<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%-*-tex-*-
\ifundefined{writestatus} \input status \relax \fi %
\chcode{matsym}


\def\cqu{\cquote{{\leftskip=0ptplus1fill\def\\%
{\endgraf}Yet what are all such gaieties to me\\Whose thoughts are full of
indices and surds?\\$x^2+7x+53$\\ $\displaystyle={11\over3}$.\\ }}{
Four~Riddles, Lewis~Carroll (1832-1898)}}


\chapterhead{matsym}{MATHEMATICS:\cr SYMBOLS\cr FONTS}
\tex\ supplies a vast array of mathematical symbols, both in fixed and
extensible forms. For the most part, the symbols mentioned are available only
in mathematics mode. Other symbols that are useful in 
regular text mode have been listed in Section \ref{specialtextsymbols}
This chapter describes the symbols available. These
symbols can be used in any font size that is on the system. Although it is
possible to build your own symbols from simpler ones, it takes a little care
to make sure that they are the correct size for all possible uses. 

\shead{mathfonts}{Ordinary Fonts in Mathematics}
The mathematics symbol font contains characters that are very similar to
to text italic font. The major difference is the spacing. {\it For instance} is
in ordinary italic but $For\ instance$, is in math italic and was invoked with
|$For\|\]|instance$|, where |\|\]  forces the  space. 
However, it is an easy matter to put
roman or {\bf bold} letters in  mathematics by just using 
|$x={\bf the\ spot}$| to obtain $x={\bf the\ spot}$. The real fun
comes  with something like the following:

\def\xbf{{\vec{\bf x}}}
$$ \xbf = 2^{\xbf^\xbf} $$
which is given by
\begintt
\def\xbf{{\vec{\bf x}}}
$$ 
\xbf = 2^{\xbf^\xbf} 
$$
\endtt
Note that the bold face is scaled correctly up into the superscripts. The
extra $\{\ldots\}$ are necessary so that the third |\xbf| will look like a
single token to the second |\@|.

\shead{mathtypes}{Mathematical Symbol Types}
Some symbols, even though they may look the same are given different command
sequences in \tex. The reason for this is that their use demands different
spacing and line breaking when in mathematics mode.
For this reason there are four mathematics symbols for a vertical
bar $\mid$, namely |\mid|, |\left\vert|, |\right\vert|, and |\vert|. The
first  is a {\it relation} -- called {\it Rel} -- like the $=$; the second
and third are opening and 
closing brackets like ( and ) --- \tex\ calls these {\it openings} and {\it
closings}; and the last is an {\it ordinary} --- called {\it Ord} --- symbol
like $\infty$. Two other types are {\it large operators} --- called {\it Op}
--- like $\sum$ and {\it binary operators} --- called {\it Bin} --- like $+$.
Finally there is the {\it punctuation} --- called {\it Punct} --- like the
period. 

\shead{greekletters}{Greek Letters}
All the Greek letters are of type {\it Ord}. They are all assumed to be in
mathematics mode.

