Nhap mon latex, go cong thuc toan bang latex,latex symbols.pdf

NHP MN LATEX, G CNG

THC TON TRONG LATEX,LATEXSYMBOLS

Mc lcNhp Mn LaTeX 2

G cng thc ton bng LaTeX 27

LaTeX Symbols 44

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Nhp mn LATEX H Quc Trung Ph. D.

y l ti liu hng dn (handout) i km vi slides dnh cho kha hc nhp mn LATEX .Mi c nhn hc theo cc tc khc nhau. Nu gp kh khn, thc hin mt phn cc bi tp e . Nu bn c

th hc nhanh, hy lm cc bi tp ! . Li gii cho cc bi tp c cung cp km theo.

1 To mt ti liu

Mt file cha m LATEX2 bt u bng command

\documentclass[options]{class}

khai bo loi ti liu s s dng. V d:

\documentclass[a4paper,12pt]{article}

Ni dung ti liu cn c cha trong mi trng document :

\begin{document}

nh ngha im bt u

\end{document}

nh ngha im kt thc.Tt c cc ti liu LATEX cn c 3 dng:

\documentclass[a4paper,12pt]{article}

\begin{document}

\end{document}

Phn ch nghing c th khc nhau, phn cn li cn c g ng nh trn.

Bi tp 1

Bt u l mt bi tp n gin. Hnh v 1 trang 2 hin th ni dung ca tp exercise1.tex. Th to tp nybng cch s dng cc hng dn trong slides. Ch : ti liu s dng kh A4, 12 point text (mc nh 10).

Sau khi g xong m ngun, hy LATEX m!. Cc tp sau y s c trong th mc hin ti: exercise1.aux,exercise1.log and exercise1.dvi. Tp .aux l tp auxiliary , LATEX dng cha cc thng tin tham chiu chov trch dn. Tp .log cha thng tin v tt c nhng g xy ra khiLATEX m ngun. Tp .dvi l u ra c th chin th bng mt chng trnh dvi viewer. g ting Vit TCVN, hy thm

\begin{document}\usepackage[tcvn]{vietnam}

vo ti liu.

T kim tra

Liu bn t mt du cch sau lnh \LaTeX? Liu bn c thy text c canh l t ng, s trang t ng xut hin cui trang?

Nhp mn LATEX FIT-HUT 1

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% exercise1.tex - a simple example file illustrating basic input

e \documentclass[a4paper,12pt]{article}

e \begin{document}

e This is a simple example that shows how easy it is to create ae \LaTeX\ file.

Note that the page is automatically numbered at the bottom.Anything following a percent sign (\%) is ignored up to the end ofthe line. %This is a comment, and will not appear in the document.

e Todays date is \today.e

This is the start of the second paragraph.e Footnotes are easy\footnote{Here is a footnote}.

It is possible to have notes in the margin\marginpar{like this}.

e \end{document}

Hnh 1: Ni dung ca tp exercise1.tex. Nu cm thy kh, hy lm nhng phn nh du e

Bng 1: Cc lnh thay i font

Lnh M Hin th\textrm{text} \textrm{Roman} Roman\textsf{text} \textsf{Sans serif} Sans serif

Family

\texttt{text} \texttt{typewriter} typewriter\textmd{text} \textmd{medium} medium

Weight

\textbf{text} \textbf{bold} bold\textup{text} \textup{upright} upright\textit{text} \textit{italic} italic\textsl{text} \textsl{slanted} slanted

Shape

\textsc{text} \textsc{Small Caps} Small Caps\emph{text} \emph{emphasized} emphasized

\textnormal{text} \textnormal{default} default

Bng 2: Khai bo chuyn Font

Khai bo M Hin th\rmfamily \rmfamily Roman Roman\sffamily \sffamily Sans serif Sans serif

Family

\ttfamily \ttfamily typewriter typewriter\mdseries \mdseries medium medium

Weight

\bfseries \bfseries bold bold\upshape \upshape upright upright\itshape \itshape italic italic\slshape \slshape slanted slanted

Shape

\scshape \scshape Small Caps Small Caps\em \em emphasized emphasized

\normalfont \normalfont default default


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Bng 3: Khai bo c Font

\tiny tiny text\scriptsize script sized text\footnotesize footnote sized text\small small text\normalsize normal sized text\large large text\Large even larger\LARGE larger still\huge huge\Huge really huge

Bng 4: Mi trng Font

rmfamily sffamily ttfamily mdseries bfseriesupshape itshape slshape scshape emtiny scriptsize footnotesize small normalsizelarge Large LARGE huge Huge

2 Cc lnh thay i font n gin

C th thay i font ca mt on text s dng lnh (Table 1), hoc chuyn sang mt font khc dng khai bo(Table 2 trang 2). C th chuyn sang kch thc font khc bng Table 3 trang 3. C th s dng cc mi trngtng ng khng s dng du gch cho (Table 4 trang 3).

Bi tp 2

Tr li bi tp lm trong Bi tp 1 g on th nht thnh font sans serif ln, g on th 2 font Romanthng. Sau , nhn mnh ch Footnotes, ri thay i ton b font ca on th 2 thnh nghing.

T kim tra

Bn lm th no thay i font on 1. Dng lnh? Dng mi trng? Khai bo? Thay i? Cch nott nht thc hin?

Khi chuyn on 2 sang font nghing, phn nhn mnh chuyn thnh font g?

3 Du chm, du phy, trng m

Cc k hiu chun c trong bng Table 5. Ch l du cch sau k hiu c gch cho b b qua (v.d. \pounds 10cho 10) cn ( \# 10 cho # 10).

Du trng m c b trn k hiu bng cc lnh trong(Table 6). Cc ch ghp v tin t c trong bng Table 7.

Bi tp 3

To ra vn bn LATEX ging Hnh v 2.


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Bng 5: Symbols

\pounds \_ _ \P - -\copyright c \$ $ \S -- \ddag \{ { \ldots . . . --- \dag \} } \i ? ?\textregistered R \# # \j ! !\texttrademark TM \% % \yen U \& &

Bng 6: Trng m

Example ExampleDefinition Input Output Definition Input Output\{object} \{c} c \={object} \={c} c\{object} \{c} c` \.{object} \.{c} c\^{object} \^{c} c \~{object} \~{c} c\"{object} \"{c} c \v{object} \v{c} c\u{object} \u{c} c \H{object} \H{c} c\t{object} \t{cc} cc \c{object} \c{c} c\d{object} \d{c} c. \b{object} \b{c} c

T kim tra

Bn c thy cc ch ghp fl, ff v fi ? Bn c thy cc ch ghp and ?

4 Kiu ti liu v section

C nhiu kiu ti liu. Cc kiu c bn l: article, report, book, slides and letter. Cc section c bn ca ti liu l:

\part[short title]{Title} \subsubsection[short title]{Title}\chapter[short title]{Title} \paragraph[short title]{Title}\section[short title]{Title} \subparagraph[short title]{Title}\subsection[short title]{Title}

Mt vi kiu khng c ht cc section (v.d. the article khng c \chapter.)Cc phn c nh s t ng. Vicnh s c th b qua bng du sao cui lnh(v.d. \chapter*{Acknowledgements}). Tm tt cn phi t trongmi trng abstract.

Tiu ca ti liu c khai bo dng cc lnh:

\author{Author Names}\title{Document Title}\date{Date}

Bng 7: Ch ghp v n v tin t

\AE \ae \OE \oe \AA A \aa a \L L \l l\O \o \SS SS \ss


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Nave people who eat fluffy pt end up manuvring themselves towards the bathroom,resulting in an ger.

The facade is my pice de resistance, do you like it? It cost 1000.00 10% underbudget.

Its a bit pass.! He had a feeling of dj vu on seeing Chloe Muller, ne Hrebcek, wearing a fine ngligto the matine.! ?Dnde esta el cuarto de bano?

Hnh 2: Bi tp 3 Trng m, k hiu, du cauu. ! dnh cho nhng ngi dng cm.

Sau c hin th bng:

\maketitle

Chuyn sang ch ph lc, dng:

\appendix

Sau vn tip tc dng \chapter, \section .... Mc lc to ra bng:

\tableofcontents

thng i sau \maketitle.S trang c nh t ng. Kiu trang c th c thay i s dng

\pagestyle{style}

t y v sau, hoc

\thispagestyle{style}

cho trang hin ti. Cc kiu thng dng: plain, empty and headings.Kiu nh s c th thay i bng:

\pagenumbering{style}

trong style l mt trong: arabic, roman, Roman, alph or Alph. ng thi, s trang t thnh 1.

Bi tp 4

Copy tp sectioning.tex, ging nh trong slides. M tp: c mt ti liu ngn cha c cc lnh chia section.to ra tiu , mc lc, chng v cc mc trong chng. Dng mi trng abstract to ra tm tt ca ti liu. V.Dnu thy ch gii

% CHAPTER : Introduction

chn

\chapter{Introduction}

vo dng tip theo.! Thm ph lc ri thay i kiu trang. Lnh\thispagestyle hiu lc hn \pagestyle nn mt s trang khng

b nh hng bi \pagestyle. Cc phn u vn bn (mc lc, li ni u) dng s la m thng , whereas themain document usually has Arabic numbering. See if you can change the page numbering style.

5 nh dng on vn

Cc on vn c canh l u mc nh tuy nhin c th thay i c bng lnh khai bo hoc mi trng .


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Khai bo \raggedright \raggedleft \centeringMi trng flushleft flushright center

C th canh gia mt dng text bng:

\centerline{text}

C th ct dng m khng thay i canh l dng \\[text] hoc \newline. Cc cch dng canh l to ra bi\linebreak[n] trong n t 0 ti 4. C th chng cch dng bng \nolinebreak[n]. Trong n l s nguyn t0 n 4. Nu mun c mt du cch, v khng mun cch dng ti du cch , dng: ~.

C th khai bo tng minh mt cch trang:

\newpage

hoc:

\pagebreak[n]

hoc chng cch trang:

\nopagebreak[n]

Lnh

\clearpage

ging \newpage, nhng hin th tt c cc i tng treo - float (xem Section 19).

Bi tp 5

To vn bn ging hnh Hnh v 3. Hy la chn dng khai bo, chuyn font hay mi trng.

This paragraph here is fully justified, but note how we haveforced the end of the line to occur before the word forced.

These linesare centred.

Here we have someleft justifiedtext.

Here we have someright justified

text.

This text is centered.

Hnh 3: Bi tp 5 nh dng


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T kim tra

Bn c thy cch dng dng 1? C thy dng 1 c canh l u, c nhiu khong cch gia cc t? Ktqu ca qu trnh LATEX l

Underfull \hbox (badness 6316) in paragraph at lines 7--9[]\OT1/cmr/m/n/10 This para-graph here is fully jus-ti-fied,but note how we have

y l thng bo ca LATEX v vic c nhiu khaongr trng. Cc dng kt thc u nu khng bt bucxung dng?

Cch thc tt nht canh gia dng cui cng? S dng \centering, hay mi trng center, hoc\centerline{}?

6 nh ngha lnh mi

nh ngha lnh mi dng:

\newcommand{cmd-name}[nargs][default]{text}

trong cmd-name l tn lnh mi (ng qun gch cho), nargs s lng tham s (mc nh 0), default l gi trmc ng ca tham s th nht, text l thao tc LATEX s lm thc hin lnh. Nu nargs ln hn 0, cc tham sc biu din tronglnh bng #1, #2, ....

Cc lnh c sn c th c nh ngha li dng \renewcommand thay cho \newcommand, nhng Khng bao ginh ngha li lnh cha r tnh nng. Ch nh ngha li cc lnh nu mun thay i hiu ng ca lnh. V d numun thay i ngy thng hin th bi cu lnh \today cn nh ngha li \today, nhng nu mun c lnh hinth ngy c bit, cn nh ngha mt lnh mi.

Bi tp 6

To ra file mi exercise6.tex, ri nh ngha \timeofday. (Ch : m bo gi lnh \timeofday, ch khngphi \time c sn.) Lnh cn c hai tham s, gi v pht. The command \timeofday{10}{25} hin th 10:25. (Ttnht l nh ngha lnh trong phn u ca ti liu preamble.) Sau khi nh ngha lnh, hy dng lnh tora u ra ging trong Hnh v 4 trong thi gian c hin th bng lnh \timeofday. Sau , thay i lnh thi gian c hin th bng ch m, dng 10-00 thay cho 10:00).

The Introductory LATEX course runs for four days from 10:00 to 13:00, with a coffee breakat 11:30.

Hnh 4: Output for Exercise 6

T kim tra

Bn c nh cch nh ngha lnh \timeofday? Bn dng lnh bao nhiu ln?) Bn bin i ni dung ca lnh \timeofday nh dng li thi gian cha?


