\ifthenelse {\boolean{paper}}
{
\abstract{
This paper 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
to CPU expensive area operations on the concrete diagrams. 
}
}
{ %% Introduction
\section{Introduction}
This paper 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
to CPU expensive area operations on the concrete diagrams. 
}


\section{Introduction : Euler Diagram }

Classical Euler diagrams consist of closed curves in the plane which are used to represent sets.
The spaitial relationship between the curves defines the set theoretic relationships, as defined below.
\begin{itemize}
\item Intersection - if the curves defining the area within curves overlap
\item Sub-set      - if a curve is enclosed by another
\item disjoint     - if the curves are separate
\end{itemize}


\section{Defining `pure intersection' and `enclosure'}
\begin{figure}[h]
 \centering
 \includegraphics[width=200pt,keepaspectratio=true]{./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}
 \label{fig:eulerg1}
\end{figure}

The set theory term `intersection' can apply to both the curves overlapping and to the sub-set case.
For instance in diagram \ref{fig:euler1} the intersection between
$A$ and $B$ exists.

$$ A \cup B \neq \emptyset $$

as does the intersection $D$ and $E$

$$ D \cup E \neq \emptyset $$

Clearly though these intersections are different, because
in the $A$, $B$ case 
$$ A \backslash B = \emptyset \wedge B \backslash A \neq \emptyset $$.
This is not the case for $D$, $E$ where:
$$ D \backslash E \neq \emptyset \wedge E \backslash D \neq \emptyset $$

\paragraph{Enclosure}
To distinguish between these we can term the $A$, $B$ case to be 
$A$ `enclosed' by $B$. We can express this as a directed relationship.

$$ B {\enc} A $$


\paragraph{Pure 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'.
We can express this as a non directed relationship.

$$ D \pin E $$


\paragraph{Mutual exclusivity of `pure intersection' and `enclosure'}

Clearly these two properties are mutually exclusive. No
contour can be both purely 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{figure}[h]
 \centering
 \includegraphics[width=200pt,keepaspectratio=true]{./eulerg_enc.jpg}
 % eulerg_enc.jpg: 315x269 pixel, 72dpi, 11.11x9.49 cm, bb=0 0 315 269
 \caption{Enclosure, a transitive relationship}
 \label{fig:eulerg_enc}
\end{figure}

$$ B {\enc} A \wedge A {\enc} C \implies B {\enc} C $$

\section{Representing Euler Diagrams as sets of relationships}

The diagram in figure \ref{fig:eulerg1} can be represented by the foillowing relationships.

$$ B {\enc} A $$
$$ D {\pin} E $$


The diagram in figure \ref{fig:eulerg_enc} can be represented by the following relationships.

$$ B {\enc} A $$
$$ A {\enc} C $$


\section{The Pure Intersection chain}

Contours may be connected via `pure intersection' relationships to form 
`chains' of contours reachable by pure intersection.

Figure \ref{fig:eulerg_pic} shows a pure intersection chain consisting of contours $M,N,O,P$ and $Q$.


\textbf{rule:}
If any contour in a pure intersection chain is enclosed by any contour, all countours within the 
pure intersection chain will be enclosed by it.