\bshortcomlist
\@|\alpha {\rm A} A|&amp;$\alpha$ A $A$ \cr
\@|\beta {\rm B} B|&amp;$\beta$ B $B$\cr 
\@|\gamma \Gamma {\mit\Gamma}|&amp;$\gamma$  $\Gamma$  ${\mit\Gamma}$ \cr
\@|\delta \Delta {\mit\Delta} |&amp;$\delta$ $\Delta$ ${\mit\Delta}$ \cr
\@|\epsilon {\rm E} E|&amp;$\epsilon$  E $E$ \cr
\@|\varepsilon|&amp;$\varepsilon$\cr
\@|\zeta {\rm Z} Z|&amp;$\zeta$ Z $Z$ \cr
\@|\eta {\rm H} H|&amp;$\eta$  H $H$ \cr
\@|\theta \Theta {\mit\Theta} |&amp;$\theta$  $\Theta$ $\mit\Theta$ \cr
\@|\vartheta|&amp;$\vartheta$ \cr
\@|\iota {\rm I} I|&amp;$\iota$ I $I$ \cr
\@|\kappa {\rm K} K |&amp;$\kappa$ K $K$ \cr
\@|\lambda \Lambda {\mit\Lambda}  |&amp;$\lambda$  $\Lambda$ $\mit\Lambda$ \cr
\@|\mu {\rm M} M |&amp;$\mu$ M $M$ \cr
\@|\nu {\rm N} N|&amp;$\nu$ N $N$ \cr
\@|\xi \Xi {\mit\Xi} |&amp;$\xi$ $\Xi$ $\mit\Xi$  \cr
 |o {\rm O} O|&amp;$o$ O $O$ \cr 
\@|\pi \Pi {\mit\Pi}|&amp;$\pi$ $\Pi$ $\mit\Pi$ \cr
\@|\varpi|&amp;$\varpi$\cr
\@|\rho {\rm R} R|&amp;$\rho$ R $R$\cr
\@|\sigma \Sigma {\mit\Sigma}|&amp;$\sigma$ $\Sigma$ $\mit\Sigma$ \cr
\@|\tau {\rm T} T|&amp;$\tau$ T $T$ \cr
\@|\upsilon \Upsilon {\mit\Upsilon}|&amp;$\upsilon$ $\Upsilon$ $\mit\Upsilon$ \cr 
\@|\phi \Phi {\mit\Phi}|&amp;$\phi$ $\Phi$ $\mit\Phi$ \cr
\@|\varphi|&amp;$\varphi$ \cr
\@|\chi {\rm X} X|&amp;$\chi$ X $X$ \cr
\@|\psi \Psi {\mit\Psi}|&amp;$\psi$ $\Psi$ $\mit\Psi$\cr
\@|\omega \Omega {\mit\Omega}|&amp;$\omega$ $\Omega$ $\mit\Omega$\cr
\eshortcomlist

\shead{caligraphic}{Calligraphic Capitals}
These look like ${\cal A} \ldots{\cal Z}$ and are obtained in math mode by
the construction |{\cal &lt;capital letter&gt;}| while in math 
mode.