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7 Cc dng u ra khc nhau

LATEX to u ra mc ingj l DVI (device independent). Tuy nhin, c th to ra cc u ra dng khc. TrongTeXnicCenter DVI c th c bin i thnh cc dng khc bng cch chn LaTeX => PS ri chn Build FilePostScript thu c c th c gi n my in PostScript , hoc c hin th bi GSviewCc ti liu PDF chin th bng Adobe Acrobat Reader. File PostScript c th c chuyn i thnh PDF bng ps2pdf. Mt khc, ccfile PDF c th thu c t file LATEX bng PDFLaTeX thay cho LATEX. Dng dvips hoc PDFLATEXc th:

Nhng nh vo ti liu have color

rotate

reflect or resize text.Vi PDFLATEX, c th c cc lin kt (hyperlinks) trong ti liu.

Bi tp 7

Chuyn i ti liu (sectioning.tex) trong Bi tp 4 thnh PostScript hin th s dng GSview. ng thi tofile PDF s dng PDFLATEX.

8 Danh sch

C 3 loi mi trng to ra cc danh sch: itemize (danh sch khng nh s), enumerate (danh sch nh s)and description (danh sch t tn). Trong mi mt mi trng, cn c cc lnh\item mi khi mun c mt phnt mi trong danh sch. C th lng cc danh sch vi nhau. Lnh itemize dng cc k hiu nh du cc phnt, lnh enumerate dng s hoc ch ci sp xp cc phn t, cn description mc nh khng nh du . C ththay i cu hnh bng cc ty bin cho lnh \item.

Bi tp 8

To ti liu ging ??, v ghi vo exercise8.tex.! Sau , i tt c cc enumeratethnh itemize. Nhn mc nh cho mi phn t l: and . Cc nhn

c th c hin th bng lnh: \labelitemi, \labelitemii, \labelitemiii and \labelitemiv. nh ngha li\labelitemi cc phn t itemize ngoi c gn nhn vi k hiu thay cho , v nh ngha li \labelitemii cc phn t mc tip theo c nh bng k hiu

9 Bng

Vn bn c th c sp xp trong hng v ct s dng mi trng tabular. Mi trng ny c mt tham s chnhl thng s v ct v canh l ct. Cc canh l c bn l : l (tri), r (phi) v c (gia). K hiu | ch dng k dccng c th c t y. Cc dng k ngang c t bng

\hline

cho ng k ngang tt c cc ct, hoc

\cline{n-m}

cho ng k ngang t ct n n m. Cc phn t gia cc ct c phn bit s dng &. V d:


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\begin{tabular}{|l|c|}\hlineItem & Cost \\ \hline\hlineCD & \pounds 11.75\\Video & \pounds 14.10\\ \hlineTotal & \pounds 25.85\\ \hline\end{tabular}

Item CostCD 11.75Video 14.10Total 25.85

Text c th c hin th trong vi ct s dng:

\multicolumn{n}{alignment}{text}

trong n l s ct cn s dng, alignment ch ra cch thc canh l (mt ch ci l, r hoc c vi k hiu ng kdc |) v text l ni dung ca .

Bi tp 9

Khi to bng, tt nht l to ra t mt bng n gin, sau b sung thm cc khc.e To vn bn nh trong Hnh v 5! Bin i m ngun vn bn c dng nhHnh v 6. Ch l 4 ct cui cng canh l phi, tr 2 hng u.

Cc phn t trong dng tiu c th c thay i s dng \multicolumn.Ch bng kt qu cui cng dng mt mi trng tabular trong dng 3 v 7 chim c chiu ngang bng.

Sau th canh l gia c bng.

Year 1 Year 2 Year 3 Total() () () ()

Books 100.00 150.00 200.00 450.00Stationary 50.00 45.00 55.00 150.00Software 500.00 600.00 300.00 1400.00US Conference 2000.00 2500.00 3000.00 7500.00EU Conference 500.00 500.00 500.00 1500.00Total 3150.00 3795.00 4055.00 11000.00

Hnh 5: Bi tp 9 e To bng

Year 1 Year 2 Year 3 Total() () () ()

Equipment ExpenditureBooks 100.00 150.00 200.00 450.00Stationary 50.00 45.00 55.00 150.00Software 500.00 600.00 300.00 1400.00Travel ExpenditureUS Conference 2000.00 2500.00 3000.00 7500.00EU Conference 500.00 500.00 500.00 1500.00Total 3150.00 3795.00 4055.00 11000.00

Hnh 6: Bi tp 9 ! To bng

10 Cn bn cng thc ton hc

C hai loi cng thc ton hc: trong dng and ring bit. Cng thc trong dng c hin th bng mi trngmath v c gn cht vi cc phn t xung quanh. Cng thc ring bit c hin th trn mt dng ring bit.C hai loi: mi trng displaymath v equation. Mi trng math c hai k hiu thu gn: $ . . . $ v $ . . . $. Mitrng displaymath v equation hin th trong mt dng.


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Ch s c hin th bng _{subscript}, v s m c hin th bng lnh ^{superscript}. Phn s c hin thbng:

\frac{numerator}{denominator}

Cn c hin th bng:

\sqrt[n]{maths}

Tn hm v.d. log or sin, c hin th bng \log, \sin. Cc ch ci Hy lp thng nh hoc , c hin thbng \alpha, \beta. Cc ch ci Hy lp hoa nh hoc , c hin th bng \Gamma, \Delta. Ch c cc lnhcho cc ch ci khc bng ch ci Latin (v.d. khng c \omicron hoc \Alpha). Mt s k hiu c kch thc phthuc vo ch trong dng hoc ring r c hin th trong Table 8.

Bng 8: K hiu ton hc hai kch thc

\bigcap

\bigcup

\bigodot

\bigotimes

\bigoplus

\bigsqcup

\biguplus

\bigvee

\bigwedge

\coprod

\int

\oint

\prod

\sum

Bi tp 10

e to ti liu nh Hnh v 7, or ! to ti liu nh Hnh v 8.Cc cu lnh s cn n:

e and ! \Delta ! \rightarrow or \to

Sau chuyn cc cng thc thnh nh s bng mi trng equationthay cho displaymath.

y

x=y1 y0x1 x0

Hnh 7: Exercise 10 e Basic Mathematics

Given a function f(x), the derivative of f with respect to x can be found from firstprinciples using:

df

dx= lim

x0f(x+ x) f(x)

x

Hnh 8: Bi tp 10 ! Cn bn cng thc ton hc

11 Mng v cc du ngoc

Cc du phn cch-ngoc c to ra bi cc lnh:

\leftsymbol\rightsymbol


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trong symbol l mt trong cc k hiu phn cch (xem Table 9 trang 11). Ch cho mi \left, Cn c \righttng ng, nhng c th c mt du phn cch khng hin th bng cch thay symbol bng du chm.

Bng 9: Cc du phn cch-ngoc

( ( ) ) [ [ ] ]\{ { \} } | | \| / / \backslash \ \langle \rangle \lfloor b \rfloor c \lceil d \rceil e\uparrow \downarrow \Uparrow \Downarrow \updownarrow l \Updownarrow m

Mng c to ra dng mi trng array. Mi trng ny c c php ging tabular, nhng ch c dng trongch ton hc.


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Bi tp 11

e To vn bn nh trong hnh Hnh v 9, hoc ! To vn bn nh trong hnh Hnh v 10. Cc cu lnh s cnn:

\cdots \vdots ...\ddots

. . . \neq 6=

Here we have a matrix equation:

2

(5 32 6

)=

(10 64 12

)

Hnh 9: Bi tp 11 e Mng

This is an augmented matrix: a11 a1n b1... . . . ... ...an1 ann bn

The Kronecker delta is defined as follows:

ij =

{1 i = j0 i 6= j

Hnh 10: Bi tp 11 ! Mng

12 Multiline Formul- Cc cng thc nhiu dng

Mi trng displaymath v equationch cho php cc cng thc mt dng. Mt trong nhng cch to cng thcnhiu dng l dng mi trng eqnarray, to ra 3 ct, ct u canh l phi, ct 2 canh l gia, ct 3 canh l phi.Mi trng cng cho php cc dng c nh s cnh k hiu = hoc tng t. Cc dng c nh s mc nh,tuy nhin c th b s bng lnh \nonumber trn dng tng ng, hoc trn tt c cc dng bng: eqnarray*. Gingnh vi tabular v array, cc ct tch bit nhau bng &, cn cc dng cch bit nhau bng \\. V d:\begin{eqnarray}f & = & x^2 + x\\f & = & 2x + 1\end{eqnarray}

f = x2 + x (1)f = 2x+ 1 (2)

Bi tp 12

e To vn bn nh hnh Hnh v 11, hoc ! To vn bn nh hnh Hnh v 12. Cc lnh s cn n:

\approx \pm \partial \leq \varepsilon


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f(x) = x2 x 2= (x 2)(x+ 1)

Hnh 11: Bi tp 12 e Cng thc a dng

For |x| where is small:

f(x) = sin(x)

x x3

6

Some partial differentiation:

f(x, y) = x2 + 2xy + y2

f

x= 2x+ 2y

Hnh 12: Bi tp 12 ! Cng thc a dng

13 Tham chiu cho

LATEX nh s t ng cc phn v cc i tng trong vn bn nh chng v cng thc.C th tham chiu ti cci tng bng cch gn nhn cho cc i tng bng:

\label{text}

text c th l bt c on vn bn no, tuy nhin, tt hn c l chn ngn v d nh, v.d. sec:intro hoc ch:results.Lnh \label cn c t ngay sau lnh sinh ra phn hoc i tng, hoc trong mi trng sinh ra i tng.V d:

\section{Introduction}\label{sec:intro}

or

\begin{equation}E = mc^2\label{eqn:einstein}\end{equation}

Lnh \label khng to ra bt c mt hin th no.Sau c th tham chiu ti cc nhn s dng:

\ref{label}

trong label l nhn c nh ngha bi lnh \label. V d, \ref{sec:intro} s hin th s th t ca mcsec:intro. Mt khc, c th tham chiu ti s trang s dng:

\pageref{label}

C th tham chiu ti cc nhn nh ngha sau, nhng cn LATEX m ngun hai ln c cc tham chiu cp nht.Tham chiu ti nhn cha nh ngha c hin th bng hai du hi ??

Bi tp 13

To vn bn nh hnh Hnh v 13. ng chn cc s vo bng tay m s dng \label v \ref.! Thm cc mc mi gia phn gii thiu v nh lut Bayes v cc cng thc b sung thy LATEX cp nht

cc tham chiu cho nh th no. Thm tiu v mc lc.


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1 Introduction

The Rev Thomas Bayes was an 18th century cleric who was also a mathematician. His the-orem, known as Bayes Theorem and described in Section 2, was published posthumuously,and it is only thanks to LaPlace that it is so well known, as many other academics at thetime failed to take the theorem seriously.

2 Bayes Theorem

Bayes Theorem is given in Equation 1, and simply states that the posterior of an event Bgiven that event A has occurred is proportional to the prior for event B.

P (B|A) = P (A|B)P (B)P (A)

(1)

Equation 1 is fundamental to the whole study and implementation of Bayesian beliefnetworks.

Hnh 13: Exercise 13 Cross referencing

14 Gi

Gi l cc file vi phn m rng .sty trong nh nghiawc cc lnh mi hoc nh ngha li cc lnh c sn. s dng gi, cn khai bo:

\usepackage[options]{package-name}

trong package-name l tn ca file cha gi, khng c m rng .sty, , options l danh sch phn bit bng duphy cc tham s s truyn vo cho gi. Lnh ny ch c dng trong phn khai bo (preamble) ca ti liu. Tt ccc lnh trong gi u c th c s dng trong ti liu.

Cc gi mi c to ra lin tc, gi m ti liu cn c th cha c ci. Khi gi cn c ti v. Rt nhiugi c ng gi di dng file .dtx, c th c mt kch bn ci t file .ins (hoc i khi .drv). Ti liuj v gic th thu c bng cch LATEX hoc PDFLATEXfile .dtx. Gi hoc lp c to ra bng cch LATEXkch bn cit.


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Bi tp 14

Nu ti liu a ngn ng, c th s dng gi babel. C rt nhiu ngn ng la chn. Ngn ng c la chnbng ty bin ca gi. V d cho ting Php:

\usepackage[francais]{babel}

S dng gi babel trong sectioning.tex Bi tp 4, v xem cc tiu thay i th no.Mtkhc, tr li tp trong Bi tp 13, v dng gi hyperref to ra cc tham chiu-link active links. S dng

PDFLATEX thay cho LATEX to file PDF . Ty bin colorlinks cho php hin th cc lin kt bng mu thay vbng cc box. Ch cn xa file auxiliary (.aux) trc khi dch bng LATEX cho hyperref cp nht cc thamchiu.

