\newcommand{\slideTitle}[2]{{\begin{center} #2 #1 \end{center}}  } 
\newcommand{\vt}[1]{$\backslash #1$} 
 
\begin{slide}{} 
\begin{center} 
\Huge More On \LaTeX\\  
By : Shy Artzi\\ 
\today 
\end{center} 
\end{slide} 
 
\begin{slide}{} 
{\Large In this lecture we will discuss the following subjects: } 
\Large  
\begin{itemize} 
    \item How to prepare a slide show 
    \item Math in \LaTeX 
\end{itemize} 
\end{slide} 
 
\begin{slide}{} 
\slideTitle{How To Prepare a Slide Show \\ Or \\ SLI\TeX}{\huge}   
\large 
\begin{enumerate} 
\item What Is SLI\TeX\ ? 
\item The Root File 
\item The Slide File 
\item Making Only Some Slides 
\end{enumerate} 
\end{slide} 
 
\begin{slide}{} 
\begin{center} 
{\Huge What Is SLI\TeX} 
\end{center} 
 
SLI\TeX\ is an extension of \LaTeX\ which provides us with a simple way of making black-and-white or color slides. 
This mechanism is needed because slides differ from the other supplied environments, such as article, book, letter, report,  
in the following points: 
\begin{itemize} 
\item Font Size 
\item Separate Pages 
\item Overlays 
\item Notes 
\item Centering  
\item Selective Printing 
\end{itemize}  
\end{slide} 
 
\begin{slide}{} 
\begin{center} 
\Huge The Root File 
\end{center} 
The input to SLI\TeX\ consists of a root file and a separate slide file. 
The root file begins with the usual  
\verb+\documentstyle{slides}+. The {\tt slides} document style is the standard for producing slides (there are no standard options). 
The \verb+\documentstyle{slides}+ is followed by the preamble which, as usual, contains only declarations, followed in turn by the \verb+\begin{document}+ command . 
\end{slide} 
\begin{slide}{} 
Any text after the \verb+\begin{document}+ is treated as ``front matter'' and not as slide material. The text for the slides is contained in a separate slides-file we will see later on. Black-and-white slides are generated by placing the command \verb+\blackandwhite{file_name}+ in the root file. 
As usual, the root file ends with the \verb+\end{document}+ command.  
 \end{slide}  
 
\begin{slide}{} 
{\large This is a demonstration of a slide root file:} 
\begin{verbatim} 
\documentstyle{slides} 
\begin{document} 
\My lecture about \LaTeX \\ 
\This page is a demonstrative ``front matter'' 
\blackandwhite{myslides} 
\end{document} 
\end{verbatim} 
In fact, this is the root file which was used to produce this lecture. 
\end{slide} 
 
\begin{slide}{} 
\begin{center} 
\Huge The Slide File 
\end{center} 
\begin{itemize} 
\item Slides 
\item Overlays 
\item Notes 
\item Page Styles for Slides 
\end{itemize} 
\end{slide} 
 
\begin{slide}{} 
\begin{center} 
\Huge Slides 
\end{center} 
Each slide is produced by the {\bf slide} environment in the following manner: 
\begin{verbatim} 
 
   \begin{slide}{}  
   ... 
   \end{slide} 
      
\end{verbatim}  
 
The text appearing on a slide is written with ordinary \LaTeX\ commands. 
Output generated by SLI\TeX\ differs from ordinary \LaTeX\ output in two ways: 
\begin{enumerate} 
\item Centering 
\item Font size  
\end{enumerate} 
\end{slide} 
 
\begin{slide}{} 
\begin{center} 
\Huge Overlays 
\end{center} 
The {\bf overlay} environment is for making a slide which is meant to be placed on top of another. It is the same as the {\bf slide} environment except for the way the pages are numbered. The first overlay following slide number 9 is numbered ``9-a'', the second is numbered ``9-b'' and so on. 
The slide and the overlay should be identical except that text visible in one is invisible in the other. 
\end{slide} 
\begin{slide}{} 
\begin{center} 
\large Invisible Text 
\end{center} 
We will use a new SLI\TeX\ command \verb+\invisible+. This command is a special color declaration for invisible text. Invisible text will not appear in the text, but space will be reserved for it so we can lay out overlays properly. 
   