\shead{opsrels}{Ords, Operators, and Relations}
The following is a composite list of the symbols of
type ordinary, relation and operator supplied by
{\it plain}. The type is specified along with the symbol. The meaning of the
type abbreviations has been given in Section \ref{mathtypes}.
\medbreak
\beginmulticolumnformat
\hfuzz=20pt
\parindent=0pt
\def\bin{\leavevmode\hbox to 30pt{\it Bin\hss}}
\def\ord{\leavevmode\hbox to 30pt{\it Ord\hss}}
\def\rel{\leavevmode\hbox to 30pt{\it Rel\hss}}
\def\op{\leavevmode\hbox to 30pt{\it Op\hss}}
\intercolumnsep = {\hskip 3em}
\numberofcolumns = 2
\obeylines
\bin\hbox to 20pt{$+$\quad\hss}\@|+|
\bin\hbox to 20pt{$-$\quad\hss}\@|-|
\rel\hbox to 20pt{$=$\quad\hss}\@|=|
\rel\hbox to 20pt{$\not=$\quad\hss}\@|\not=| |\ne| |\neq|
\rel\hbox to 20pt{$&lt;$\quad\hss}\@|&lt;|
\rel\hbox to 20pt{$\not&lt;$\quad\hss}\@|\not&lt;|
\rel\hbox to 20pt{$&gt;$\quad\hss}\@|&gt;|
\rel\hbox to 20pt{$\not&gt;$\quad\hss}\@|\not&gt;|
\ord\hbox to 20pt{$\aleph$\quad\hss}\@|\aleph| 
\ord\hbox to 20pt{$\amalg$\quad\hss}\@|\amalg|
\ord\hbox to 20pt{$\angle$\quad\hss}\@|\angle|
\rel\hbox to 20pt{$\approx$\quad\hss}\@|\approx|
\rel\hbox to 20pt{$\not\approx$\quad\hss}\@|\not\approx|
\bin\hbox to 20pt{$\ast$\quad\hss}\@|\ast| or *
\rel\hbox to 20pt{$\asymp$\quad\hss}\@|\asymp|
\ord\hbox to 20pt{$\backslash$\quad\hss}\@|\backslash|
\op\hbox to 20pt{$\bigcap$\quad\hss}\@|\bigcap|
\bin\hbox to 20pt{$\bigcirc$\quad\hss}\@|\bigcirc|
\op\hbox to 20pt{$\bigcup$\quad\hss}\@|\bigcup|
\op\hbox to 20pt{$\bigodot$\quad\hss}\@|\bigodot|
\op\hbox to 20pt{$\bigoplus$\quad\hss}\@|\bigoplus|
\op\hbox to 20pt{$\bigotimes$\quad\hss}\@|\bigotimes|
\op\hbox to 20pt{$\bigsqcup$\quad\hss}\@|\bigsqcup|
\bin\hbox to 20pt{$\bigtriangledown$\quad\hss}\@|\bigtriangledown|
\bin\hbox to 20pt{$\bigtriangleup$\quad\hss}\@|\bigtriangleup|
\op\hbox to 20pt{$\biguplus$\quad\hss}\@|\biguplus|
\op\hbox to 20pt{$\bigvee$\quad\hss}\@|\bigvee|
\op\hbox to 20pt{$\bigwedge$\quad\hss}\@|\bigwedge|
\ord\hbox to 20pt{$\bot$\quad\hss}\@|\bot|
\rel\hbox to 20pt{$\bowtie$\quad\hss}\@|\bowtie|
\bin\hbox to 20pt{$\bullet$\quad\hss}\@|\bullet|
\bin\hbox to 20pt{$\cap$\quad\hss}\@|\cap|
\bin\hbox to 20pt{$\cdot$\quad\hss}\@|\cdot|
\bin\hbox to 20pt{$\circ$\quad\hss}\@|\circ|
\ord\hbox to 20pt{$\clubsuit$\quad\hss}\@|\clubsuit|
\rel\hbox to 20pt{$\cong$\quad\hss}\@|\cong|
\rel\hbox to 20pt{$\not\cong$\quad\hss}\@|\not\cong|
\op\hbox to 20pt{$\coprod$\quad\hss}\@|\coprod|
\bin\hbox to 20pt{$\cup$\quad\hss}\@|\cup|
\bin\hbox to 20pt{$\dagger$\quad\hss}\@|\dagger|
\rel\hbox to 20pt{$\dashv$\quad\hss}\@|\dashv|
\bin\hbox to 20pt{$\ddagger$\quad\hss}\@|\ddagger|
\ord\hbox to 20pt{$\diamondsuit$\quad\hss}\@|\diamondsuit|
\balancecolumnsize = 470pt
\bin\hbox to 20pt{$\diamond$\quad\hss}\@|\diamond|
\bin\hbox to 20pt{$\div$\quad\hss}\@|\div|
\rel\hbox to 20pt{$\doteq$\quad\hss}\@|\doteq|
\rel\hbox to 20pt{$\Downarrow$\quad\hss}\@|\Downarrow|
\rel\hbox to 20pt{$\downarrow$\quad\hss}\@|\downarrow| 
\ord\hbox to 20pt{$\ell$\quad\hss}\@|\ell|
\ord\hbox to 20pt{$\emptyset$\quad\hss}\@|\emptyset|
\rel\hbox to 20pt{$\equiv$\quad\hss}\@|\equiv|