! Sau , hin th tiu dng \maketitle lnh. Nu khng dng \date, ngy hin ti s c s dng (khidng lnh \today). Thay i nh dnh ngy s dng gi datetime. (Cn ti gi v v ci t.)

15 Trch dn

Danh mc cc ti liu tham kho c th c to ra bng mi trng thebibliography Mi trng ny tng t nhcc mi trng danh sch Section 8, nhng thay v dng lnh \item cn s dng:

\bibitem[label]{key}

trong key l kha trch dn duy nht, cn label dng thay i cch hin th trch dn (thng thng [1], [2],...). Ti liu sau c th c trch dn bng :

\cite[text]{key-list}

trong key-list l danh sch cc kha ca ti liu nh ngha bng \bibitem, phn tch bng du phy, v.d.\cite{lamport,kopka}, cn text l thng tin b sung v ti liu tham kho, v.d. \cite[Chapter 4]{lamport}.Mi trng thebibliography c mt tham s l nhn di nht trong thebibliography.

Nhiu ngi mun to thebibliography dng phn mm BibTEX. Cn to ra mt file vi phn m rng .bib, chadanh mc tt c cc ti liu c th s dng (CSDL ti liu tham kho), cn BibTEX lc ra cc ti liu no ctrch dn trong ti liu. BibTEX cng sp xp cc phn t, nh dng chng mt cch thng nht.

c th s dng BibTEX, cn khai bo tn CSDL dng trong ti liu LATEX bng lnh:

\bibliography{filename}

trong filename l tn ca CSDL khng c the .bib. Lnh ny cn t ni m danh mc ti liu tham kho cnc hin th, thng l cui vn bn \end{document}).

C nhiu kiu nh dng ti liu tham kho. V d: theo th t ABC, theo th t trch dn, cc tham chiu nhs, hoc gn nhn theo tc gi v nm xut bn. Kiu nh dng c khai bo bng lnh:

\bibliographystyle{style}

trong style l tn ca kiu nh dng. Cc kiu thng dng l: plain, unsrt, alpha v abbrv.Cc phn t ca CSDL c th c nhp vo theo dng:

@entry type{keyword,field = "text",...field = "text"

}

V d:

@book{lamport,author = "Leslie Lamport",title = "\LaTeX\ : a document preparation system",publisher = "Addison-Wesley",


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edition = 2,year = 1994}

Cc tc gi cn c tch bit bng and. V d:

@book{goossens,author = "Michel Goossens and Frank Mittlebach and Alexander Samarin",title = "The \LaTeX\ Companion",publisher = "Addison-Wesley",year = 1994}

sau khi to CSDL, thm cc lnh \bibliographystyle v \bibliography, cn LATEX m ngun, ri BibTEXfileauxilary (khng c m rng .aux extension) ri LATEX m ngun 2 ln m bo cc tham chiu cho c cpnht.

Bi tp 15

To CSDL BibTEX c cc tham chiu nh trong Hnh v 14 trang 17, v to vn bn nhu trong hnh v . Cnto 2 files: exercise15.tex (ti liu) v exercise15.bib (CSDL.) Cn cc phn t u vo nh sau:

Entry Type Fieldsarticle author, title, journal, volume, number, pages, month, yearinproceedings author, title, booktitle, volume, address, pages, month, yearbook author, title, edition, publisher, yearincollection author, title, booktitle, chapter, pages, editor, publisher, year

Thay i cc kiu nh dng cc phn t c hin th theo th t trch dn .(kiu nh dng unsrt). Th cckiu khc: alpha, abbrv v acm, v so snh kt qu. Khi trch dn nhiu ti liu cng lc [3,2,4], c th hin th theokhong [24]. Cn dng gi citesort, nh ngha li \cite. S dng kiu nh dng unsrt.

! Th to kiu nh dng khc dng makebst. Th g

Ilatex makebstti dng lnh, ri theo cc hng dn. Sau thay i cc ty bin ca \bibliographystyle dng ti liu mi.

16 Chiu di

LATEX c cc lnh biu din chiu di (e.g. \textwidth). C hai loi chiu di:

rigid Chiu di c nh, v.d. 4in

rubber Chiu di vi mm do (v.d. 2in plus 0.1in minus 0.1in). cho php LaTeX c th thay i cho phhp vi bi cnh

Chiu di c th c thay i s dng cc lnh:

\setlength{cmd}{value}

gn cho cmd gi tr value. C th ko di chiu di dng:

\addtolength{cmd}{value}

Thm value vo cmd. Nu value, cmd b tr i. C th gn cc chiu di rng su cho bin chiu di bng:

\settowidth{cmd}{text}\settoheight{cmd}{text}\settodepth{cmd}{text}

Text trong text khng hin th, ch dng tnh cc chiu tng ng. Lnh \settowidth c s dng trong phnsau.


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There is a very useful mathematics text book for engineering students written by ErwinKreyszig [2]. For those of you interested in the hazard from Clostridium botulinum associatedwith Alaskan native foods should refer to [5]. For information on a fast index assignmentalgorithm that uses quadratic assignment, see [3, 1, 4].

Ti liu

[1] Gavin Cawley and Nicola Talbot. A fast index assignment algorithm for vector quan-tization over noisy transmission channels. I.E.E. Electronic Letters, 32(15):13431344,July 1996.

[2] Erwin Kreyszig. Advanced Engineering Mathematics. Wiley, 6th edition, 1988.

[3] Nicola Talbot and Gavin Cawley. A quadratic index assignment algorithm for vector quan-tisation over noisy transmission channels. In Proceedings of the Institute of Acoustics Au-tumn Conference on Speech and Hearing, volume 18, pages 195199, November 1996.

[4] Nicola Talbot and Gavin Cawley. A fast index assignment algorithm for robust vectorquantisation of image data. In Proceedings of the I.E.E.E. International Conference onImage Processing, Santa Barbara, California, USA, October 1997.

[5] Robert B. Wainwright. Hazards from Northern native foods. In Andreas H. W. Hauschildand Karen L. Dodds, editors, Clostridium botulinum: Ecology and control in Foods, chap-ter 12, pages 305322. Marcel Dekker, Inc, 1993.

Hnh 14: Bi tp 15 Trch dn v ti liu tham kho

C nhiu bin chiu di, tuy nhin c th nh ngha cc bin chiu di mi bng

\newlengthcmd

V d: \newlength\mylen to ra chiu di c tn l \mylen. Gi tr (points) ca chiu di c hin th s dnglnh:

\thecmd

V d: \the\mylen.Giao din ca trang c th c thay i bng cch thay i gi tr ca chiu di: (v.d. \textwidth) tuy nhin

c th s dng gi geometry.

Bi tp 16

tr li ti liu trong Bi tp 1, thay i (\parindent) v 0pt, ri thay i khong cch gia cc on (\parskip)thnh 3ex.

17 Boxes v Minipages

Bt c ni dung no c th chia thnh cc hp-khung nh. Khung nh nht cha mt k t. Mi khung c chiu di,chiu rng, v chiu su. Chiu cao l khong cch t ng gc-chn n nh ca khung, cn chiu su l khongcch t chn n y ca khung. Xem minh ha trong Hnh v 15.

yheightdepth

width

aheightwidth

Hnh 15: Minh ha cc chiu rng su v cao.


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Mi mt box c x l nh mt i tng n, v khng tch dng. Nh vy c th trnhs tch dng bng ccht text vo mt khung. Cc loi khung c bn bao gm:

\mbox{text}

t kch c v canh l trong box dng:

\makebox[width][justification]{text}

Canh l c th l l (tri), r (phi) hoc c (gia). Khung c th c t xung quanh box bng:

\fbox{text}\framebox[width][justification]{text}

Gi fancybox cung cp cc lnh ging nh \fbox:

\ovalbox{text}\Ovalbox{text}\doublebox{text}\shadowbox{text}

Cc box trn ch c th cha khng qu mt dng. c th c on vn trong box, dng:

\parbox[alignment][height]{width}{contents}

hoc dng mi trng:

\begin{minipage}[alignment][height]{width}{contents}

C th lm ni hoc lm chm:

\raisebox{lift}[depth][height]{contents}

C th to ra cc hnh ch nht en - rules - bng:

\rule[lift]{width}{height}

V d:\newlength\mylen\settowidth{\mylen}{Some Text}%\makebox[0pt][l]{\rule[0.5ex]{\mylen}{1pt}}%Some Text

Some Text

C th ghi mt vi on text vo mt savebox. u tin cn nh ngha mt savebox mi:

\newsavebox{cmd}

trong cmd l tn ca box mi. Text c th c ghi vo box bng lnh:

\sbox{cmd}{text}

hoc bng mi trng:

\begin{lrbox}{cmd}text\end{lrbox}

Ni dung ca savebox c th c hin th bng:

\usebox{cmd}


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Bi tp 17

e Ti to li hnh Hnh v 16, hoc ! Ti to li hnh Hnh v 17. Gi : lnh \vfill to ra mt du cch ng in y trang, cn \hfill du cch ngang. Lnh \dotfill hin th cc du chm n cui dng.

Once upon atime there weretwo minipages.The first was 1inch wide

whereas the other was two incheswide, and they lived happily sideby side each inside an fbox.

Hnh 16: Bi tp 17 e Boxes v Minipages

C th to khung xung quanh box:

This is a minipage thatis one and a quarterinches wide. . . . . . . . . . . . .

And two inches ////high inheight.

Hnh 17: Bi tp 17 ! Boxes v Minipages

T kim tra

Thay i canh l ca minipage. Dng \parbox thay cho minipage. So snh\parbox v minipage! Th cc loi khung xung quanh minipage. Lm ni v lm chm bng \raisebox. ! Lm th no khai bo chiu cao ca minipage?

18 Lm vic vi nh

C th to ra cc nh s dng cc lnh LATEX cho v hnh. Gi pstricks hoc pgf c th to ra nhng hnh nh phctp, tuy nhin hi kh khn vi mt s ngi. Cch n gin hn l dng cc chng trnh x l nh, ri nhng votrong ti liuLATEX bng lnh:

\includegraphics[options]{filename}

trong gi graphicx: Cc loi nh ph thuc vo loi u ra. thng l PS v EPS (vi dvips) v PDF hay PNG (viPDFLATEX). V d cu lnh:

\includegraphics{shapes}


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dng shapes.ps nu dng LATEX v shapes.pdf nu dng PDFLATEX.Tham s ph option-list dng qun l nh. Cc tham s hay dng c trong Table 10 trang 20

Bng 10: Cc tham s thng dng cho \includegraphics

angle=x quay nh mt gc x

width=len phng/thu v chiu rng len.height=len phng/thu v chiu cao len.scale=x phng/thu.trim=lx by rx ty ct nh gc di bn tri l (lx, by) v gc trn bn phi l (rx, ty).draft khng hin th nh, ch hin th khung xung quanh.

Gi graphicx cn cung cp cc lnh sau:

\rotatebox{angle}{text}

Quay text mt gc angle.

\scalebox{h scale}[v scale]{text}

Phng/thu text. Nu ch c h scale t l tng i c gi.

\reflectbox{text}

Phn x text

\resizebox{h length}{v length}{text}

Thay i kch thc text c kch thc h length v chiu cao v length. Du ! m bo t l tng i khi ch cmt kch thc c a ra.

Bi tp 18

Copy cc files shapes.ps and shapes.pdf, as described in the slides. to ti liu exercise18.tex cha nhshapes.ps hoc shapes.pdf, ph thuc vo engine Canh l gia dng, \centerline. ng khung dng \fbox.Phng to/thu nh v xoay.

19 Hnh v v bng

Hnh v v bng l cc i tng floats c floated-treo vo mt v tr trogn ti liu. Hnh v v bng c tiu v c s th t i km. Tiu c hin th bng:

\caption[short caption]{caption text}

LATEX nh s cc i tng treo t ng, cc i tn treo c tham chiu bng: \label v \ref.Hnh v c to ra bng: figure. Bng c to ra bng mi trng table. C figure v table u khng th cha

ngt trang.Cc hnh con c to bi lnh:

\subfigure[caption]{contents}

c nh ngha trong gi subfigure.Danh sch hnh v v danh sch bng hin th bng:

\listoffigures\listoftables

Thng thng sau mc lc.