\end{slide} 
 
\begin{slide}{} 
  This is the {\invisible Overlay} example page. 
  {\invisible Which contains a little surprise.} 
  {\invisible put something funny here} 
\end{slide} 
 
\begin{overlay}{} 
  {\invisible This is the}  
  Overlay  
  {\invisible example page. 
   Which contains a little surprise. 
   put something funny here} 
\end{overlay} 
 
\begin{overlay}{} 
  {\invisible This is the Overlay  example page.} 
  Which contains a little surprise. 
  {\invisible put something funny here} 
\end{overlay} 
 
 
 
\begin{overlay}{} 
  {\invisible This is the Overlay example page. 
  Which contains a little surprise.} 
   put something funny here 
\end{overlay} 
 
\begin{slide}{} 
\begin{center} 
\Large Overlay Example 
\end{center} 
This slide is an example for overlay text. 
The following SLI\TeX\ commands will generate the overlays: 
\small 
\begin{verbatim}  
\begin{slide}{} 
  This is the {\invisible Overlay} example page. 
  {\invisible Which contains a little surprise.} 
  {\invisible put something funny here} 
\end{slide} 
 
\begin{overlay}{} 
  {\invisible This is the} Overlay \invisible example page. 
  {\invisible Which contains a little surprise.} 
  {\invisible put something funny here} 
\end{overlay} 
 
\begin{overlay}{} 
  {\invisible This is the Overlay  example page.} 
  Which contains a little surprise. 
  {\invisible put something funny here} 
\end{overlay} 
 
\begin{overlay}{} 
  {\invisible This is the Overlay example page. 
   Which contains a little surprise.} 
   put something funny here 
\end{overlay} 
\end{verbatim} 
\end{slide}  
  
\begin{slide}{} 
\begin{center} 
\huge Notes 
\end{center} 
\begin{itemize} 
\item An environment for producing a one-page note. 
\item A note appears only in black-and-white version of the slides.  
\item Numbered like overlays (sequentially) but with numbers, i.e, a note after 15 will be numbered 15.1\. 
\item No slide's frame. 
\end{itemize}  
\end{slide} 
 
\begin{slide}{} 
\begin{center} 
\Large Note Example 
\end{center} 
The following code will produce the next page which is a note : 
\begin{verbatim} 
\begin{note} 
This is a note to myself, perhaps reminding 
 me to check and see if anybody is awake. 
\end{note} 
\end{verbatim} 
\end{slide} 
 
\begin{note} 
This is a note to myself, perhaps reminding me to check and see if anybody is awake. 
\end{note}  
 
\begin{slide}{} 
\begin{center} 
\large Page Styles for Slides 
\end{center} 
 
There are three different page styles for slides: 
\begin{description} 
\item [headings] - Numbering and alignment style (default). 
\item [plain] - Numbering and no alignments.               
\item [empty] - No numbering and no alignments. 
\end{description} 
\end{slide} 
 
\begin{slide}{} 
\begin{center} 
\Huge Making Only Some Slides 
\end{center} 
It is sometimes convenient to generate only some of the slides from the input file. 
There are several way to do this: 
 
\begin{itemize} 
\item Numbered slides: \verb+\onlyslides{4,8}+ 
\item Ranged slides: \verb+\onlyslides{4,8,9-12}+ 
\item Nonexistent slides: \verb+\onlyslides{10-9999}+ 
\item Only Notes: \verb+\onlynotes{6,7}+ 
\end{itemize} 
 
These commands must be placed in the root file. 
 
