.

eulerg/eulerg.tex

@@ -17,7 +17,7 @@ and time saving procedures, implemented in the FMMD analysis tool.
 }
 { %% Introduction
 \section{Introduction}
-This paper discusses representing Euler Diagrams as graphs, or sets of relationships.
+This chapter discusses representing Euler Diagrams as graphs, or sets of relationships.
 By representing Euler diagrams in this way,
 algorithms to invesigate properties of the diagrams, are possible, without
 having to resort
@@ -43,7 +43,7 @@ and write down set theory equations.
 The interest here though, is to define relationships between the contours, that allow
 processing and parsing of the diagram without resorting to extra area operations in the concerete plane.
 
-\section{Defining `pure intersection' and `enclosure'}
+\section{Defining `pair-wise intersection' and `enclosure'}
 
 %\begin{figure}[htp]
 %  \begin{center}
@@ -60,7 +60,7 @@ processing and parsing of the diagram without resorting to extra area operations
 % \centering
 % \includegraphics[width=200pt,keepaspectratio=true]{./eulerg/eulerg1.jpg}
 % % eulerg1.jpg: 513x215 pixel, 72dpi, 18.10x7.58 cm, bb=0 0 513 215
-% \caption{An Euler Diagram showing enclosure and Pure Intersection}
+% \caption{An Euler Diagram showing enclosure and Pair-wise Intersection}
 % \label{fig:eulerg1}
 %\end{figure}
 %
@@ -84,7 +84,7 @@ processing and parsing of the diagram without resorting to extra area operations
 %   \label{fig:subfig3}
 % }
 
- \caption{An Euler Diagram showing enclosure and Pure Intersection}
+ \caption{An Euler Diagram showing enclosure and Pair-wise Intersection}
  \label{fig:eulerg1}
 
 \end{figure}
@@ -117,20 +117,20 @@ $A$ `enclosed' by $B$. We can express this as a directed relationship.
 $$ B {\enc} A $$
 
 
-\paragraph{Pure Intersection}
+\paragraph{Pair-wise Intersection}
 In the $D$, $E$ case we have 
 
 We can say that where the areas defined by the curves intersect but no one curve encloses the 
-other, we can term this `pure intersection'.
+other, we can term this `pair-wise intersection'.
 We can express this as a non directed relationship.
 
 $$ D \pin E $$
 
 
-\paragraph{Mutual exclusivity of `pure intersection' and `enclosure'}
+\paragraph{Mutual exclusivity of `pair-wise intersection' and `enclosure'}
 
 Clearly these two properties are mutually exclusive. No
-contour can be both purely intersected and enclosed with the same contour.
+contour can be both pair-wisely intersected and enclosed with the same contour.
 Also enclosure, is transitive. That is to say if B encloses A, and A encloses C
 then B encloses C, see figure \ref{fig:eulerg_enc}.
 
@@ -183,8 +183,8 @@ an {\em enclosure} relationship as a directed vertice and
 In graph theory a node is said to be reachable from another node
 if you can start at the one node, travel via the edges
 and arrive at the other. 
-Contours may be connected via `pure intersection' relationships to form 
+Contours may be connected via `pair-wise intersection' relationships to form 
-`chains' of contours reachable by pure intersection.
+`chains' of contours reachable by pair-wise intersection.
 These are termed {\pic}s.
 Figure \ref{fig:eulerg_pic} shows a {\pic} consisting of contours $M,N,O,P$ and $Q$.
 
@@ -205,7 +205,7 @@ all the countours within the
 
 This is because a contour 
 enclosing which bisects another contour in a {\pic}
-becomes part of the pure~intersection~chain.
+becomes part of the pair-wise~intersection~chain.
 % Hmmmm thats true but a better way to say it ????
 %The diagram in figure \ref{fig:eulerg_enc} can be represented by the following relationships.
 