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Bi tp 19

Copy circle.ps (hoc circle.pdf) v rectangle.ps (hoc rectangle.pdf). To ra ti liu exercise19.texHnh 18 v 19, v Table 11. Tham chiu ti cc i tng trn bng \ref.

Hnh 18: Some shapes

(a) A Circle (b) A Rectangle

Hnh 19: (a) A Circle, (b) A Rectangle

Bng 11: An example table

A BI 0.5 1.0II 12 14

20 Slides

C nhiu lp to slides. Lp n gin nht l slides. Cc lp phc tp beamer v prosper c th to ra cc slidesrt chuyn nghip. y chng ta dng lp seminar.

Mi slide c to ra bng mi trng slide (landscape) hoc slide* (portrait). Cc on text nm ngoi slide uc coi l ch thch. C th chuyn i slides thnh bi bo(v.d. cho handouts) dng ty bin article ca lp.

C th trn cc slide ng v ngang vi nhau, nhng phi in chng mt cch ring bit. C th hin th ringbit cc slide bng:

\landscapeonly\portraitonly

tt c cc lnh ny phi t trong phn khai bo ca ti liu.C th hin th ring slide hoc ch thch s dng ty bin slidesonly hoc notesonly.Khung ca slide c th thay i bng:

\slideframe{style}


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Cc kiu khung c bn: none and plain. Gi fancyboxcung cp cc kiu khung khc: shadow, double, oval andOval. Kiu trang mi cho slide c th c nh ngha bi:

\newpagestyle{name}{header}{footer}

Bi tp 20

Lm li vi slide hc. m bo cc slide c tiu , thm cc ch thch, ngoi mi trng slide. Th cc tybin article cho lp seminar : \documentclass[article]{seminar}.

21 nh ngha mi trng mi

Mi trng mi nh ngha bng lnh:

\newenvironment{env-name}[n][default]{begin-code}{end-code}

trong n l s cc tham s default l gi tr mc nh ca tham s u tin, nu tham s b sung l cn thit.Ging lnh \newcommand cc tham s c biu din bng #1, #2 ..., v cc bin ny ch s dng c trong phnbegin-code.

Bi tp 21

nh ngha mt mi trng mi, k mt ng k ngang u v cui mt section.Ch thch: Lnh \noindent xa qui nh v tht u dng.

22 Con m

Tt c cc s th t do LATEX to ra t ng u c lu vo cc bin con m. Bin con m mi khai bo bng:

\newcounter{ctr-name}[outer-ctr]

V d: \newcounter{exercise}. Tham s b sung outer-ctr khi to li con m mi khi outer-ctr tng mt n v.V d: \newcounter{exercise}[chapter] s khi ng li con m exercise ti u mi chng.

Cc bin con m l cc bin nguycn, c gi tr thay i bng:

\stepcounter{ctr}

tng ctr 1.

\refstepcounter{ctr}

tng 1, cho php tham chiu n bng \ref v \label.

\setcounter{ctr}{value}

t gi tr con m l value.

\addtocounter{ctr}{value}

cng value vo con m. Nu gi tr ca con m khc cn hco value dng

\value{ctr}

Gi tr ca con m c hin th bng \thectr. V d: \thepage hin th con m page. Mc nh, \thectr hinth cc gi tr bng ch s rp, tuy nhin c th nh ngha li bng cc lnh:


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\arabic{ctr}\roman{ctr}\Roman{ctr}\alph{ctr}\Alph{ctr}\fnsymbol{ctr}

(Ch \fnsymbol ch c th c dng trong ch cng thc ton.) V d:

\renewcommand{\thechapter}{\Roman{chapter}}

chuyn s chng sang ch s La m hoa.

Bi tp 22

Bin i mi trng to ra trong Bi tp 21 hin th s th t cc bi tp nh trong handouts. To ra tithiu 2 mi trng exercise thy con m tng ng. Tham chiu n cc bi tp dng \ref.

Gi , khi bt u exercise con m cn phi tng trc khi gi tr c in ra.Sau , nh ngha li \theexercise hin th theo s La m .! Bin i s th t ph thuc vo section.

23 Cc lnh iu kin

TEX c cc lnh iu kin nh ngha bi gi ifthen :

\ifthenelse{test}{then text}{else text}\whiledo{test}{do text}

test, l biu thc Boolean. Cc bin Boolean c nh ngha bng:

\newboolean{name}

Bin boolean c gn gi tr bng:

\setboolean{name}{value}

trong value l true hoc false. Gi tr ca bin boolean variable c kim tra bng:

\boolean{name}

Kim tra hai chui c ging nhau khng bng:

\equal{string 1}{string 2}

Cc s c th c so snh bng . so snh cc con m dng \value{counter-name}. Kim tra chn l:

\isodd{counter-name}

V d:

\ifthenelse{\isodd{page}}{This page is odd}{This page is even}

hin th thnh: This page is evenCc chiu di c nh gi bng:

\lengthtest{relation}

V d:

This page is in\ifthenelse{\lengthtest{\textwidth > \textheight}}{landscape}{portrait}orientation.

hin th: This page is in portrait orientation.


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24 Vit gi hoc lp mi

Gi mi c lu tr trong tp c m rng .sty. Dng u tin ca file l:

\NeedsTeXFormat{LaTeX2e}

Tn ca gi c khai bo bng:

\ProvidesPackage{name}

cn ging tn file .sty. Dng cui cng ca file:

\endinput

Cc tham s c nh ngha bng

\DeclareOption{option name}{code}

Thao tc mc nh vi bt c ty bin cha nh ngha no c khai bo bng:

\DeclareOption*{default-code}

Trong default-code, Cc lnh sau c th s dng:

\CurrentOption tn ca ty bin\OptionNotUsed ty bin khng c x l.

sau khi c nh ngha cc ty bin c th c x l bi cc lnh:

\ExecuteOptions{options} hin th cc ty bin mc nh\ProcessOptions x l theo trt t nh ngha\ProcessOptions* x l theo trt t gn gi tr.

C th chuyn cc gi tr ca cc ty bin cho cc gi khc s dng

\PassOptionsToPackage{option-list}{package-name}

Cc gi cn s dng khai bo bng:

\RequirePackage{package-name}

Vit mt lp mi tng t nh gi mi, nhng phn m rng ca file l .cls thay cho .sty, \ProvideClassthay cho \ProvidePackage, v c th dng lnh:

\PassOptionsToClass{option-list}{class-name}

khi phi ti lp cn thit bng:

\LoadClass{class-name}

C th ti mt lp, nhng c th s dng nhiu gi(\RequirePackage).K hiu @ c cc ngha khc nhau trong gi, lp, v ti liu. trong ti liu .tex, @ ging nh cc du chm

phy khc, tuy nhin, trong mt lp hay gi, @ c coi nh mt ch ci. C ngha l trong .sty hay .cls file, tnlnh c th cha @ trong khi trong .texth khng th. V d trong .sty hay .cls file, \c@page l lnh, cn trong.tex file, \c@page hin th thnh: @page. Cc lnh cha @ l cc lnh ni b.

Tiu ca Sections, subsections thay i bng cch nh ngha li \section, \subsection. Cc lnh ny cnc:

\@startsection{type}{level}{indent}{before-skip}{after-skip}{style}

Tiu chng v phn thay i bng cch nh ngha li: \@makechapterhead (chng nh s), \@makechapterhead*(chng khng nh s), \@part (phn nh s) and \@spart (phn khng nh s). Cch n gin nht l copym chun trong cc file lp, gi chun, sau bin i theo mnh.


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Bi tp 23

Vit mt lp ti lp report v thay i tiu chng sao cho c mt ng k trn v di tiu , tiu inbng ch thng canh gia, thay i tiu mc (section) thnh font sans-serif ln.

Dng lp trong ti liu ca Bi tp 4 .

Recommended Reading

[1] A Guide to LATEX2: document preparation for beginners and advanced users, Helmut Kopka and Patrick W.Daly, Addison-Wesley (1995).

[2] The LATEX Companion, Michel Goossens, Frank Mittelbach and Alexander Samarin (Addison-Wesley, 1994).

[3] The LATEX Graphics Companion, Michel Goossens, Sebastian Rahtz and Frank Mittelbach, Addison-Wesley(1997).

[4] The LATEX Web Companion, Michel Goossens and Sebastian Rahtz, Addison-Wesley (1999).


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Short Math Guide for LATEX

Michael Downes

American Mathematical Society

Version 1.09 (2002-03-22), currently available athttp://www.ams.org/tex/short-math-guide.html

1. Introduction This is a concise summary of recommended features in LATEX and acouple of extension packages for writing math formulas. Readers needing greater depthof detail are referred to the sources listed in the bibliography, especially [Lamport], [LUG],[AMUG], [LFG], [LGG], and [LC]. A certain amount of familiarity with standard LATEXterminology is assumed; if your memory needs refreshing on the LATEX meaning of command,optional argument, environment, package, and so forth, see [Lamport].

The features described here are available to you if you use LATEX with two extensionpackages published by the American Mathematical Society: amssymb and amsmath. Thus,the source file for this document begins with

\documentclass{article}

\usepackage{amssymb,amsmath}

The amssymb package might be omissible for documents whose math symbol usage is rela-tively modest; the easiest way to test this is to leave out the amssymb reference and see ifany math symbols in the document produce Undefined control sequence messages.

Many noteworthy features found in other packages are not covered here; see Section 10.Regarding math symbols, please note especially that the list given here is not intended to becomprehensive, but to illustrate such symbols as users will normally find already present intheir LATEX system and usable without installing any additional fonts or doing other setupwork.

If you have a need for a symbol not shown here, you will probably want to consult TheComprehensive LATEX Symbols List (Pakin):

http://www.ctan.org/tex-archive/info/symbols/comprehensive/.

2. Inline math formulas and displayed equations

2.1. The fundamentals Entering and leaving math mode in LATEX is normally done withthe following commands and environments.

inline formulas displayed equations

$ . . . $$ . . . $

\[...\] unnumbered

\begin{equation*}

. . .\end{equation*}

unnumbered

\begin{equation}

. . .\end{equation}

automaticallynumbered

Note. Alternative environments \begin{math} . . . \end{math}, \begin{displaymath} . . . \end{displaymath}are seldom needed in practice. Using the plain TEX notation $$ . . . $$ for displayed equations is not recom-mended. Although it is not expressly forbidden in LATEX, it is not documented anywhere in the LATEX bookas being part of the LATEX command set, and it interferes with the proper operation of various featuressuch as the fleqn option.

Environments for handling equation groups and multi-line equations are shown in Table 1.

1

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Short Math Guide for LATEX, version 1.09 (2002-03-22) 2

Table 1: Multi-line equations and equation groups (vertical lines indicating nominal mar-gins).\begin{equation}\label{xx}

\begin{split}

a& =b+c-d\\

& \quad +e-f\\

& =g+h\\

& =i

\end{split}

\end{equation}

a = b+ c d+ e f

= g + h

= i

(2.1)

\begin{multline}

a+b+c+d+e+f\\

+i+j+k+l+m+n

\end{multline}

a+ b+ c+ d+ e+ f

+ i+ j + k + l +m+ n (2.2)

\begin{gather}

a_1=b_1+c_1\\

a_2=b_2+c_2-d_2+e_2

\end{gather}

a1 = b1 + c1 (2.3)

a2 = b2 + c2 d2 + e2 (2.4)

\begin{align}

a_1& =b_1+c_1\\

a_2& =b_2+c_2-d_2+e_2

\end{align}

a1 = b1 + c1 (2.5)

a2 = b2 + c2 d2 + e2 (2.6)

\begin{align}

a_{11}& =b_{11}&

a_{12}& =b_{12}\\

a_{21}& =b_{21}&

a_{22}& =b_{22}+c_{22}

\end{align}

a11 = b11 a12 = b12 (2.7)

a21 = b21 a22 = b22 + c22 (2.8)

\begin{flalign*}

a_{11}& =b_{11}&

a_{12}& =b_{12}\\

a_{21}& =b_{21}&

a_{22}& =b_{22}+c_{22}

\end{flalign*}

a11 = b11 a12 = b12

a21 = b21 a22 = b22 + c22

Note 1. The split environment is something of a special case. It is a subordinate environment that canbe used as the contents of an equation environment or the contents of one line in a multiple-equationstructure such as align or gather.

Note 2. The eqnarray and eqnarray* environments described in [Lamport] are not recommended becausethey produce inconsistent spacing of the equal signs and make no attempt to prevent overprinting of theequation body and equation number.

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2.2. Automatic numbering and cross-referencing To get an auto-numbered equa-tion, use the equation environment; to assign a label for cross-referencing, use the \labelcommand:

\begin{equation}\label{reio}

...