\end{slide} 
 
\begin{slide}{} 
\slideTitle{Summary}{\huge} 
SLI\TeX\ is a very powerful instrument allowing us to easily create slides with \LaTeX, adding many slides features such as: 
\begin{enumerate} 
\item Overlays 
\item Notes 
\item Automatic centered text 
\item Alignment marks \& slide numbering  
\end{enumerate} 
\end{slide} 
 
\begin{slide}{} 
\slideTitle{Math in \LaTeX}{\Huge} 
{\Large This part of the lecture will consist mainly of simple and complex mathematical formulas in \LaTeX. 
We will discuss several topics:}   
\begin{itemize} 
\item Some Common Structures 
\item Mathematical Symbols 
\item Putting One Thing Above Another 
\end{itemize}   
\end{slide} 
 
\begin{slide}{} 
\slideTitle{Some Common Structures}{\huge} 
The common structures we will see are the following: 
\begin{itemize} 
\item Subscripts and Superscripts 
\item Fractions 
\item Roots 
\item Ellipsis                  
\end{itemize} 
\end{slide}    
 
\begin{slide}{}  
\slideTitle{Subscripts and Superscripts}{\large} 
 
Subscripts and superscripts are made with the \_ and \^\ commands. These commands can be combined to make complicated subscripts and superscripts expressions.\\ 
\\ 
\begin{tabular}{cl|cl|cl}  
$x^{2y}$ & \verb+x^{2y}+ & $x^{y^{z^{t}}}$ & \verb+x^{y^{z^{t}}}+ & $x^{y}_{1}$& \verb+x^{y}_{1}+ \\ 
$x_{2y}$ & \verb+x_{2y}+ & $x^{y_{z_{t}}}$ & \verb+x^{y_{z_{t}}}+ & $x_{y}^{1}$& \verb+x_{y}^{1}+ \\ 
\end{tabular} 
\end{slide} 
 
\begin{slide}{} 
\slideTitle{Fractions}{\Large} 
Fractions are made with the / symbol or the \verb+\frac+ command. Frac is used for large fractions in displayed formulas. It has two arguments: the numerator and the denominator. 
\end{slide} 
 
\begin{slide}{} 
\slideTitle{Fractions Example}{\Large} 
\begin{itemize} 
\item {\large / example:}\\ 
Typing:  \verb+ Dividing 3 by 2 gives $3/2$+ \\ 
Will produce:  Dividing 3 by 2 gives $3/2$ \\ 
\item {\large {\it Frac} examples:} \\ 
\begin{tabular}{ll} \\ 
$x = \frac{y+z/2}{y^{2}+1}$ & \verb?\[ x = \frac{y+z/2}{y^{2}+1} \]? \\ \\ 
$ \frac{x+y}{1 + \frac{y}{z+1}} $ & \verb?\[ \frac{x+y}{1+\frac{y}{z+1} \]? 
\end{tabular} 
\end{itemize} 
\end{slide} 
 
\begin{slide}{} 
\slideTitle{Roots}{\Large} 
 
The \verb+\sqrt+ command produces the square root of its argument. 
There is an optional first argument for the other roots.\\ 
\\ For example:  \verb? 
A square root \( \sqrt{x+y} \) 
and an {\it n}th root  \(\sqrt[n]{2} \).? \\ 
 
Will produce:   
A square root \( \sqrt{x+y} \) and an {\it n}th root  \(\sqrt[n]{2} \). 
\end{slide} 
 
\begin{slide}{} 
\slideTitle{Ellipses}{\Large} 
 
The command \verb+\ldots+ and \verb+\cdots+ produce two different kinds of ellipses(\ldots). 
 
\begin{itemize} 
\item \verb+\cdots+ -  Centered Ellipses:\\ 
      \begin{tabular}{ll} 
      $a + \cdots + z$ & \verb?$a + \cdots + z$? 
      \end{tabular} 
\item \verb+\ldots+ -  Low Ellipses:\\ 
      \begin{tabular}{ll} 
      $x_{1}, \ldots ,x_{n}$ & \verb?$x_{1}, \ldots  ,x{n}$? 
      \end{tabular}       
\end{itemize} 
 