\rel\hbox to 20pt{$\not\equiv$\quad\hss}\@|\not\equiv|
\ord\hbox to 20pt{$\exists$\quad\hss}\@|\exists|
\ord\hbox to 20pt{$\flat$\quad\hss}\@|\flat|
\ord\hbox to 20pt{$\forall$\quad\hss}\@|\forall|
\ord\hbox to 20pt{$\frown$\quad\hss}\@|\frown|
\rel\hbox to 20pt{$\geq$\quad\hss}\@|\ge| \@|\geq|
\rel\hbox to 20pt{$\not\geq$\quad\hss}\@|\not\geq|
\rel\hbox to 20pt{$\gg$\quad\hss}\@|\gg|
\rel\hbox to 20pt{$\not\gg$\quad\hss}\@|\not\gg|
\ord\hbox to 20pt{$\hbar$\quad\hss}\@|\hbar|
\ord\hbox to 20pt{$\heartsuit$\quad\hss}\@|\heartsuit|
\rel\hbox to 20pt{$\hookleftarrow$\quad\hss}\@|\hookleftarrow| 
\rel\hbox to 20pt{$\hookrightarrow$\quad\hss}\@|\hookrightarrow|
\ord\hbox to 20pt{$\Im$\quad\hss}\@|\Im|
\ord\hbox to 20pt{$\imath$\quad\hss}\@|\imath|
\ord\hbox to 20pt{$\infty$\quad\hss}\@|\infty|
\op\hbox to 20pt{$\int$\quad\hss}\@|\int|
\rel\hbox to 20pt{$\in$\quad\hss}\@|\in|
\rel\hbox to 20pt{$\not\in$\quad\hss}\@|\not\in|
\ord\hbox to 20pt{$\jmath$\quad\hss}\@|\jmath|
\rel\hbox to 20pt{$\Leftarrow$\quad\hss}\@|\Leftarrow|
\rel\hbox to 20pt{$\leftarrow$\quad\hss}\@|\leftarrow| 
\rel\hbox to 20pt{$\leftharpoondown$\quad\hss}\@|\leftharpoondown| 
\rel\hbox to 20pt{$\leftharpoonup$\quad\hss}\@|\leftharpoonup| 
\rel\hbox to 20pt{$\Leftrightarrow$\quad\hss}\@|\Leftrightarrow| 
\rel\hbox to 20pt{$\leftrightarrow$\quad\hss}\@|\leftrightarrow| 
\rel\hbox to 20pt{$\leq$\quad\hss}\@|\le| \@|\leq|
\rel\hbox to 20pt{$\not\leq$\quad\hss}\@|\not\leq|
\rel\hbox to 20pt{$\ll$\quad\hss}\@|\ll|
\rel\hbox to 20pt{$\not\ll$\quad\hss}\@|\not\ll|
\rel\hbox to 20pt{$\Longleftarrow$\quad\hss}\@|\Longleftarrow| 
\rel\hbox to 20pt{$\longleftarrow$\quad\hss}\@|\longleftarrow| 
\rel\hbox to 20pt{$\Longleftrightarrow$\quad\hss}\@|\Longleftrightarrow| 
\rel\hbox to 20pt{$\longleftrightarrow$\quad\hss}\@|\longleftrightarrow|  
\rel\hbox to 20pt{$\longmapsto$\quad\hss}\@|\longmapsto| 
\rel\hbox to 20pt{$\Longrightarrow$\quad\hss}\@|\Longrightarrow| 
\rel\hbox to 20pt{$\longrightarrow$\quad\hss}\@|\longrightarrow|
\rel\hbox to 20pt{$\mapsto$\quad\hss}\@|\mapsto|
\rel\hbox to 20pt{$\mid$\quad\hss}\@|\mid|
\rel\hbox to 20pt{$\models$\quad\hss}\@|\models|
\bin\hbox to 20pt{$\mp$\quad\hss}\@|\mp|
\ord\hbox to 20pt{$\nabla$\quad\hss}\@|\nabla|
\ord\hbox to 20pt{$\natural$\quad\hss}\@|\natural|
\rel\hbox to 20pt{$\nearrow$\quad\hss}\@|\nearrow| 
\rel\hbox to 20pt{$\nwarrow$\quad\hss}\@|\nwarrow|
\ord\hbox to 20pt{$\neg$\quad\hss}\@|\neg| \@|\lnot|
\rel\hbox to 20pt{$\ni$\quad\hss}\@|\ni| \@|\owns|
\bin\hbox to 20pt{$\odot$\quad\hss}\@|\odot|
\op\hbox to 20pt{$\oint$\quad\hss}\@|\oint|
\bin\hbox to 20pt{$\ominus$\quad\hss}\@|\ominus|
\bin\hbox to 20pt{$\oplus$\quad\hss}\@|\oplus|
\bin\hbox to 20pt{$\oslash$\quad\hss}\@|\oslash|
\bin\hbox to 20pt{$\otimes$\quad\hss}\@|\otimes|
\rel\hbox to 20pt{$\parallel$\quad\hss}\@|\parallel|
\ord\hbox to 20pt{$\partial$\quad\hss}\@|\partial|
\bin\hbox to 20pt{$\perp$\quad\hss}\@|\perp|
\bin\hbox to 20pt{$\pm$\quad\hss}\@|\pm|
\rel\hbox to 20pt{$\preceq$\quad\hss}\@|\preceq|
\rel\hbox to 20pt{$\not\preceq$\quad\hss}\@|\not\preceq|