\end{equation}

To get a cross-reference to an auto-numbered equation, use the \eqref command:

... using equations \eqref{ax1} and \eqref{bz2}, we

can derive ...

The above example would produce something like

using equations (3.2) and (3.5), we can derive

In other words, \eqref{ax1} is equivalent to (\ref{ax1}).To give your equation numbers the form m.n (section-number.equation-number), use

the \numberwithin command in the preamble of your document:

\numberwithin{equation}{section}

For more details on custom numbering schemes see [Lamport, 6.3, C.8.4].The subequations environment provides a convenient way to number equations in a

group with a subordinate numbering scheme. For example, supposing that the currentequation number is 2.1, write

\begin{equation}\label{first}

a=b+c

\end{equation}

some intervening text

\begin{subequations}\label{grp}

\begin{align}

a&=b+c\label{second}\\

d&=e+f+g\label{third}\\

h&=i+j\label{fourth}

\end{align}

\end{subequations}

to geta = b+ c (2.9)

some intervening text

a = b+ c (2.10a)

d = e+ f + g (2.10b)

h = i+ j (2.10c)

By putting a \label command immediately after \begin{subequations} you can get areference to the parent number; \eqref{grp} from the above example would produce (2.10)while \eqref{second} would produce (2.10a).

3. Math symbols and math fonts

3.1. Classes of math symbols The symbols in a math formula fall into different classesthat correspond more or less to the part of speech each symbol would have if the formulawere expressed in words. Certain spacing and positioning cues are traditionally used forthe different symbol classes to increase the readability of formulas.

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Classnumber Mnemonic

Description(part of speech) Examples

0 Ord simple/ordinary (noun) A 0 1 Op prefix operator

2 Bin binary operator (conjunction) + 3 Rel relation/comparison (verb) = < 4 Open left/opening delimiter ( [ { 5 Close right/closing delimiter ) ] } 6 Pun postfix/punctuation . , ; !

Note 1. The distinction in TEX between class 0 and an additional class 7 has to do only with font selectionissues and is immaterial here.

Note 2. Symbols of class Bin, notably the minus sign , are automatically coerced to class 0 (no space) ifthey do not have a suitable left operand.

The spacing for a few symbols follows tradition instead of the general rule: although /is (semantically speaking) of class 2, we write k/2 with no space around the slash ratherthan k / 2. And compare p|q p|q (no space) with p\mid q p | q (class-3 spacing).

The proper way to define a new math symbol is discussed in LATEX2 font selection[LFG]. It is not really possible to give a useful synopsis here because one needs first tounderstand the ramifications of font specifications.

3.2. Some symbols intentionally omitted here The following math symbols thatare mentioned in the LATEX book [Lamport] are intentionally omitted from this discussionbecause they are superseded by equivalent symbols when the amssymb package is loaded.If you are using the amssymb package anyway, the only thing that you are likely to gain byusing the alternate name is an unnecessary increase in the number of fonts used by yourdocument.

\Box, see \square \Diamond, see \lozenge \leadsto, see \rightsquigarrow

\Join, see \bowtie ./\lhd, see \vartriangleleft C

\unlhd, see \trianglelefteq E\rhd, see \vartriangleright B

\unrhd, see \trianglerighteq D

Furthermore, there aremany, many additional symbols available for LATEX use aboveand beyond the ones included here. This list is not intended to be comprehensive. For amuch more comprehensive list of symbols, including nonmathematically oriented ones suchas phonetic alphabetic or dingbats, see The Comprehensive LATEX Symbols List (Pakin):

http://www.ctan.org/tex-archive/info/symbols/comprehensive/.

3.3. Latin letters and Arabic numerals The Latin letters are simple symbols, class 0.The default font for them in math formulas is italic.

ABC DE F GH I J K LM N OP QRS T U V W X Y Za b c d e f g h i j k lmn o p q r s t u v w x y z

When adding an accent to an i or j in math, dotless variants can be obtained with \imathand \jmath:

\imath \jmath \hat{\jmath}

Arabic numerals 09 are also of class 0. Their default font is upright/roman.

0 1 2 3 4 5 6 7 8 9

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3.4. Greek letters Like the Latin letters, the Greek letters are simple symbols, class 0.For obscure historical reasons, the default font for lowercase Greek letters in math formu-las is italic while the default font for capital Greek letters is upright/roman. (In otherfields such as physics and chemistry, however, the typographical traditions are somewhatdifferent.) The capital Greek letters not present in this list are the letters that have thesame appearance as some Latin letter: A for Alpha, B for Beta, and so on. In the list oflowercase letters there is no omicron because it would be identical in appearance to Latin o.In practice, the Greek letters that have Latin look-alikes are seldom used in math formulas,to avoid confusion.

\Gamma \Delta \Lambda \Phi \Pi \Psi \Sigma \Theta \Upsilon \Xi \Omega

\alpha \beta \gamma \delta \epsilon \zeta \eta \theta \iota \kappa \lambda \mu \nu

\xipi \pi \rho \sigma \tau \upsilon \phi \chi \psi \omega

z \digamma \varepsilon \varkappa \varphi$ \varpi% \varrho \varsigma \vartheta

3.5. Other alphabetic symbols These are also class 0.

\alephi \bethk \daleth \gimel

{ \complement` \ell \eth~ \hbar

} \hslashf \mho \partial \wp

s \circledSk \Bbbk` \Finva \Game

= \Im< \Re

3.6. Miscellaneous simple symbols These symbols are also of class 0 (ordinary) whichmeans they do not have any built-in spacing.

# \#& \& \angle8 \backprimeF \bigstar \blacklozenge \blacksquareN \blacktriangleH \blacktriangledown \bot

\clubsuit \diagdownupslope \diagup \diamondsuit \emptyset \exists[ \flat \forall \heartsuit \infty

\lozenge] \measuredangle \nabla\ \natural \neg@ \nexists \prime] \sharp \spadesuit^ \sphericalangle

\square\surd

> \top4 \triangleO \triangledown \varnothing

Note 1. A common mistake in the use of the symbols and # is to try to make them serve as binaryoperators or relation symbols without using a properly defined math symbol command. If you merely usethe existing commands \square or \# the inter-symbol spacing will be incorrect because those commandsproduce a class-0 symbol.

Note 2. Synonyms: \lnot

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3.7. Binary operator symbols

*+ + -q \amalg \astZ \barwedge \bigcirc5 \bigtriangledown4 \bigtriangleup \boxdot \boxminus \boxplus \boxtimes \bullet \cape \Cap

\cdot \centerdot \circ~ \circledast} \circledcirc \circleddash \cupuniondbl \Cupg \curlyveeuprise \curlywedge \dagger \ddagger \diamond \div> \divideontimesu \dotplus

[ \doublebarwedgem \gtrdot \intercalh \leftthreetimesl \lessdotn \ltimes \mp \odot \ominus \oplus \oslash \otimes \pmi \rightthreetimeso \rtimes\ \setminus

r \smallsetminusu \sqcapunionsq \sqcup? \star \times/ \triangleleft. \trianglerightunionmulti \uplus \veeY \veebar \wedgeo \wr

Synonyms: \land, \lor, uniondbl \doublecup, e \doublecap

3.8. Relation symbols: < = > and variants

< >

\approxu \approxeq \asympv \backsimw \backsimeql \bumpeqm \Bumpeq$ \circeq= \cong2 \curlyeqprec3 \curlyeqsucc.= \doteq

+ \doteqdotP \eqcirch \eqsim1 \eqslantgtr

0 \eqslantless \equiv; \fallingdotseq \geq= \geqq> \geqslant \gg \ggg \gnapprox

\gneq \gneqq \gnsim' \gtrapproxR \gtreqlessT \gtreqqless \gtrless& \gtrsim \gvertneqq \leq

5 \leqq6 \leqslant/ \lessapproxQ \lesseqgtrS \lesseqqgtr \lessgtr. \lesssim \ll \lll \lnapprox \lneq \lneqq \lnsim \lvertneqq \ncong6= \neq \ngeq \ngeqq \ngeqslant

\ngtr \nleq \nleqq

\nleqslant \nless \nprec \npreceq \nsim \nsucc \nsucceq \precw \precapprox4 \preccurlyeq \preceq \precnapprox \precneqq \precnsim- \precsim: \risingdotseq

\sim' \simeq \succv \succapprox< \succcurlyeq \succeq \succnapprox \succneqq \succnsim% \succsim \thickapprox \thicksim, \triangleq

Synonyms: 6= \ne, \le, \ge, + \Doteq, \llless, \gggtr

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3.9. Relation symbols: arrows See also Section 4.

\circlearrowleft \circlearrowrightx \curvearrowlefty \curvearrowright \downdownarrows \downharpoonleft \downharpoonright \hookleftarrow \hookrightarrow \leftarrow \Leftarrow \leftarrowtail \leftharpoondown \leftharpoonup \leftleftarrows \leftrightarrow \Leftrightarrow \leftrightarrows \leftrightharpoons! \leftrightsquigarrow

W \Lleftarrow \longleftarrow= \Longleftarrow \longleftrightarrow \Longleftrightarrow7 \longmapsto \longrightarrow= \Longrightarrow" \looparrowleft# \looparrowright \Lsh7 \mapsto( \multimap: \nLeftarrow< \nLeftrightarrow; \nRightarrow \nearrow8 \nleftarrow= \nleftrightarrow9 \nrightarrow

\nwarrow \rightarrow \Rightarrow \rightarrowtail \rightharpoondown \rightharpoonup \rightleftarrows

\rightleftharpoons \rightrightarrows \rightsquigarrowV \Rrightarrow \Rsh \searrow \swarrow \twoheadleftarrow \twoheadrightarrow \upharpoonleft \upharpoonright \upuparrows

Synonyms: \gets, \to, \restriction

3.10. Relation symbols: miscellaneous

\backepsilon \becauseG \betweenJ \blacktriangleleftI \blacktriangleright./ \bowtiea \dashv_ \frown \in| \mid|= \models3 \ni- \nmid/ \notin \nparallel. \nshortmid/ \nshortparallel* \nsubseteq

" \nsubseteqq+ \nsupseteq# \nsupseteqq6 \ntriangleleft5 \ntrianglelefteq7 \ntriangleright4 \ntrianglerighteq0 \nvdash1 \nVdash2 \nvDash3 \nVDash \parallel \perpt \pitchfork \proptop \shortmidq \shortparallela \smallfrown

` \smallsmile^ \smile@ \sqsubsetv \sqsubseteqA \sqsupsetw \sqsupseteq \subsetb \Subset \subseteqj \subseteqq( \subsetneq$ \subsetneqq \supsetc \Supset \supseteqk \supseteqq) \supsetneq% \supsetneqq

\thereforeE \trianglelefteqD \trianglerighteq \varpropto \varsubsetneq& \varsubsetneqq! \varsupsetneq' \varsupsetneqqM \vartriangleC \vartriangleleftB \vartriangleright` \vdash

\Vdash \vDash \Vvdash

Synonyms: 3 \owns

3.11. Cumulative (variable-size) operators\int\oint\bigcap\bigcup

\bigodot\bigoplus\bigotimes\bigsqcup

\biguplus\bigvee\bigwedge\coprod

\prod

\smallint\sum

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3.12. Punctuation

. ./ /| |, ,

; ;: \colon: :! !

? ? \dotsb. . . \dotsc \dotsi

\dotsm. . . \dotso

. . . \ddots

... \vdots

Note 1. The : by itself produces a colon with class-3 (relation) spacing. The command \colon producesspecial spacing for use in constructions such as f\colon A\to B f : A B.

Note 2. Although the commands \cdots and \ldots are frequently used, we recommend the more seman-tically oriented commands \dotsb \dotsc \dotsi \dotsm \dotso for most purposes (see 4.6).

3.13. Pairing delimiters (extensible) See Section 6 for more information.()( )[ ][ ]{}\lbrace \rbrace \lvert \rvert

\lVert \rVert\langle \rangle \lceil \rceil \lfloor \rfloor

\lgroup \rgroup \lmoustache \rmoustache

3.14. Nonpairing extensible symbols \vert \Vert / / \backslash \arrowvert www \Arrowvert \bracevertNote 1. Using \vert, |, \Vert, or \| for paired delimiters is not recommended (see 6.2).