There are also vertical and diagonal ellipses (for arrays) :\\ 
\begin{tabular}{clcl} 
$\vdots$ & \verb+$\vdots$+ & $\ddots$ & \verb+$\ddots$+  
\end{tabular} 
 
\end{slide} 
 
\begin{slide}{} 
\slideTitle{Mathematical Symbols}{\huge} 
\begin{itemize} 
\item \TeX\ commands allow us to produce almost any mathematical symbol.\\ 
\item Most of these commands can be used only in math mode. 
\item We will see several topics: 
      \begin{enumerate} 
      \item Greek Letters 
      \item Calligraphic Letters 
      \item A Menagerie of Mathematical Symbols 
      \item Log-like Functions       
      \end{enumerate} 
\end{itemize} 
   
\end{slide} 
 
\begin{slide}{} 
\slideTitle{Greek Letters}{\Large} 
\begin{itemize} 
\item Lowercase Greek letters by adding a \verb+\+ to the letter name. 
\item Uppercase Greek letters by capitalizing the first letter of the command name. 
\item Variant forms of several Greek letters. 
\end{itemize} 
{\bf Example:}\\ 
\begin{verbatim} 
 Making lowercase $\phi$ is easy as 
 making uppercase $\Phi$ and also easy  
 as making another form of $\varphi$. 
\end{verbatim}  
   
 Making lowercase $\phi$ is easy as making uppercase $\Phi$ and also easy as making another form of $\varphi$. 
\end{slide} 
 
\begin{slide}{} 
%\drawItem{\alpha} 
\end{slide} 
 
\begin{slide}{} 
\slideTitle{Calligraphic Letters}{\Large} 
\TeX\ provides twenty-six uppercase calligraphic letters, which can be produce by the \verb+\cal+ declaration.\\ 
{\bf Example:} $\cal A$,$\cal B$,$\cal C$,$\cal D$,\ldots,$\cal Z$.  
\end{slide} 
 
\begin{slide}{} 
\slideTitle{A Menagerie of Mathematical Symbols}{\Large} 
\TeX\ can make lots of special mathematical symbols. 
In the following slides we will see some examples. Additional examples can be found in ``\LaTeX\ --- User's Guide and Reference Manual'' by Leslie Lamport.  
\end{slide} 
 
\begin{slide}{} 
\  
\tiny 
\begin{tabular}{||cl|cl|cl|cl||} \hline 
$\pm$ &  \verb+\pm+   & $\cap$ & \verb+\cap+ & $\diamond$ & \verb+\diamond+  & $\oplus$ & \verb+\oplus+ \\ 
$\mp$ & \verb+\mp+ & $\cup$ & \verb+\cup+ & $\bigtriangleup$  & \verb+\bigtriangleup+& $\ominus$ & \verb+\ominus+ \\ 
$\times$ & \verb+\times+ & $\uplus$ & \verb+\uplus+ & $\bigtriangledown$  & \verb+\bigtriangledown+& $\otimes$ & \verb+\otimes+ \\ 
$\div$ & \verb+\div+ & $\sqcap$ & \verb+\sqcap+ & $\triangleleft$  & \verb+\triangleleft+& $\oslash$ & \verb+\oslash+ \\ 
$\ast$ & \verb+\ast+ & $\sqcup$ & \verb+\sqcup+ & $\triangleright$  & \verb+\trianglerght+& $\odot$ & \verb+\odot+ \\ 
$\star$ & \verb+\star+ & $\vee$ & \verb+\vee+ & $\lhd$  & \verb+\lhd+& $\bigcirc$ & \verb+\bigcirc+ \\ 
$\circ $ & \verb+\circ+ & $\wedge$ & \verb+\wedge+ & $\rhd$  & \verb+\rhd+& $\dagger$ & \verb+\dagger+ \\ 
$\bullet$ & \verb+\bullet+ & $\setminus$ & \verb+\setminus+ & $\unlhd$  & \verb+\unlhd+& $\ddagger  $ & \verb+\ddagger+ \\ 
$\cdot$ & \verb+\cdot+ & $\wr$ & \verb+\wr+ & $\unrhd$  & \verb+\unrhd+& $\amalg$ & \verb+\amalg+ \\ \hline 
 \end{tabular} 
 
