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%TCIDATA{<META NAME="Title" CONTENT="Functions and Relations">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:08:58 1997}
%TCIDATA{Language=American English}
%TCIDATA{CSTFile=MathHelp.cst}
%TCIDATA{<LINK REL="author" HREF="DM0-1.tex">}
%TCIDATA{<LINK REL="first" HREF="DM2-1.TEX">}
%TCIDATA{<LINK REL="contents" HREF="DM0-0.tex">}
%TCIDATA{<LINK REL="copyright" HREF="DM0-2.tex">}
%TCIDATA{<LINK REL="end" HREF="DM2-7.tex">}
%TCIDATA{<LINK REL="index" HREF="DMIndex.tex">}
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\begin{document}


\section{Functions and Relations\label{Absolute value}}

Numbers to be computed with should be entered in mathematics mode and appear
red on your screen. If that is not the case, select the number and click 
\FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{text.wmf}{\special{language
"Scientific Word";type "GRAPHIC";display "PICT";valid-file "F";width
0.3009in;height 0.3009in;depth 0.0701in;original-width
16.0625pt;original-height 16.0625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Text.wmf';file-properties "XNPEU";}} to change
the numbers to mathematics.

\subsection{\ Absolute Value}

The \emph{absolute value } 
\index{Absolute value@Absolute value}of a number $z$ is the distance of $z$
from zero. The absolute value of $z$ is denoted $\left| z\right| $.

\begin{quote}
$\blacktriangleright $ To put vertical bars around an expression
\end{quote}

\begin{enumerate}
\item[1.]  Select the expression.

\item[2.]  Click 
\FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{fences.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.3009in;height 0.3009in;depth 0.0701in;original-width
24.0625pt;original-height 22.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'FENCES.WMF';file-properties "XNPEU";}}.

\item[3.]  Choose the vertical bar.

\item[4.]  Choose \textsf{OK}.
\end{enumerate}

\begin{quote}
$\blacktriangleright $ To find an absolute value
\end{quote}

\begin{enumerate}
\item[1.]  Place the insertion point in an expression enclosed between
vertical bars.

\item[2.]  Choose \textsf{Evaluate}.
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left| -7\right| =\,7\vspace{4pt}$

$\vspace{4pt}\left| -11.3\right| =\,11.3$

$\left| 43\right| =\,43\vspace{4pt}$

$\vspace{4pt}\left| 21-13\right| =\allowbreak 8$
\end{quotation}

\strut

\subsection{\ Maximum and Minimum \label{Maximum}}

The functions $\max $ and $\min $ find the largest and smallest numbers in a
list of numbers separated by commas and enclosed in parentheses. These
functions can be chosen from the list provided by \FRAME{itbpF}{0.2992in}{%
0.2793in}{0.0692in}{}{}{function.wmf}{\special{language "Scientific
Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid-file
"F";width 0.2992in;height 0.2793in;depth 0.0692in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop
"1.020382";cropright "0.9851";cropbottom "0.033782";filename
'Function.wmf';file-properties "XNPEU";}}or \textsf{Insert + Math Name}, or
they can be entered from the keyboard while in mathematics mode. You can
also find the largest or smallest of several numbers by inserting the binary
operations join $\vee $ or meet $\wedge $ between the numbers. You will find
these symbols on the \textsf{Binary Operations} panel under \FRAME{itbpF}{%
0.3425in}{0.2992in}{0.0692in}{}{}{Figure }{\special{language "Scientific
Word";type "GRAPHIC";display "PICT";valid-file "F";width 0.3425in;height
0.2992in;depth 0.0692in;original-width 24.8125pt;original-height
23.3125pt;cropleft "0";croptop "0.983641";cropright "1.0256";cropbottom
"0.031341";filename 'binop.wmf';file-properties "XNPEU";}}. Leave the
insertion point in the expression and click \FRAME{itbpF}{0.3001in}{0.3001in%
}{0.0701in}{}{}{evaluate.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid-file "F";width
0.3001in;height 0.3001in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Evaluate.wmf';file-properties "XNPEU";}}or
choose \textsf{Evaluate} to get the following.\ 