Synonyms: \|

3.15. Extensible vertical arrowsx \uparrow ~ww \Uparrow y \downarrow ww \Downarrow xy \updownarrow ~w \Updownarrow3.16. Accents

x \acute{x}x` \grave{x}x \ddot{x}x \tilde{x}

x \bar{x}x \breve{x}x \check{x}x \hat{x}

~x \vec{x}x \dot{x}x \ddot{x}...x \dddot{x}

xxx \widetilde{xxx}xxx \widehat{xxx}

3.17. Named operators These operators are represented by a multiletter abbreviation.

arccos \arccosarcsin \arcsinarctan \arctan

arg \argcos \coscosh \coshcot \cotcoth \coth

csc \cscdeg \degdet \detdim \dimexp \expgcd \gcdhom \hominf \inf

inj lim \injlimker \kerlg \lg

lim \limlim inf \liminflim sup \limsup

ln \lnlog \log

max \maxmin \minPr \Pr

proj lim \projlimsec \secsin \sinsinh \sinhsup \sup

tan \tantanh \tanhlim \varinjlimlim \varprojlimlim \varliminf

lim \varlimsup

To define additional named operators outside the above list, use the \DeclareMathOperatorcommand; for example, after

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\DeclareMathOperator{\rank}{rank}

\DeclareMathOperator{\esssup}{ess\,sup}

one could write

\rank(x) rank(x)\esssup(y,z) ess sup(y, z)

The star form \DeclareMathOperator* creates an operator that takes limits in a displayedformula like sup or max.

When predefining such a named operator is problematic (e.g., when using one in thetitle or abstract of an article), there is an alternative form that can be used directly:

\operatorname{rank}(x) rank(x)

3.18. Math font switches Not all of the fonts necessary to support comprehensivemath font switching are commonly available in a typical LATEX setup. Here are the resultsof applying various font switches to a wide range of math symbols when the standard setof Computer Modern fonts is in use. It can be seen that the only symbols that respondcorrectly to all of the font switches are the uppercase Latin letters. In fact, nearly all mathsymbols apart from Latin letters remain unaffected by font switches; and although thelowercase Latin letters, capital Greek letters, and numerals do respond properly to somefont switches, they produce bizarre results for other font switches. (Use of alternative mathfont sets such as Lucida New Math may ameliorate the situation somewhat.)

default \mathbf \mathsf \mathit \mathcal \mathbb \mathfrak

X X X X X X Xx x x x x x0 0 0 0 0 0[ ] [ ] [ ] [ ] [ ] [ ] [ ]

+ + + + + + +

= = = = = = =

q q q q q q q< < < < < < 0 z

n1kn(1 qk)

\frac{\sum_{n > 0} z^n}

{\prod_{1\leq k\leq n} (1-q^k)}

Using \dfrac instead of \frac wouldnt change anything in this case; if you want the sumand product symbols to appear full size, you need the \displaystyle command:

n>0

zn1kn

(1 qk)\frac{{\displaystyle\sum_{n > 0} z^n}}

{{\displaystyle\prod_{1\leq k\leq n} (1-q^k)}}

And if you want full-size symbols but with limits on the side, use the nolimits commandalso:

n>0zn

1kn(1 qk)

\frac{{\displaystyle\sum\nolimits_{n> 0} z^n}}

{{\displaystyle\prod\nolimits_{1\leq k\leq n} (1-q^k)}}

There are similar commands \textstyle, \scriptstyle, and \scriptscriptstyle, toforce LATEX to use the symbol size and spacing that would be applied in (respectively)inline math, first-order subscript, or second-order order subscript, even when the currentcontext would normally yield some other size.

Note: These commands belong to a special class of commands referred to in the LATEXbook as declarations. In particular, notice where the braces fall that delimit the effect ofthe command:

Right:{\displaystyle ...}Wrong:\displaystyle{...}

10. Other packages of interest Many other LATEX packages that address some aspectof mathematical formulas are available from CTAN (the Comprehensive TEX Archive Net-work). To recommend a few examples:

accents Under accents and accents using arbitrary symbols.

amsthm General theorem and proof setup.

bm Bold math package, provides a more general and more robust implementation of\boldsymbol.

cases Apply a large brace to two or more equations without losing the individual equationnumbers.

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delarray Delimiters spanning multiple rows of an array.

kuvio Commutative diagrams and other diagrams.

xypic Commutative diagrams and other diagrams.

rsfs Ralph Smiths Formal Script, font setup.

The TEX Catalogue,

http://www.tex.ac.uk/tex-archive/help/Catalogue/catalogue.html,

is a good place to look if you know a packages name.

11. Other documentation of interest

References

[AMUG] American Mathematical Society: Users Guide for the amsmath package,amsldoc.tex, 1999.

[CLSL] Pakin, Scott: The Comprehensive LATEX Symbols List, http://www.ctan.org/tex-archive/info/symbols/comprehensive/, July 2001.

[Lamport] Lamport, Leslie: LATEX: a document preparation system, 2nd edition, Addison-Wesley, 1994.

[LC] Goossens, Michel; Mittelbach, Frank; Samarin, Alexander: The LATEX Com-panion, Addison-Wesley, 1994.

[LFG] LATEX3 Project Team: LATEX2 font selection, fntguide.tex, 1994.

[LGC] Goossens, Michel; Rahtz, Sebastian; Mittelbach, Frank: The LATEX GraphicsCompanion, Addison Wesley Longman, 1997.

[LGG] Carlisle, D. P.: Packages in the graphics bundle, grfguide.tex, 1995.

[LUG] LATEX3 Project Team: LATEX2 for authors, usrguide.tex, 1994.

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The Great, Big List of LATEX Symbols

David Carlisle Scott Pakin Alexander Holt

February 7, 2001

List of Tables

1 LATEX 2 Escapable Special Char-acters . . . . . . . . . . . . . . . . . 2

2 LATEX 2 Commands Defined toWork in Both Math and Text Mode 2

3 Non-ASCII Letters (Excluding Ac-cented Letters) . . . . . . . . . . . . 2

4 Greek Letters . . . . . . . . . . . . . 25 Punctuation Marks Not Found in OT1 36 Predefined LATEX 2 Text-Mode

Commands . . . . . . . . . . . . . . 37 Binary Operation Symbols . . . . . . 38 Relation Symbols . . . . . . . . . . . 49 Punctuation Symbols . . . . . . . . . 410 Arrow Symbols . . . . . . . . . . . . 411 Miscellaneous Symbols . . . . . . . . 512 Variable-sized Symbols . . . . . . . . 513 Log-like Symbols . . . . . . . . . . . 514 Delimiters . . . . . . . . . . . . . . . 515 Large Delimiters . . . . . . . . . . . 616 Math-Mode Accents . . . . . . . . . 617 Some Other Constructions . . . . . . 618 textcomp Symbols . . . . . . . . . . . 619 AMS Delimiters . . . . . . . . . . . . 820 AMS Arrows . . . . . . . . . . . . . 821 AMS Negated Arrows . . . . . . . . 822 AMS Greek . . . . . . . . . . . . . . 823 AMS Hebrew . . . . . . . . . . . . . 824 AMS Miscellaneous . . . . . . . . . . 825 AMS Commands Defined to Work in

Both Math and Text Mode . . . . . 9

26 AMS Binary Operators . . . . . . . 927 AMS Binary Relations . . . . . . . . 928 AMS Negated Binary Relations . . . 1029 stmaryrd Delimiters . . . . . . . . . . 1030 stmaryrd Arrows . . . . . . . . . . . 1031 stmaryrd Extension Characters . . . 1032 stmaryrd Binary Operators . . . . . . 1133 stmaryrd Large Binary Operators . . 1134 stmaryrd Binary Relations . . . . . . 1135 stmaryrd Negated Binary Relations . 1136 wasysym Math-Mode Symbols . . . . 1137 wasysym General Symbols . . . . . . 1238 wasysym Electrical and Physical

Symbols . . . . . . . . . . . . . . . . 1239 wasysym Polygons and Stars . . . . . 1240 wasysym Musical Notes . . . . . . . . 1241 wasysym Circles . . . . . . . . . . . . 1242 wasysym Phonetic Symbols . . . . . 1243 wasysym Astrological and Zodiacal

Symbols . . . . . . . . . . . . . . . . 1344 wasysym APL Symbols . . . . . . . . 1345 wasysym APL Modifiers . . . . . . . 1346 pifont Commands for Using Zapf

Dingbats . . . . . . . . . . . . . . . . 1347 marvosym Astrological and Zodiacal

Symbols . . . . . . . . . . . . . . . . 1448 marvosym Digits . . . . . . . . . . . 1449 marvosym Euro Signs . . . . . . . . . 1550 marvosym Miscellaneous . . . . . . . 1551 Math Alphabets . . . . . . . . . . . 16

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Table 1: LATEX 2 Escapable Special Characters

$ \$ % \% \_ } \} & \& # \# { \{

Table 2: LATEX 2 Commands Defined to Work in Both Math and Text Mode

$ \$ \_ \ddag { \{ \P cO \copyright . . . \dots } \} \S \dag \pounds

(Where two symbols are present, the left one is the faked symbol that LATEX 2 providesby default, and the right one is the true symbol that textcomp makes available.)

Table 3: Non-ASCII Letters (Excluding Accented Letters)

a \aa \DH L \L \o \ssA \AA \dh l \l \O SS \SS \AE \DJ \NG \OE \TH

\ae \dj \ng \oe \th

= Not available in the OT1 font encoding. Use the fontenc package to select an alternate fontencoding, such as T1.

Table 4: Greek Letters

\alpha \theta o o \tau \beta \vartheta pi \pi \upsilon \gamma \iota $ \varpi \phi \delta \kappa \rho \varphi \epsilon \lambda % \varrho \chi \varepsilon \mu \sigma \psi \zeta \nu \varsigma \omega \eta \xi

\Gamma \Lambda \Sigma \Psi \Delta \Xi \Upsilon \Omega \Theta \Pi \Phi

(The remaining Greek majuscules can be produced with ordinary Latin letters. The symbolM, for instance, is used for both an uppercase m and an uppercase .)

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Table 5: Punctuation Marks Not Found in OT1

\guillemotleft \guilsinglleft \quotedblbase " \textquotedbl

\guillemotright \guilsinglright \quotesinglbase

(To get these symbols, use the fontenc package to select an alternate font encoding, suchas T1.)

Table 6: Predefined LATEX 2 Text-Mode Commands

\textasciicircum < \textless \textasciitilde a \textordfeminine* \textasteriskcentered o \textordmasculine

\ \textbackslash \textparagraph| \textbar \textperiodcentered{ \textbraceleft \textquestiondown} \textbraceright \textquotedblleft \textbullet \textquotedblright

cO \textcopyright \textquoteleft \textdagger \textquoteright \textdaggerdbl rO \textregistered$ \textdollar \textsection

. . . \textellipsis \textsterling \textemdash TM \texttrademark \textendash \textunderscore \textexclamdown \textvisiblespace> \textgreater


Table 7: Binary Operation Symbols

\pm \cap \diamond \oplus \mp \cup a \bigtriangleup \ominus \times unionmulti \uplus ` \bigtriangledown \otimes \div u \sqcap / \triangleleft \oslash \ast unionsq \sqcup . \triangleright \odot? \star \vee \lhd \bigcirc \circ \wedge \rhd \dagger \bullet \ \setminus unlhd \unlhd \ddagger \cdot o \wr unrhd \unrhd q \amalg+ + -

Not predefined in LATEX 2. Use one of the packages latexsym, amsfonts, amssymb, or wasysym.