\rel\hbox to 20pt{$\prec$\quad\hss}\@|\prec|
\rel\hbox to 20pt{$\not\prec$\quad\hss}\@|\not\prec|
\ord\hbox to 20pt{$\prime$\quad\hss}\@|\prime|
\op\hbox to 20pt{$\prod$\quad\hss}\@|\prod|
\rel\hbox to 20pt{$\propto$\quad\hss}\@|\propto|
\ord\hbox to 20pt{$\Re$\quad\hss}\@|\Re|
\rel\hbox to 20pt{$\Rightarrow$\quad\hss}\@|\Rightarrow| 
\rel\hbox to 20pt{$\rightarrow$\quad\hss}\@|\rightarrow| 
\rel\hbox to 20pt{$\rightharpoondown$\quad\hss}\@|\rightharpoondown|  
\rel\hbox to 20pt{$\rightharpoonup$\quad\hss}\@|\rightharpoonup|  
\rel\hbox to 20pt{$\rightleftharpoons$\quad\hss}\@|\rightleftharpoons|
\bin\hbox to 20pt{$\setminus$\quad\hss}\@|\setminus|
\ord\hbox to 20pt{$\sharp$\quad\hss}\@|\sharp|
\rel\hbox to 20pt{$\simeq$\quad\hss}\@|\simeq|
\rel\hbox to 20pt{$\not\simeq$\quad\hss}\@|\not\simeq|
\rel\hbox to 20pt{$\sim$\quad\hss}\@|\sim|
\rel\hbox to 20pt{$\not\sim$\quad\hss}\@|\not\sim|
\rel\hbox to 20pt{$\smile$\quad\hss}\@|\smile|
\ord\hbox to 20pt{$\spadesuit$\quad\hss}\@|\spadesuit|
\bin\hbox to 20pt{$\sqcap$\quad\hss}\@|\sqcap|
\bin\hbox to 20pt{$\sqcup$\quad\hss}\@|\sqcup|
\rel\hbox to 20pt{$\sqsubseteq$\quad\hss}\@|\sqsubseteq|
\rel\hbox to 20pt{$\not\sqsubseteq$\quad\hss}\@|\not\sqsubseteq|
\rel\hbox to 20pt{$\sqsupseteq$\quad\hss}\@|\sqsupseteq|
\rel\hbox to 20pt{$\not\sqsupseteq$\quad\hss}\@|\not\sqsupseteq|
\ord\hbox to 20pt{$\star$\quad\hss}\@|\star|
\rel\hbox to 20pt{$\subseteq$\quad\hss}\@|\subseteq|
\rel\hbox to 20pt{$\not\subseteq$\quad\hss}\@|\not\subseteq|
\rel\hbox to 20pt{$\subset$\quad\hss}\@|\subset|
\rel\hbox to 20pt{$\not\subset$\quad\hss}\@|\not\subset|
\rel\hbox to 20pt{$\succeq$\quad\hss}\@|\succeq|
\rel\hbox to 20pt{$\not\succeq$\quad\hss}\@|\not\succeq|
\rel\hbox to 20pt{$\succ$\quad\hss}\@|\succ|
\rel\hbox to 20pt{$\not\succ$\quad\hss}\@|\not\succ|
\op\hbox to 20pt{$\sum$\quad\hss}\@|\sum|
\rel\hbox to 20pt{$\supseteq$\quad\hss}\@|\supseteq|
\rel\hbox to 20pt{$\not\supseteq$\quad\hss}\@|\not\supseteq|
\rel\hbox to 20pt{$\supset$\quad\hss}\@|\supset|
\rel\hbox to 20pt{$\not\supset$\quad\hss}\@|\not\supset|
\ord\hbox to 20pt{$\surd$\quad\hss}\@|\surd|
\rel\hbox to 20pt{$\searrow$\quad\hss}\@|\searrow| 
\rel\hbox to 20pt{$\swarrow$\quad\hss}\@|\swarrow|
\bin\hbox to 20pt{$\times$\quad\hss}\@|\times|
\bin\hbox to 20pt{$\top$\quad\hss}\@|\top|
\bin\hbox to 20pt{$\triangleleft$\quad\hss}\@|\triangleleft|
\bin\hbox to 20pt{$\triangleright$\quad\hss}\@|\triangleright|
\ord\hbox to 20pt{$\triangle$\quad\hss}\@|\triangle|
\bin\hbox to 20pt{$\uplus$\quad\hss}\@|\uplus|
\rel\hbox to 20pt{$\Uparrow$\quad\hss}\@|\Uparrow| 
\rel\hbox to 20pt{$\uparrow$\quad\hss}\@|\uparrow| 
\rel\hbox to 20pt{$\Updownarrow$\quad\hss}\@|\Updownarrow| 
\rel\hbox to 20pt{$\updownarrow$\quad\hss}\@|\updownarrow|         
\ord\hbox to 20pt{$\Vert$\quad\hss}\@|\Vert| or |\|\vrt 
\ord\hbox to 20pt{$\vert$\quad\hss}\@|\vert| or \vrt
\rel\hbox to 20pt{$\vdash$\quad\hss}\@|\vdash|
\bin\hbox to 20pt{$\vee$\quad\hss}\@|\lor| \@|\vee|
\bin\hbox to 20pt{$\wedge$\quad\hss}\@|\land| \@|\wedge|
\ord\hbox to 20pt{$\wp$\quad\hss}\@|\wp|
\bin\hbox to 20pt{$\wr$\quad\hss}\@|\wr|
\endmulticolumnformat

