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eulerg/eulerg.tex

@@ -134,6 +134,12 @@ 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}.
 
+\begin{definition}
+\label{def:mutex}
+No contour can be both pair-wisely intersected and enclosed with the same contour.
+\end{definition}
+
+
 \begin{figure}[h]
  \centering
 
@@ -155,7 +161,8 @@ then B encloses C, see figure \ref{fig:eulerg_enc}.
 $$ B {\enc} A \wedge A {\enc} C \implies B {\enc} C $$
 
 \begin{definition}
-Enlcosure relationships are transitive
+\label{def:enctrans}
+Enlcosure relationships are transitive.
 \end{definition}
 
 \section{Representing Euler Diagrams \\ as sets of relationships}
@@ -198,13 +205,14 @@ Figure \ref{fig:eulerg_pic} shows a {\pic} consisting of contours $M,N,O,P$ and
 
 %\textbf{rule:}
 \begin{definition}
+\label{def:encpic}
 If any contour in a {\pic} is enclosed by any contour not belonging to the chain, 
 all the countours within the 
 {\pic} will be enclosed by it. 
 \end{definition}
 
-This is because a contour 
-enclosing which bisects another contour in a {\pic}
+Note that any contour 
+which bisects another contour in a {\pic}
 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.
@@ -272,13 +280,14 @@ shows contour A enclosing all elements in $PIC1$
  \label{fig:eulerg_pic_g_a}
 \end{figure}
 
-\pagebreak[0]
+\pagebreak[1]
+\subsection{Reducing clutter in the graph}
 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
+to it, encloses all elements of the {\pic}, see definition \ref{def:encpic}, we can draw this in a less cluttered way
 see figure \ref{fig:eulerg_pic_g_a_unc}.
 
-\pagebreak[0]
-\subsection{Reducing clutter in the graph}
+
+
 \begin{figure}[h]
  \centering
  \includegraphics[width=200pt,bb=0 0 330 162]{./eulerg/eulerg_pic_g_a_unc.jpg}
@@ -308,7 +317,7 @@ The number of available zones within a {\pic} $P$ does not change
 when other contours are added or removed from the diagram
 that are not, or would not become members of the {\pic} $P$.
 \end{definition}
-his is to say, the the number of zones within a {\pic} is not affected by changes in the diagram 
+That is to say, the the number of zones within a {\pic} is not affected by changes in the diagram 
 that do not alter the {\pic}.
 This allows us to analyses {\pic}s separately, thus reducing the $2^N$ overhead of analysing an Euler diagram for available zones.