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Table 8: Relation Symbols

\leq \geq \equiv |= \models \prec \succ \sim \perp \preceq \succeq ' \simeq | \mid \ll \gg \asymp \parallel \subset \supset \approx ./ \bowtie \subseteq \supseteq = \cong 1 \Join< \sqsubset = \sqsupset 6= \neq ^ \smilev \sqsubseteq w \sqsupseteq .= \doteq _ \frown \in 3 \ni \propto = =` \vdash a \dashv < < > >: :


Table 9: Punctuation Symbols

, , ; ; : \colon . \ldotp \cdotp

Table 10: Arrow Symbols

\leftarrow \longleftarrow \uparrow \Leftarrow = \Longleftarrow \Uparrow \rightarrow \longrightarrow \downarrow> \Rightarrow => \Longrightarrow \Downarrow \leftrightarrow \longleftrightarrow l \updownarrow \Leftrightarrow > \Longleftrightarrow m \Updownarrow7 \mapsto 7 \longmapsto \nearrow \hookleftarrow \hookrightarrow \searrow \leftharpoonup \rightharpoonup \swarrow \leftharpoondown \rightharpoondown \nwarrow

\rightleftharpoons ; \leadsto


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Table 11: Miscellaneous Symbols

. . . \ldots \cdots ... \vdots . . . \ddots \aleph \prime \forall \infty~ \hbar \emptyset \exists 2 \Box \imath \nabla \neg 3 \Diamond \jmath

\surd [ \flat 4 \triangle

` \ell > \top \ \natural \clubsuit \wp \bot ] \sharp \diamondsuit< \Re \| \ \backslash \heartsuit= \Im \angle \partial \spadesuit0 \mho . . | |


Table 12: Variable-sized Symbols\sum

\bigcap

\bigodot

\prod

\bigcup

\bigotimes\coprod

\bigsqcup

\bigoplus

\int

\bigvee

\biguplus\oint

\bigwedge

Table 13: Log-like Symbols

\arccos \cos \csc \exp \ker \limsup \min \sinh

\arcsin \cosh \deg \gcd \lg \ln \Pr \sup

\arctan \cot \det \hom \lim \log \sec \tan

\arg \coth \dim \inf \liminf \max \sin \tanh

Table 14: Delimiters

( ( ) ) \uparrow \Uparrow[ [ ] ] \downarrow \Downarrow{ \{ } \} l \updownarrow m \Updownarrowb \lfloor c \rfloor d \lceil e \rceil \langle \rangle / / \ \backslash| | \|

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Table 15: Large Delimiters \rmoustache \lmoustache \rgroup \lgroup \arrowvert ww \Arrowvert \bracevertTable 16: Math-Mode Accents

a \hat{a} a \acute{a} a \bar{a} a \dot{a} a \breve{a}a \check{a} a` \grave{a} ~a \vec{a} a \ddot{a} a \tilde{a}

Table 17: Some Other Constructions

abc \widetilde{abc} abc \widehat{abc}abc \overleftarrow{abc}

abc \overrightarrow{abc}

abc \overline{abc} abc \underline{abc}abc \overbrace{abc} abc \underbrace{abc}abc \sqrt{abc} n

abc \sqrt[n]{abc}

f f abcxyz \frac{abc}{xyz}

Table 18: textcomp Symbols1

\textacutedbl \textlquill

\textascendercompwordmark m \textmarried

\textasciiacute M \textmho

\textasciibreve = \textminus

\textasciicaron \textmu

\textasciidieresis n \textmusicalnote

` \textasciigrave \textnaira

\textasciimacron 9 \textnineoldstyle

* \textasteriskcentered \textnumero

\textbaht W \textohm

\textbardbl \textonehalf

O \textbigcircle 1 \textoneoldstyle

\textblank \textonequarter

b \textborn \textonesuperior

\textbrokenbar \textopenbullet

(continued on next page)

1These symbols are also available in math mode through the use of the mathcomp package. See the mathcompdocumentation for usage information.

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(continued from previous page)

\textbullet a \textordfeminine

\textcapitalcompwordmark o \textordmasculine

\textcelsius \textparagraph

\textcent \textperiodcentered

\textcentoldstyle \textpertenthousand

\textcircledP \textperthousand

\textcolonmonetary \textpeso

\textcopyleft \textpilcrow

cO \textcopyright \textpm

\textcurrency ' \textquotesingle

\textdagger \textquotestraightbase

\textdaggerdbl \textquotestraightdblbase

- \textdblhyphen > \textrangle

\textdblhyphenchar ] \textrbrackdbl

\textdegree \textrecipe

d \textdied \textreferencemark

\textdiscount rO \textregistered \textdiv \textrightarrow

c \textdivorced \textrquill

$ \textdollar \textsection

\textdollaroldstyle \textservicemark

\textdong 7 \textsevenoldstyle

_ \textdownarrow 6 \textsixoldstyle

8 \texteightoldstyle \textsterling

\textestimated \textsurd

\texteuro 3 \textthreeoldstyle

5 \textfiveoldstyle \textthreequarters

\textflorin \textthreequartersemdash

4 \textfouroldstyle \textthreesuperior

/ \textfractionsolidus ~ \texttildelow

\textgravedbl \texttimes

\textguarani TM \texttrademark

\textinterrobang \texttwelveudash

\textinterrobangdown 2 \texttwooldstyle

< \textlangle \texttwosuperior

[ \textlbrackdbl ^ \textuparrow

l \textleaf \textwon

\textleftarrow \textyen

\textlira 0 \textzerooldstyle

\textlnot


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Table 19: AMS Delimiters

p \ulcorner q \urcorner x \llcorner y \lrcorner

Table 20: AMS Arrows

99K \dashrightarrow L99 \dashleftarrow \leftleftarrows \leftrightarrowsW \Lleftarrow \twoheadleftarrow \leftarrowtail " \looparrowleft \leftrightharpoons x \curvearrowleft \circlearrowleft \Lsh \upuparrows \upharpoonleft \downharpoonleft ( \multimap! \leftrightsquigarrow \rightrightarrows \rightleftarrows \rightrightarrows \rightleftarrows \twoheadrightarrow \rightarrowtail # \looparrowright

\rightleftharpoons y \curvearrowright \circlearrowright \Rsh \downdownarrows \upharpoonright \downharpoonright \rightsquigarrow

Table 21: AMS Negated Arrows

8 \nleftarrow 9 \nrightarrow : \nLeftarrow ; \nRightarrow= \nleftrightarrow < \nLeftrightarrow

Table 22: AMS Greek

z \digamma \varkappa

Table 23: AMS Hebrew

i \beth k \daleth \gimel

Table 24: AMS Miscellaneous

~ \hbar } \hslash M \vartriangle O \triangledown \square \lozenge s \circledS \angle] \measuredangle @ \nexists 0 \mho ` \Finva \Game k \Bbbk 8 \backprime \varnothingN \blacktriangle H \blacktriangledown \blacksquare \blacklozengeF \bigstar ^ \sphericalangle { \complement \ethupslope \diagup \diagdown

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Table 25: AMS Commands Defined to Work in Both Math and Text Mode

X \checkmark r \circledR z \maltese

Table 26: AMS Binary Operators

u \dotplus r \smallsetminus e \Cap uniondbl \CupZ \barwedge Y \veebar [ \doublebarwedge \boxminus \boxtimes \boxdot \boxplus > \divideontimesn \ltimes o \rtimes h \leftthreetimes i \rightthreetimesuprise \curlywedge g \curlyvee \circleddash ~ \circledast} \circledcirc \centerdot \intercal

Table 27: AMS Binary Relations

5 \leqq 6 \leqslant 0 \eqslantless . \lesssim/ \lessapprox u \approxeq l \lessdot \lll \lessgtr Q \lesseqgtr S \lesseqqgtr + \doteqdot: \risingdotseq ; \fallingdotseq v \backsim w \backsimeqj \subseteqq b \Subset < \sqsubset 4 \preccurlyeq2 \curlyeqprec - \precsim w \precapprox C \vartriangleleftE \trianglelefteq \vDash \Vvdash ` \smallsmilea \smallfrown l \bumpeq m \Bumpeq = \geqq> \geqslant 1 \eqslantgtr & \gtrsim ' \gtrapproxm \gtrdot \ggg \gtrless R \gtreqlessT \gtreqqless P \eqcirc $ \circeq , \triangleq \thicksim \thickapprox k \supseteqq c \Supset= \sqsupset < \succcurlyeq 3 \curlyeqsucc % \succsimv \succapprox B \vartriangleright D \trianglerighteq \Vdashp \shortmid q \shortparallel G \between t \pitchfork \varpropto J \blacktriangleleft \therefore \backepsilonI \blacktriangleright \because

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Table 28: AMS Negated Binary Relations

\nless \nleq \nleqslant \nleqq \lneq \lneqq \lvertneqq \lnsim \lnapprox \nprec \npreceq \precnsim \precnapprox \nsim . \nshortmid - \nmid0 \nvdash 2 \nvDash 6 \ntriangleleft 5 \ntrianglelefteq* \nsubseteq ( \subsetneq \varsubsetneq $ \subsetneqq& \varsubsetneqq \ngtr \ngeq \ngeqslant \ngeqq \gneq \gneqq \gvertneqq \gnsim \gnapprox \nsucc \nsucceq \nsucceq \succnsim \succnapprox \ncong/ \nshortparallel \nparallel 2 \nvDash 3 \nVDash7 \ntriangleright 4 \ntrianglerighteq + \nsupseteq # \nsupseteqq) \supsetneq ! \varsupsetneq % \supsetneqq ' \varsupsetneqq

Table 29: stmaryrd DelimitersH \Lbag I \Rbag * \lbag + \rbagV \llceil W \rrceil T \llfloor U \rrfloorJ \llbracket K \rrbracketTable 30: stmaryrd Arrows

=\ \Longmapsfrom Z=> \Longmapsto \ \Mapsfrom Z > \Mapsto1 \nnearrow 0 \nnwarrow % \ssearrow $ \sswarrow \shortdownarrow \shortuparrow \shortleftarrow \shortrightarrow [ \longmapsfrom [ \mapsfrom ^ \leftarrowtriangle _ \rightarrowtriangle \lightning M \rrparenthesis - \leftrightarroweq ] \leftrightarrowtriangle

Note that wasysym also defines a \lightning symbol. The differenceother than vs. is that the stmaryrd version (above) is limited to math mode.

Table 31: stmaryrd Extension CharactersY \Arrownot \ \Mapsfromchar Z \MapstocharX \arrownot [ \mapsfromchar

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Table 32: stmaryrd Binary Operators \Ydown \Yleft \Yright \Yup \baro \bbslash N \binampersand O \bindnasrepma \boxast \boxbar \boxbox \boxbslash \boxcircle \boxdot \boxempty \boxslash. \curlyveedownarrow / \curlyveeuparrow ' \curlywedgedownarrow & \curlywedgeuparrow) \fatbslash # \fatsemi ( \fatslash 9 \interleave2 \leftslice ! \merge \minuso \mooC \nplus : \obar @ \oblong ; \obslash= \ogreaterthan < \olessthan > \ovee ? \owedge3 \rightslice \sslash 8 \talloblong , \varbigcirc \varcurlyvee \varcurlywedge \varoast \varobar \varobslash \varocircle \varodot 5 \varogreaterthan4 \varolessthan \varominus \varoplus \varoslash \varotimes 6 \varovee 7 \varowedge " \vartimesTable 33: stmaryrd Large Binary Operatorse

\bigboxb

\bigcurlyveec

\bigcurlywedgeg\biginterleave

p\bignplus

f\bigparalleld

\bigsqcap`

\bigtriangledowna

\bigtriangleup

Table 34: stmaryrd Binary RelationsA \inplus B \niplus D \subsetplus F \subsetpluseqE \supsetplus G \supsetpluseq P \trianglelefteqslant Q \trianglerighteqslantTable 35: stmaryrd Negated Binary RelationsR \ntrianglelefteqslant S \ntrianglerighteqslantTable 36: wasysym Math-Mode Symbols

2 \Box > \apprle \logof unlhd \unlhd3 \Diamond t \iiint # \ocircle unrhd \unrhd1 \Join s \iint v \oiint r \varint \LHD \invneg \rhd u \varoint \RHD ; \leadsto < \sqsubset \wasypropto? \apprge \lhd = \sqsupset

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Table 37: wasysym General Symbols

1 \Bowtie - \blacksmiley / \frownie \recorderL \DOWNarrow | \brokenvert \invdiameter , \smiley \LEFTarrow \cent 6 \kreuz \sun \RIGHTarrow \checked \lightning \varangleK \UParrow \clock \male \wasylozenge0 \agemO \currency h \permil \wasythereforem \ataribox \diameter \phone \bell \female \pointerNote that stmaryrd also defines a \lightning symbol. The differenceother than vs.is that the wasysym version (above) gives the correct character only in text mode.