\end{slide} 
 
\begin{slide}{}  
\tiny  
\begin{tabular}{||cl|cl|cl|cl||} \hline 
$\aleph$ &  \verb+\aleph+   & $\prime$ & \verb+\prime+ & $\forall$ & \verb+\forall+  & $\infty$ & \verb+\infty+ \\ 
$\hbar$ & \verb+\hbar+ & $\emptyset$ & \verb+\emptyset+ & $\exists$  & \verb+\exists+& $\Box$ & \verb+\Box+ \\ 
$\imath$ & \verb+\imath+ & $\nabla$ & \verb+\nabla+ & $\neg$  & \verb+\neg+& $\Diamond$ & \verb+\Diamond+ \\ 
$\jmath$ & \verb+\jmath+ & $\surd$ & \verb+\surd+ & $\flat$  & \verb+\flat+& $\triangle$ & \verb+\triangle+ \\ 
$\ell$ & \verb+\ell+ & $\top$ & \verb+\top+ & $\natural$  & \verb+\natural+& $\clubsuit$ & \verb+\clubsuit+ \\ 
$\wp$ & \verb+\wp+ & $\bot$ & \verb+\bot+ & $\sharp$  & \verb+\sharp+& $\diamondsuit$ & \verb+\diamondsuit+ \\ 
$\Re $ & \verb+\Re+ & $\|$ & \verb+\|+ & $\backslash$  & \verb+\backslash+& $\heartsuit$ & \verb+\heartsuit+ \\ 
$\Im$ & \verb+\Im+ & $\angle$ & \verb+\angle+ & $\partial$  & \verb+\partial+& $\spadesuit  $ & \verb+\spadesuit+ \\ 
$\mho$ & \verb+\mho+ & $\sum$ & \verb+\sum+ & $\prod$ & \verb+\prod+ & $\coprod$ & \verb+\coprod+ \\ \hline 
 \end{tabular} 
 
\end{slide} 
 
\begin{slide}{} 
\tiny 
\tt  
\begin{tabular}{||cl|cl|cl||} \hline 
$\leftarrow$ &  \verb+\leftarrow+   & $\longleftarrow$ & \verb+\longleftarrow+ & $\uparrow$ & \verb+\uparrow+ \\ 
$\Leftarrow$ &  \verb+\Leftarrow+   & $\Longleftarrow$ & \verb+\Longleftarrow+ & $\Uparrow$ & \verb+\Uparrow+ \\ 
$\rightarrow$ &  \verb+\rightarrow+   & $\longrightarrow$ & \verb+\longrightarrow+ & $\downarrow$ & \verb+\downarrow+ \\ 
$\Rightarrow$ &  \verb+\Rightarrow+   & $\Longrightarrow$ & \verb+\Longrightarrow+ & $\Downarrow$ & \verb+\Downarrow+ \\ 
$\leftrightarrow$ &  \verb+\leftrightarrow+   & $\longleftrightarrow$ & \verb+\longleftrightarrow+ & $\updownarrow$ & \verb+\updownarrow+ \\ 
$\Leftrightarrow$ &  \verb+\Leftrightarrow+   & $\Longleftrightarrow$ & \verb+\Longleftrightarrow+ & $\Updownarrow$ & \verb+\Updownarrow+ \\ 
$\mapsto$ &  \verb+\mapsto+   & $\hookrightarrow$ & \verb+\hookrightarrow+ & $\nearrow$ & \verb+\nearrow+ \\ 
$\leftharpoonup$ &  \verb+\leftharpoonup+   & $\rightharpoonup$ & \verb+\rightharpoonup+ & $\searrow$ & \verb+\searrow+ \\ 
$\leftharpoondown$ &  \verb+\leftharpoondown+   & $\rightharpoondown$ & \verb+\rightharpoondown+ & $\swarrow$ & \verb+\swarrow+ \\ 
$\hookleftarrow$ &  \verb+\hookleftarrow+   & $\hookrightarrow$ & \verb+\hookrightarrow+ & $\nwarrow$ & \verb+\nwarrow+ \\ 
$\rightleftharpoons$ & \verb+\rightleftharpoons+ & $\leadsto$ & \verb+\leadsto+  & & \\ \hline 
 \end{tabular} 
 