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\max (27,\sqrt{236},\frac{65}{2},-14)=\,\frac{65}{2}$\medskip

$\min (27,\sqrt{236},\frac{65}{2},-14)=\,-14$\medskip

$\max \left( 27,\min \left( \sqrt{236},\max \left( \frac{65}{2},-14\right)
\right) \right) =27$\medskip

$27\vee \sqrt{236}\vee \frac{65}{2}\vee -14=\allowbreak \frac{65}{2}$\medskip

$27\wedge \sqrt{236}\wedge \frac{65}{2}\wedge -14=\allowbreak -14$\medskip

$27\vee \left( \sqrt{236}\wedge \left( \frac{65}{2}\vee -14\right) \right) =$
$27$
\end{quotation}

\strut

You can often determine the maximum or minimum by inspection after applying 
\textsf{Evaluate Numerically} to each of the numbers.

\strut

\subsection{\ Greatest and 
\index{Floor function@Floor function}\label{Floor} Smallest Integer Functions
}

You can find the greatest integer 
\index{Greatest integer@Greatest integer} less than or equal to a number by
using the floor function, denoted $\left\lfloor z\right\rfloor $ .

\begin{quote}
$\blacktriangleright $ To put floor brackets around an expression
\end{quote}

\begin{enumerate}
\item[1.]  Select the expression.

\item[2.]  Click 
\FRAME{itbpF}{0.2992in}{0.2992in}{0.0692in}{}{}{fences.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid-file "F";width 0.2992in;height 0.2992in;depth
0.0692in;original-width 24.0625pt;original-height 22.5625pt;cropleft
"0";croptop "1.0179";cropright "1.0179";cropbottom "0";filename
'FENCES.WMF';file-properties "XNPEU";}}.

\item[3.]  Click the left bracket \FRAME{itbpF}{0.1747in}{0.3157in}{0.0701in%
}{}{}{leftflor.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid-file "F";width
0.1747in;height 0.3157in;depth 0.0701in;original-width
13.5625pt;original-height 26.375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'leftflor.wmf';file-properties "XNPEU";}} .

\item[4.]  Choose \textsf{OK}.
\end{enumerate}

\begin{quote}
$\blacktriangleright $ To find a greatest integer value
\end{quote}

\begin{enumerate}
\item[1.]  Place the insertion point in an expression enclosed between floor
brackets.

\item[2.]  Choose \textsf{Evaluate}.
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left\lfloor 5.6\right\rfloor =5\medskip $

$\left\lfloor -11.3\right\rfloor =-12\medskip $

$\left\lfloor \frac{43}{5}\right\rfloor =8\medskip $

$\left\lfloor \pi +e\right\rfloor =5$
\end{quotation}

\strut

You can find the smallest integer 
\index{Smallest integer@Smallest integer} 
\index{Ceiling function@Ceiling function}greater than or equal to a number
by using the 
\index{Ceiling function}\label{Ceiling}ceiling function, denoted $%
\left\lceil z\right\rceil $.\ 

\begin{quote}
$\blacktriangleright $ To put ceiling brackets around an expression
\end{quote}

\begin{enumerate}
\item[1.]  Select the expression.

\item[2.]  Click 
\FRAME{itbpF}{0.2992in}{0.2992in}{0.0692in}{}{}{fences.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid-file "F";width 0.2992in;height 0.2992in;depth
0.0692in;original-width 24.0625pt;original-height 22.5625pt;cropleft
"0";croptop "1.0179";cropright "1.0179";cropbottom "0";filename
'FENCES.WMF';file-properties "XNPEU";}}.