@@ -226,18 +226,18 @@ $$ A {\enc} Q $$
 }
 
 To form the {\pic} we can follow
-reachable pure intersection relationships.
+reachable pair-wise intersection relationships.
 
 $ M {\pin} N {\pin} O {\pin} P $ are part of the same chain.
 following from $O$, $O {\pin} Q$.
-Thus by the definition of being reachable by pure instersection relationships,$M,N,O,P,Q$
+Thus by the definition of being reachable by pair-wise instersection relationships,$M,N,O,P,Q$
 are in the same {\pic}, even though $Q$ encloses $P$.
 
 We can define this {\pic} as $PIC1$ as a set of contours.
 
 $$ PIC1 = \{ M,N,O,P,Q \} $$
 
-Contour $A$, by virtue of not bisecting any contour in the pure instersection
+Contour $A$, by virtue of not bisecting any contour in the pair-wise instersection
 chain $PIC1$, does not belong to  $PIC1$. Because it encloses one of the contours, it
 encloses all contours in the chain. 
 Knowing this can save on unecessary area operations on the concrete diagram.
@@ -246,15 +246,15 @@ Knowing this can save on unecessary area operations on the concrete diagram.
  \centering
  \includegraphics[width=200pt,bb=0 0 330 158,keepaspectratio=true]{./eulerg/eulerg_pic_g.jpg}
  % eulerg_pic_g.jpg: 330x158 pixel, 72dpi, 11.64x5.57 cm, bb=0 0 330 158
- \caption{Pure Intersection chain PIC1 as a graph}
+ \caption{Pair-wise Intersection chain PIC1 as a graph}
  \label{fig:eulerg_pic_g}
 \end{figure}
-%  \subsection{The Pure intersection chain PIC1}
+%  \subsection{The Pair-wise intersection chain PIC1}
 %  \begin{figure}[h]
 %   \centering
 %   \includegraphics[width=200pt,bb=0 0 955 286,keepaspectratio=true]{./eulerg_pic_g.jpg}
 %   % eulerg_pic.jpg: 955x286 pixel, 72dpi, 33.69x10.09 cm, bb=0 0 955 286
-%   \caption{The pure Intersection PIC1 as a graph}
+%   \caption{The pair-wise Intersection PIC1 as a graph}
 %   \label{fig:eulerg_pic1}
 %  \end{figure}
 
@@ -263,8 +263,7 @@ Figure \ref{fig:eulerg_pic_g} only shows the {\pic}, but does not show the conto
 enclosing $PIC1$. Figure \ref{fig:eulerg_pic_g_a}
 shows contour A enclosing all elements in $PIC1$
 
-\pagebreak[0]
+\subsection{Enclosure and pair-wise \\ intersection in the graph}
-\subsection{Enclosure and pure \\ intersection in the graph}
 \begin{figure}[h]
  \centering
  \includegraphics[width=200pt,bb=0 0 330 162,keepaspectratio=true]{./eulerg/eulerg_pic_g_a.jpg}
@@ -273,6 +272,7 @@ shows contour A enclosing all elements in $PIC1$
  \label{fig:eulerg_pic_g_a}
 \end{figure}
 
+\pagebreak[0]
 Because we know that a contour enclosing a contour within a {\pic} but not belonging
 to it, encloses all elements of the {\pic}, we can draw this in a less cluttered way
 see figure \ref{fig:eulerg_pic_g_a_unc}.

style.tex

@@ -76,7 +76,8 @@
 \newcommand{\bcs}{\em base~components}
 \newcommand{\enc}{\ensuremath{\stackrel{enc}{\longrightarrow}}}
 \newcommand{\pin}{\ensuremath{\stackrel{pi}{\longleftrightarrow}}}
-\newcommand{\pic}{\em pure~intersection~chain}
+%\newcommand{\pic}{\em pure~intersection~chain}
+\newcommand{\pic}{\em pair-wise~intersection~chain}
 
 %----- Display example text (#1) in typewriter font