in italic.

Then there is the definition style which gives name and number in boldface and body in

roman. And ¬nally there is the remark style which gives number and name in italics and

body in roman.

For example if you put in the preamble

\usepackage{amsthm}

\newtheorem{thm}{Theorem}[section]

\theoremstyle{definition}

\newtheorem{dfn}{Definition}[section]

\theoremstyle{remark}

\newtheorem{note}{Note}[section]

\theoremstyle{plain}

\newtheorem{lem}[thm]{Lemma}

and then type somewhere in your document

\begin{dfn}

A triangle is the figure formed by joining each pair

of three non collinear points by line segments.

\end{dfn}

\begin{note}

A triangle has three angles.

\end{note}

\begin{thm}

The sum of the angles of a triangle is $180ˆ\circ$.

\end{thm}

\begin{lem}

The sum of any two sides of a triangle is greater than or equal to the third.

\end{lem}

112 TYPESETTING THEOREMS

IX.

then you get

De¬nition IX.2.1. A triangle is the ¬gure formed by joining each pair of three non collinear

points by line segments.

A triangle has three angles. 1 note

Note IX.2.1.

The sum of the angles of a triangle is 180—¦ .

Theorem IX.2.1.

Lemma The sum of any two sides of a triangle is greater than or equal to the third.

IX.2.2.

Note how the \theoremstyle command is used to switch between various styles, espe-

cially the last \theoremstyle{plain} command. Without it, the previous \theoremstyle{remark}

will still be in force when lem is de¬ned and so “Lemma” will be typeset in the remark

style.

Custom made theorems

IX.2.2.