\end{slide} 
 
\begin{slide}{} 
\slideTitle{Log-like Functions }{\Large}        
\begin{itemize} 
\item Logarithm function - \verb+\log+ command.\\ 
      Example:\\ 
       \verb?\( \log xy = \log x + \log y \)? \\ 
             \( \log xy = \log x + \log y \)  
\item Other commands in the next slides. 
\item \verb+\bmod+ - binary modulo relation.\\ 
      Example: \\ 
       \verb+\( \gcd(m,n) = a \bmod b \)+\\ 
       \( \gcd(m,n) = a \bmod b \) 
\item \verb+\pmod+ - parenthesized modulo expression.\\ 
      Example: \\ 
      \verb?\( x \equiv y \pmod{a+b} \)?  \\  
      \( x \equiv y \pmod{a+b} \)         
\end{itemize} 
\end{slide} 
 
\begin{slide}{} 
\slideTitle{Log-like Functions table}{\Large} 
\begin{center} 
\begin{large} 
\begin{tabular}{||l|l|l|l||} \hline 
\vt{arccos} & \vt{cos}  & \vt{csc} & \vt{exp} \\ 
\vt{arcsin} & \vt{cosh} & \vt{deg} & \vt{gcd} \\ 
\vt{arctan} & \vt{cot}  & \vt{det} & \vt{hom} \\ 
\vt{arg}    & \vt{coth} & \vt{dim} & \vt{inf} \\ \hline 
\end{tabular} 
\end{large} 
\end{center}  
\end{slide}     
 
\begin{slide}{} 
\slideTitle{Log-like Functions table (cont'd)}{\Large} 
\begin{center} 
\begin{large} 
\begin{tabular}{||l|l|l|l||} \hline 
\vt{ker}    & \vt{lg}  & \vt{min} & \vt{sinh} \\ 
\vt{lim}    & \vt{ln}  & \vt{Pr}  & \vt{sup} \\ 
\vt{limsup} & \vt{log} & \vt{sec} & \vt{tan} \\ 
\vt{liminf} & \vt{max} & \vt{sin} & \vt{tanh} \\ \hline 
\end{tabular} 
\end{large} 
\end{center}  
\end{slide}     
 
\begin{slide}{} 
\slideTitle{Putting One Thing Above Another}{\huge} 
There are several cases when we need to place one object above another: 
\begin{enumerate} 
\item Accents. 
\item Overlining and Underlining. 
\item Stacking Symbols.              
\end{enumerate}  
\end{slide} 
 
\begin{slide}{} 
\slideTitle{Accents}{\Large} 
There are special commands for putting accents above a letter. 
They can be produced by the commands in the following table. 
\begin{center} 
\begin{large} 
\begin{tabular}{||cl|cl||} \hline 
$\hat{a}$ & \verb+\hat{a}+ & $\acute{a}$ & \verb+\acute{a}+ \\ 
$\check{a}$ & \verb+\check{a}+ & $\grave{a}$ & \verb+\grave{a}+ \\  
$\breve{a}$ & \verb+\breve{a}+ & $\tilde{a}$ & \verb+\tilde{a}+ \\ 
$\bar{a}$ & \verb+\bar{a}+ & $\vec{a}$ & \verb+\vec{a}+ \\ 
$\dot{a}$ & \verb+\dot{a}+ & $\ddot{a}$ & \verb+\ddot{a}+ \\ \hline 
\end{tabular} 
\end{large} 
\end{center} 
 