\item[3.]  Click the left ceiling bracket \FRAME{itbpF}{0.1686in}{0.3131in}{%
0.0701in}{}{}{Figure }{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid-file "F";width
0.1686in;height 0.3131in;depth 0.0701in;original-width
13.5625pt;original-height 27.125pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'leftceil.wmf';file-properties "XNPEU";}}.

\item[4.]  Choose \textsf{OK}.
\end{enumerate}

\begin{quote}
$\blacktriangleright $ To find a smallest greater integer value
\end{quote}

\begin{enumerate}
\item[1.]  Place the insertion point in an expression enclosed between
ceiling brackets.

\item[2.]  Choose \textsf{Evaluate}.
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left\lceil 5.6\right\rceil =6\medskip $

$\left\lceil -11.3\right\rceil =-11\medskip $

$\left\lceil \frac{43}{5}\right\rceil \,=9\medskip $

$\left\lceil \pi +e\right\rceil =6\medskip $
\end{quotation}

The floor and ceiling brackets are also available in the panel under \FRAME{%
itbpF}{0.2992in}{0.2992in}{0.0692in}{}{}{Figure }{\special{language
"Scientific Word";type "GRAPHIC";display "PICT";valid_file "F";width
0.2992in;height 0.2992in;depth 0.0692in;original-width
21.8125pt;original-height 27.875pt;cropleft "-0.0107";croptop
"0.9468";cropright "0.9413";cropbottom "0.0931";filename
'ceiling.wmf';file-properties "XNPEU";}} \textsf{Special Delimiters, }%
although these are not expanding brackets.

\strut

\subsection{\ Checking \label{Check equality}Equality and Inequality}

There are 
\index{Check equality@Check equality}three possible responses to \textsf{%
Check Equality}: true, false, and undecidable 
\index{Undecidable@Undecidable}\label{Undecidable}. The latter means that
the test is inconclusive and the equality may be either true or false. Maple
and Mathematica use probabilistic methods to check equality, and there is a
very small probability that an equation judged as true is actually false.
Some expressions cannot be compared by this method---hence the inconclusive
response.

\begin{quote}
$\blacktriangleright $ To check an equality
\end{quote}

\begin{enumerate}
\item[1.]  Leave the insertion point in the equation.

\item[2.]  Choose \textsf{Check Equality.}\ 
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Check Equality}
\end{quote}

\begin{quotation}
$e^{i\pi }=\,-1$ is true

$\pi =\,3.14$ is false

$\arcsin \sin t=t$ is undecidable
\end{quotation}

\ 

You can also use \textsf{Check Equality }to check an \emph{inequality}
between two numbers. Set the difference of the two numbers equal to the
absolute value of the difference, place the insertion point in the equation,
and choose \textsf{Check Equality}.

\begin{quote}
$\blacktriangleright $ \textsf{Check Equality}
\end{quote}

\begin{quotation}
$%
\frac{9}{8}-\frac{8}{9}=\left| \frac{9}{8}-\frac{8}{9}\right| $ is true

$\pi ^{e}-e^{\pi }=\left| \pi ^{e}-e^{\pi }\right| $ is false
\end{quotation}

These results verify that $\frac 98-\frac 89\geq 0$, or $\frac 98\geq \frac
89$; and that $\pi ^e-e^\pi <0$, or $\pi ^e<e^\pi $.

In most cases, you can recognize an inequality by inspection after applying 
\textsf{Evaluate Numerically} to each of the numbers.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate Numerically}
\end{quote}

\begin{quotation}
$\frac{9}{8}=\,1.125$\qquad $\qquad \qquad $

$\frac{8}{9}=\,.8888888889$

$\pi ^{e}=\,22.45915771$\qquad

$\;e^{\pi }=\,23.14069264$
\end{quotation}

\strut

\subsection{\ Union and Intersection 
\index{Union@Union}\label{Union}}

You can find the union of two or more finite sets with \textsf{Evaluate}, by
using the symbol $\cup $ between the sets.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left\{ 1,2,3\right\} \cup \left\{ a,b,c\right\} =%
\vspace{6pt}\left\{ 1,2,3,a,b,c\right\} $

