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a a a a
\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.

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