\shead{delims}{Brackets or Delimiters}
\tex\ supplies a large variety of delimiters or brackets, some of different
sizes and others that grow to be the correct size. Here is a list of the
delimiters supplied by {\it plain}. 

\bshortcomlist
\@|\big\Arrowvert|&amp;center part of double up arrow -- $\big\Arrowvert$\cr
\@|\big\arrowvert|&amp;center part of up arrow --  $\big\arrowvert$\cr
\@|\langle|&amp;left angle bracket --  $\langle$\cr
\@|\rangle|&amp;right angle bracket --  $\rangle$\cr
\@|\big\bracevert|&amp;center part of large braces -- $\big\bracevert$ \cr
|\{| or \@|\lbrace|&amp;left brace --  $\{$\cr
|\}| or \@|\rbrace|&amp;right brace --  $\}$\cr
|[| or \@|\lbrack|&amp;left square bracket --  $[$\cr
|]| or \@|\rbrack|&amp;right square bracket --  $]$\cr
\@|\lceil|&amp;left ceiling --  $\lceil$\cr
\@|\rceil|&amp;right ceiling --  $\rceil$\cr
\@|\lfloor|&amp;left floor --  $\lfloor$\cr
\@|\rfloor|&amp;right floor --  $\rfloor$\cr
\@|\big\lgroup|&amp;large left paren without middle -- $\big\lgroup$\cr
\@|\big\rgroup|&amp;large right paren without middle -- $\big\rgroup$\cr
\@|\big\lmoustache|&amp;  -- $\big\lmoustache$\cr
\@|\big\rmoustache|&amp;  -- $\big\rmoustache$\cr
|/|&amp;slash --  $/$\cr
\@|\backslash|&amp;reverse slash --  $\backslash$\cr
\vrt\ or |\vert|&amp;vertical bar --  $\vert$\cr
|\|\vrt\ or |\Vert|&amp;double vertical bar --  $\Vert$\cr
|\downarrow|&amp;down arrow --  $\downarrow$\cr
|\Downarrow|&amp;double down arrow --  $\Downarrow$\cr
|\uparrow|&amp;up arrow --  $\uparrow$\cr
|\Uparrow|&amp;double up arrow --  $\Uparrow$\cr
|\updownarrow|&amp;up-and-down arrow --  $\updownarrow$\cr
|\Updownarrow|&amp;double up-and-down arrow --  $\Updownarrow$\cr
|(|&amp;left paren --  $($\cr
|)|&amp;right paren --  $)$\cr
\eshortcomlist