$\left\{ 1,2,3\right\} \cup \left\{ 3,5\right\} \cup \left\{ 7\right\} 
\vspace{6pt}=\left\{ 1,2,3,5,7\right\} $

$\left\{ \sqrt{2},\pi ,3.9,r\right\} \cup \left\{ a,b,c\right\} =\allowbreak
\left\{ \pi ,r,a,b,c,3.\,9,\sqrt{2}\right\} $
\end{quotation}

\strut

You can find the intersection 
\index{Intersection@Intersection}\label{Intersection}of two or more finite
sets with \textsf{Evaluate}, using the symbol $\cap $ between the sets.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left\{ 1,2,3\right\} \cap \left\{ 2,4,6\right\} =\left\{ 2\right\} 
\vspace{6pt}$

$\left\{ a,b,c,d\right\} \cap \left\{ d,e,f\right\} =\left\{ d\right\} 
\vspace{6pt}$

$\left\{ 1,2,3\right\} \cap \left\{ a,b,c\right\} =\allowbreak \emptyset 
\vspace{6pt}$

$\left\{ 1,2,3\right\} \cap \left\{ {}\right\} =\allowbreak \emptyset $
\end{quotation}

\strut If two sets have no elements in common their intersection is the 
\label{Empty set}\textit{empty set}, denoted by empty brackets $\left\{
{}\right\} $ or the symbol $\emptyset $. To enter symbol for the empty set,
click \FRAME{itbpF}{0.3105in}{0.2888in}{0.0701in}{}{}{miscellaneous.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 0.3105in;height 0.2888in;depth
0.0701in;original-width 20.3125pt;original-height 18.8125pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";filename
'miscellaneous.wmf';file-properties "XNPEU";}} and select it from the 
\textsf{Miscellaneous Symbols} panel that appears.

You can evaluate combinations of union and intersection.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left\{ 1,2,3,c\right\} \cap \left( \left\{ 2,4,6\right\} \cup \left\{
a,b,c\right\} \right) =\allowbreak \left\{ 2,c\right\} \vspace{6pt}$

$\left( \left\{ 1,2,3,c\right\} \cap \left\{ 2,4,6\right\} \right) \cup
\left( \left\{ 1,2,3,c\right\} \cap \left\{ a,b,c\right\} \right)
=\allowbreak \left\{ 2,c\right\} $
\end{quotation}

\strut

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Defining Functions of One Variable}{}{}{DM5-2.tex#Define,
function (2)}

\item  \hyperref{Plotting Special Functions}{}{}{DM6-4.tex#Special Functions}

\item  \hyperref{Valid Names for Functions and Expressions}{}{}{%
DM5-1.tex#Function name}
\end{itemize}

\rule{0.67in}{0.01in}

\hyperref{\FRAME{itbpF}{38.875pt}{24.5pt}{6pt}{}{}{index.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 38.875pt;height 24.5pt;depth
6pt;original-width 36.875pt;original-height 22.5625pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'index.wmf';file-properties
"XNPEU";}}}{}{}{DMIndex.tex#Math Index}\quad {\small Index entries: 
\hyperref{Absolute value}{}{}{DMIndex.tex#Absolute}, \hyperref{Ceiling}{}{}{%
DMIndex.tex#Ceiling function}, \hyperref{Check equality}{}{}{%
DMIndex.tex#Check equality}, \hyperref{Floor}{}{}{DMIndex.tex#Floor function}%
, \hyperref{Intersection}{}{}{DMIndex.tex#Intersection}, \hyperref{Maximum}{%
}{}{DMIndex.tex#Maximum}, \hyperref{Minimum}{}{}{DMIndex.tex#Minimum}, 
\hyperref{Union}{}{}{DMIndex.tex#Union}}

\end{document}