Table 38: wasysym Electrical and Physical Symbols

: \AC @ \VHF :::: \photon :: \HF QPPPPPPR \gluon

Table 39: wasysym Polygons and Stars

2 \CheckedBox C \davidsstar 8 \octagon B \varhexstar2 \Square 7 \hexagon D \pentagon4 \XBox A \hexstar 9 \varhexagon

Table 40: wasysym Musical Notes

\eighthnote \halfnote \twonotes \fullnote \quarternote

Table 41: wasysym Circles

\CIRCLE G# \LEFTcircle H# \RIGHTcircle ! \rightturn# \Circle I \Leftcircle J \RightcircleG \LEFTCIRCLE H \RIGHTCIRCLE " \leftturn

Table 42: wasysym Phonetic Symbols

D \DH k \dh l \openo \Thorn U \inve \thorn

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Table 43: wasysym Astrological and Zodiacal Symbols

\ascnode X \jupiter \newmoon \venus \astrosun $ \leftmoon \ \pluto \vernal \descnode \mars % \rightmoon \earth ' \mercury Y \saturn# \fullmoon [ \neptune uranus \uranus

Table 44: wasysym APL Symbols

~ \APLbox ~ \APLinv E \APLstar \APLcomment p \APLleftarrowbox \APLupF \APLdown \APLlog n \APLuparrowboxo \APLdownarrowbox \APLminus \ \notbackslash} \APLinput q \APLrightarrowbox / \notslash

Table 45: wasysym APL Modifiers

\APLcirc{} \APLnot{} | \APLvert{}

Table 46: pifont Commands for Using Zapf Dingbats

! \ding{33} G \ding{71} m \ding{109} \ding{181} \ding{219}" \ding{34} H \ding{72} n \ding{110} \ding{182} \ding{220}# \ding{35} I \ding{73} o \ding{111} \ding{183} \ding{221}$ \ding{36} J \ding{74} p \ding{112} \ding{184} \ding{222}% \ding{37} K \ding{75} q \ding{113} \ding{185} \ding{223}& \ding{38} L \ding{76} r \ding{114} \ding{186} \ding{224}' \ding{39} M \ding{77} s \ding{115} \ding{187} \ding{225}( \ding{40} N \ding{78} t \ding{116} \ding{188} \ding{226}) \ding{41} O \ding{79} u \ding{117} \ding{189} \ding{227}* \ding{42} P \ding{80} v \ding{118} \ding{190} \ding{228}+ \ding{43} Q \ding{81} w \ding{119} \ding{191} \ding{229}, \ding{44} R \ding{82} x \ding{120} \ding{192} \ding{230}- \ding{45} S \ding{83} y \ding{121} \ding{193} \ding{231}. \ding{46} T \ding{84} z \ding{122} \ding{194} \ding{232}/ \ding{47} U \ding{85} { \ding{123} \ding{195} \ding{233}0 \ding{48} V \ding{86} | \ding{124} \ding{196} \ding{234}1 \ding{49} W \ding{87} } \ding{125} \ding{197} \ding{235}2 \ding{50} X \ding{88} ~ \ding{126} \ding{198} \ding{236}

(continued on next page)

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(continued from previous page)

3 \ding{51} Y \ding{89} \ding{161} \ding{199} \ding{237}4 \ding{52} Z \ding{90} \ding{162} \ding{200} \ding{238}5 \ding{53} [ \ding{91} \ding{163} \ding{201} \ding{239}6 \ding{54} \ \ding{92} \ding{164} \ding{202} \ding{241}7 \ding{55} ] \ding{93} \ding{165} \ding{203} \ding{242}8 \ding{56} ^ \ding{94} \ding{166} \ding{204} \ding{243}9 \ding{57} _ \ding{95} \ding{167} \ding{205} \ding{244}: \ding{58} ` \ding{96} \ding{168} \ding{206} \ding{245} \ding{59} a \ding{97} \ding{169} \ding{207} \ding{246} \ding{60} b \ding{98} \ding{170} \ding{208} \ding{247}= \ding{61} c \ding{99} \ding{171} \ding{209} \ding{248}> \ding{62} d \ding{100} \ding{172} \ding{210} \ding{249}? \ding{63} e \ding{101} \ding{173} \ding{211} \ding{250}@ \ding{64} f \ding{102} \ding{174} \ding{212} \ding{251}A \ding{65} g \ding{103} \ding{175} \ding{213} \ding{252} \ding{66} h \ding{104} \ding{176} \ding{214} \ding{253} \ding{67} i \ding{105} \ding{177} \ding{215} \ding{254}D \ding{68} j \ding{106} \ding{178} \ding{216}E \ding{69} k \ding{107} \ding{179} \ding{217}F \ding{70} l \ding{108} \ding{180} \ding{218}

Table 47: marvosym Astrological and Zodiacal Symbols

\Jupiter \Moon \Saturn \Venus \Mars \Neptune \Sun \Mercury \Pluto \Uranus \Aries \Cancer \Libra \Capricorn \Taurus \Leo \Scorpio \Aquarius \Gemini \Virgo \Sagittarius \Pisces

Note that \Aries . . . \Pisces can also be specified with \Zodiac{1} . . . \Zodiac{12}.

Table 48: marvosym Digits

0 \MVZero 2 \MVTwo 4 \MVFour 6 \MVSix 8 \MVEight1 \MVOne 3 \MVThree 5 \MVFive 7 \MVSeven 9 \MVNine

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Table 49: marvosym Euro Signs

\EUR d \EURcr c \EURhv e \EURtm

Table 50: marvosym Miscellaneous

\Ankh q \Cutright & \Lefttorque \Righttorque \Bat u \FAX B \Letter \Smiley" \Beam t \fax E \Lightning * \Snowflake# \Bearing v \Faxmachine ) \Lineload \Squaredot \Bicycle F \FHBOlogo $ \Loosebearing \Squarepipe \Celtcross f \FHBOLOGO \Lsteel ! \StopsignC \CEsign % \Fixedbearing \Manfront T \TelefonV \Checkedbox \Flatsteel \Manside \Tsteel. \Circles o \Football H \Mobilefone \TTsteel \Circpipe ( \Force \MVA p \VectorarrowU \Clocklogo \Frowny @ \MVAt P \VectorarrowhighK \Coffeecup \Heart \MVp \Womanface= \Corresponds I \Industry A \Pickup \Womanfront \Cross i \Info Z \Pointinghand \WomansideX \Crossedbox + \Kross \Rectpipe b \Writinghands \Cutleft R \Kutline > \Rightarrow M \Yingyangr \Cutline S \Leftscissors Q \Rightscissors

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Table 51: Math Alphabets

Required packageABCdef123 \mathrm{ABCdef123} noneABCdef123 \mathit{ABCdef123} noneABCdef \mathnormal{ABCdef123} noneABC \mathcal{ABC} noneABC \mathscr{ABC} mathrsfsABC \mathcal{ABC} euscript with option: mathcal

or \mathscr{ABC} euscript with option: mathcrABCdef123 \mathpzc{ABCdef123} none; manually defined

ABC \mathbb{ABC} amsfonts or amssymbABCdef123 \mathbb{ABCdef123} bboldABCdef12 \mathbbm{ABCdef12} bbmABCdef12 \mathbbmss{ABCdef12} bbmABCdef12 \mathbbmtt{ABCdef12} bbmABC1 \mathds{ABC1} dsfontABC1 \mathds{ABC1} dsfont with option: sansABCdef123 \mathfrak{ABCdef123} eufrakABCdef123 \textfrak{ABCdef123} yfontsABCdef123 \textswab{ABCdef123} yfonts

Put \DeclareMathAlphabet{\mathpzc}{OT1}{pzc}{m}{it} in your documents pream-ble to make \mathpzc typeset its argument in Zapf Chancery.

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Index

If youre having trouble locating a symbol, try looking under T for \text. . .. Many text-mode commands begin with that prefix.

Symbols\# . . . . . . . . . . . . . . . . 2\$ . . . . . . . . . . . . . . . . 2\% . . . . . . . . . . . . . . . . 2\& . . . . . . . . . . . . . . . . 2( . . . . . . . . . . . . . . . . 5) . . . . . . . . . . . . . . . . 5+ . . . . . . . . . . . . . . . . 3, . . . . . . . . . . . . . . . . 4- . . . . . . . . . . . . . . . . 3. . . . . . . . . . . . . . . . . 5/ . . . . . . . . . . . . . . . . 5: . . . . . . . . . . . . . . . . 4; . . . . . . . . . . . . . . . . 4< . . . . . . . . . . . . . . . . 4[ . . . . . . . . . . . . . . . . 5] . . . . . . . . . . . . . . . . 5\_ . . . . . . . . . . . . . . . . 2

A\AA . . . . . . . . . . . . . . . 2\aa . . . . . . . . . . . . . . . 2\AC . . . . . . . . . . . . . . 12accents . . . . . . . . . . . . 6\acute . . . . . . . . . . . . 6\AE . . . . . . . . . . . . . . . 2\ae . . . . . . . . . . . . . . . 2\agemO . . . . . . . . . . . 12\aleph . . . . . . . . . . . . 5\alpha . . . . . . . . . . . . 2alphabets

Greek . . . . . . . 2, 8Hebrew . . . . . . . . 8math . . . . . . . . . 16

\amalg . . . . . . . . . . . . 3AMS . . . . . . . . . . . 810amsfonts . . . . . . 35, 16amssymb . . . . . . 35, 16\angle . . . . . . . . . . 5, 8\Ankh . . . . . . . . . . . . 15APL

modifiers . . . . . . 13symbols . . . . . . . 13

\APLbox . . . . . . . . . . 13\APLcirc . . . . . . . . . 13\APLcomment . . . . . . . 13

\APLdown . . . . . . . . . 13\APLdownarrowbox . . . 13\APLinput . . . . . . . . . 13\APLinv . . . . . . . . . . 13\APLleftarrowbox . . . 13\APLlog . . . . . . . . . . 13\APLminus . . . . . . . . . 13\APLnot . . . . . . . . . . 13\APLrightarrowbox . . 13\APLstar . . . . . . . . . 13\APLup . . . . . . . . . . . 13\APLuparrowbox . . . . 13\APLvert . . . . . . . . . 13\apprge . . . . . . . . . . 11\apprle . . . . . . . . . . 11\approx . . . . . . . . . . . 4\approxeq . . . . . . . . . . 9\Aquarius . . . . . . . . . 14\arccos . . . . . . . . . . . 5\arcsin . . . . . . . . . . . 5\arctan . . . . . . . . . . . 5\arg . . . . . . . . . . . . . . 5\Aries . . . . . . . . . . . 14\Arrownot . . . . . . . . . 10\arrownot . . . . . . . . . 10arrows . . . . . . . 4, 8, 10

negated . . . . . . . . 8\Arrowvert . . . . . . . . . 6\arrowvert . . . . . . . . . 6ASCII . . . . . . . . . . . . . 2\ascnode . . . . . . . . . 13\ast . . . . . . . . . . . . . . 3astrological symbols 13, 14\astrosun . . . . . . . . . 13\asymp . . . . . . . . . . . . 4\ataribox . . . . . . . . . 12

B\backepsilon . . . . . . . 9\backprime . . . . . . . . . 8\backsim . . . . . . . . . . 9\backsimeq . . . . . . . . . 9\backslash . . . . . . . . . 5\bar . . . . . . . . . . . . . . 6\baro . . . . . . . . . . . . 11\barwedge . . . . . . . . . . 9

\Bat . . . . . . . . . . . . . 15\Bbbk . . . . . . . . . . . . . 8bbm . . . . . . . . . . . . . 16bbold . . . . . . . . . . . . 16\bbslash . . . . . . . . . 11\Beam . . . . . . . . . . . . 15\Bearing . . . . . . . . . 15\because . . . . . . . . . . 9\bell . . . . . . . . . . . . 12\beta . . . . . . . . . . . . . 2\beth . . . . . . . . . . . . . 8\between . . . . . . . . . . 9\Bicycle . . . . . . . . . 15\bigbox . . . . . . . . . . 11\bigcap . . . . . . . . . . . 5\bigcirc . . . . . . . . . . 3\bigcup . . . . . . . . . . . 5\bigcurlyvee . . . . . . 11\bigcurlywedge . . . . 11\biginterleave . . . . 11\bignplus . . . . . . . . . 11\bigodot . . . . . . . . . . 5\bigoplus . . . . . . . . . . 5\bigotimes . . . . . . . . . 5\bigparallel . . . . . . 11\bigsqcap . . . . . . . . . 11\bigsqcup . . . . . . . . . . 5\bigstar . . . . . . . . . . 8\bigtriangledown . 3, 11\bigtriangleup . . 3, 11\biguplus . . . . . . . . . . 5\bigvee . . . . . . . . . . . 5\bigwedge . . . . . . . . . . 5\binampersand . . . . . 11binary operators 3, 9, 11

large . . . . . . . . . 11binary relations . . . 9, 11

negated . . . . 10, 11\bindnasrepma . . . . . 11\blacklozenge . . . . . . 8\blacksmiley . . . . . . 12\blacksquare . . . . . . . 8\blacktriangle . . . . . 8\blacktriangledown . . 8\blacktriangleleft . . 9\blacktriangleright . 9

\bot . . . . . . . . . . . . . . 5\Bowtie . . . . . . . . . . 12\bowtie . . . . . . . . . . . 4\Box . . . . . . . . . . . 5, 11\boxast . . . . . . . . . . 11\boxbar . . . . . . . . . . 11\boxbox . . . . . . . . . . 11\boxbslash . . . . . . . . 11\boxcircle . . . . . . . . 11\boxdot . . . . . . . . 9, 11\boxempty . . . . . . . . . 11\boxminus . . . . . . . . . . 9\boxplus . . . . . . . . . . 9\boxslash . . . . . . . . . 11\boxti

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