\end{slide} 
 
\begin{slide}{} 
\slideTitle{Accents (cont'd) } {\Large} 
\begin{itemize} 
\item Wide version of \verb+\hat+ and \verb+\tilde+ - \verb+\widehat+ and \verb+\widetilde+. \\ 
      Example:  
      \begin{description} 
      \item [typing] \verb?\(\widehat{1-x} = \widetilde{-d}\)? 
      \item [results]\( \widehat{1-x} = \widetilde{-d} \) 
      \end{description} 
\item The letter i and j should lose the dots when accented. They are produced by the \verb+\imath+ and \verb+\jmath+ commands. \\ 
      Example: 
      \begin{description} 
      \item [typing] \verb?\( \vec{\imath} + \tilde{\jmath} \)? 
      \item [results] \( \vec{\imath} + \tilde{\jmath} \) 
      \end{description}     
        
       
\end{itemize} 
\end{slide} 
 
\begin{slide}{} 
\slideTitle{Overlining and Underlining}{\Large} 
\begin{center} 
\begin{itemize} 
\item Horizontal line above an argument - \verb+\overline+ \\ 
      Example: \\ 
      \begin{description} 
      \item[typing] \verb?\( \overline{\overline{y}^{5} + 9} \)? \\ 
      \item[result] \( \overline{\overline{y}^{5} + 9} \) 
      \end{description} 
\item Horizontal line under an argument - \verb+\underline+ \\ 
      Example: \\ 
      \begin{description} 
      \item[typing] \verb?\underline{Important} stuff is underlined? \\ 
      \item[result] \underline{Important} stuff is underlined 
      \end{description}       
\end{itemize} 
\end{center}  
\end{slide} 
 
\begin{slide}{} 
\slideTitle{Overlining and Underlining - (cont'd)}{\Large} 
\begin{center} 
\begin{itemize} 
\item Horizontal braces above an argument - \verb+\overbrace+ \\ 
      Example: \\ 
      \begin{description} 
      \item[typing] \verb?\( \overbrace{a + b + c + d} \)? \\ 
      \item[result] \( \overbrace{a + b + c + d} \) 
      \end{description} 
\item Horizontal braces under an argument - \verb+\underbrace+ \\ 
      Example: \\ 
      \begin{description} 
      \item[typing]  
\begin{verbatim} 
\[ \underbrace{a + \overbrace{b + 
\cdots + y}^{24} + z }_{26} \] 
\end{verbatim} 
      \item[result] \[ \underbrace{a + \overbrace{b + \cdots + y}^{24} + z }_{26} \] 
      \end{description}       
\end{itemize} 
\end{center}  
\end{slide} 
 
\begin{slide}{} 
\slideTitle{Stacking Symbols}{\Large}              
The \verb+\stackrel+ command stacks one symbol above another. \\ 
Example: 
\begin{description} 
\item[typing] \verb?\( A \stackrel{a'}{\rightarrow} B \)? 
\item[results] \( A \stackrel{a'}{\rightarrow} B \) 
\item[typing] 
\begin{verbatim} 
\( \vec{x} \stackrel{def}{=}(x_{1}, 
\ldots,x_{n})\) 
\end{verbatim} 
\item[results] \( \vec{x} \stackrel{def}{=} (x_{1}, \ldots,x_{n}) \) 
\end{description} 
\end{slide} 
 
\begin{slide}{} 
\slideTitle{Summary}{\huge} 
\TeX\ is a very powerful instrument allowing us to make almost any mathematical symbol and formula.  
\end{slide} 
 
\begin{slide}{} 
\slideTitle{THE END}{\Huge} 
\end{slide}