Two very powerful operators exist for the delimiters, namely |\left| and
|\right|. They must be used in pairs as 
\par
\hbox to \hsize{\hss\@|$\left&lt;delimiter&gt; ...\right&lt;delimiter&gt;$|,\hss} 
\noindent
where the |&lt;delimiter&gt;| is one of the above. The result
of these operators is a pair of braces that grow the correct size for the
particular formula in which they are used. This saves you from the problem of
measuring. A |\right.| implies there is nothing at the place of the right
delimiter and |\left.| similarly on the left. This allows a delimiter on only
one side of a formula but to still have the operators match in pairs.

Finally there is a second set of four pairs of operators, \@|\bigl|, |\Bigl|,
\@|\biggl|, \@|\Biggl| and the corresponding  right hand ones  \@|\bigr|, \@|\Bigr|,
\@|\biggr|, and \@|\Biggr|. These produce big, but fixed size delimiters for instance we have
\begintt
$$
 \bigl\{\quad 
\Bigl[\quad 
\biggl\lfloor\quad 
\Biggl\Updownarrow\quad\Biggr\Updownarrow
\quad\biggr\rfloor
\quad\Bigr]
\quad\bigr\}
$$
\endtt
produces
$$
 \bigl\{\quad 
\Bigl[\quad 
\biggl\lfloor\quad 
\Biggl\Updownarrow\quad\Biggr\Updownarrow
\quad\biggr\rfloor
\quad\Bigr]
\quad\bigr\}
$$

\shead{sincos}{Common Mathematical Functions}
Certain functions such as |sin|, |cos|, and |limsup| for example are shown in
mathematics text in Roman font. In addition some like |lim| sometimes take
limits that should be placed in the same way as the limits on the large
operators ({\it Op}). For example
\begintt
$$
\limsup_{t \leftrightarrow \infty} 
$$
\endtt
gives
$$
\limsup_{t \leftrightarrow \infty} 
$$
There are 32 of these already defined in {\it plain}. They are
\beginthreecolumn
\@|\arg|    
\@|\arccos| 
\@|\arcsin| 
\@|\arctan| 
\@|\cot|  
\@|\coth| 
\@|\cosh| 
\@|\cos|  
\@|\csc|
\@|\det|
\@|\deg|
\@|\dim|
\@|\exp|
\@|\gcd|
\@|\hom|
\@|\inf|
\@|\ker|   
\@|\lg|    
\@|\liminf|
\@|\limsup|
\@|\lim|   
\@|\ln|    
\@|\log|   
\@|\max|   
\@|\min| 
\@|\Pr|  
\@|\sec| 
\@|\sin| 
\@|\sinh|
\@|\sup|
\@|\tanh|
\@|\tan|
\endthreecolumn

\shead{buildown}{Build Your Own Symbols}
There are several advanced facilities in \tex\ for building your own symbols.
However, there is one rather simple form using 
\@|\buildrel &lt;superscript&gt; \over &lt;relation&gt;|. For example
|\buildrel \triangle  \over =| gives $\buildrel \triangle  \over =$.

\ejectpage
\done
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