From e7722cb1b18c2542057d41c10c700166253abe95 Mon Sep 17 00:00:00 2001 From: Robin Clark Date: Mon, 16 Aug 2010 22:44:07 +0100 Subject: [PATCH] Added euler diagrams as graphs --- eulerg/Makefile | 24 ++++++ eulerg/eulerg.tex | 162 +++++++++++++++++++++++++++++++++++++++ eulerg/eulerg.tex.backup | 119 ++++++++++++++++++++++++++++ eulerg/eulerg1.dia | Bin 0 -> 1150 bytes eulerg/eulerg1.jpg | Bin 0 -> 13028 bytes eulerg/eulerg_enc.dia | Bin 0 -> 981 bytes eulerg/eulerg_enc.jpg | Bin 0 -> 12195 bytes eulerg/eulerg_pic.dia | Bin 0 -> 1437 bytes eulerg/eulerg_pic.jpg | Bin 0 -> 24695 bytes eulerg/paper.tex | 35 +++++++++ style.tex | 2 + thesis.tex | 6 +- 12 files changed, 347 insertions(+), 1 deletion(-) create mode 100644 eulerg/Makefile create mode 100644 eulerg/eulerg.tex create mode 100644 eulerg/eulerg.tex.backup create mode 100644 eulerg/eulerg1.dia create mode 100644 eulerg/eulerg1.jpg create mode 100644 eulerg/eulerg_enc.dia create mode 100644 eulerg/eulerg_enc.jpg create mode 100644 eulerg/eulerg_pic.dia create mode 100644 eulerg/eulerg_pic.jpg create mode 100644 eulerg/paper.tex diff --git a/eulerg/Makefile b/eulerg/Makefile new file mode 100644 index 0000000..fb80b55 --- /dev/null +++ b/eulerg/Makefile @@ -0,0 +1,24 @@ + +# +# Make the propositional logic diagram a paper +# +SOURCE = eulerg.tex + +paper: paper.tex eulerg_paper.tex $(SOURCE) + #cat introduction.tex | sed s/chapter/paper/ > introduction.tex + #latex paper.tex + #dvipdf paper pdflatex cannot use eps ffs + pdflatex paper.tex + mv paper.pdf eulerg_paper.pdf + okular eulerg_paper.pdf + + +# Remove the need for referncing graphics in subdirectories +# +eulerg_paper.tex: eulerg.tex paper.tex $(SOURCE) + cat eulerg.tex | sed 's/eulerg\///' > eulerg_paper.tex + + +bib: $(SOURCE) + bibtex paper + diff --git a/eulerg/eulerg.tex b/eulerg/eulerg.tex new file mode 100644 index 0000000..ec2a1bf --- /dev/null +++ b/eulerg/eulerg.tex @@ -0,0 +1,162 @@ + + + +\ifthenelse {\boolean{paper}} +{ +\abstract{ +This paper discusses representing Euler Diagrams as graphs, or sets of relationships. +By representing Euler diagrams in this way, +algorithms to invesigate properties of the diagrams, are possible, without +having to resort +to CPU expensive area operations on the concrete diagrams. +} +} +{ %% Introduction +\section{Introduction} +This paper discusses representing Euler Diagrams as graphs, or sets of relationships. +By representing Euler diagrams in this way, +algorithms to invesigate properties of the diagrams, are possible, without +having to resort +to CPU expensive area operations on the concrete diagrams. +} + + +\section{Introduction : Euler Diagram } + +Classical Euler diagrams consist of closed curves in the plane which are used to represent sets. +The spaitial relationship between the curves defines the set theoretic relationships, as defined below. +\begin{itemize} +\item Intersection - if the curves defining the area within curves overlap +\item Sub-set - if a curve is enclosed by another +\item disjoint - if the curves are separate +\end{itemize} + + +\section{Defining `pure intersection' and `enclosure'} +\begin{figure}[h] + \centering + \includegraphics[width=200pt,keepaspectratio=true]{./eulerg1.jpg} + % eulerg1.jpg: 513x215 pixel, 72dpi, 18.10x7.58 cm, bb=0 0 513 215 + \caption{An Euler Diagram showing enclosure and Pure Intersection} + \label{fig:eulerg1} +\end{figure} + +The set theory term `intersection' can apply to both the curves overlapping and to the sub-set case. +For instance in diagram \ref{fig:eulerg1} the intersection between +$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 $Q$. + +\begin{figure}[h] + \centering + \includegraphics[width=300pt,keepaspectratio=true]{./eulerg_pic.jpg} + % eulerg_pic.jpg: 955x286 pixel, 72dpi, 33.69x10.09 cm, bb=0 0 955 286 + \caption{Pure Intersection Chain with Enclosure} + \label{fig:eulerg_pic} +\end{figure} + +\textbf{rule:} +If any contour in a pure intersection chain is enclosed by any contour not belonging to the chain, +all the countours within the +pure intersection chain will be enclosed by it. This is because a contour +enclosing which bisects(????) another contour in a pure intersection chain +becomes part of the pure~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. + + +The diagram in figure \ref{fig:eulerg_pic} can be represented by the following relationships. + + +$$ M {\pin} N $$ +$$ N {\pin} O $$ +$$ O {\pin} P $$ +$$ O {\pin} Q $$ +$$ Q {\enc} P $$ +$$ A {\enc} M $$ +$$ A {\enc} N $$ +$$ A {\enc} O $$ +$$ A {\enc} P $$ +$$ A {\enc} Q $$ + + +To form the pure intersection chain we can follow +reachable pure 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,$M,N,O,P,Q$ +are in the same pure intersection chain, even though $Q$ encloses $P$. +Contour $A$, by virtue of not bisecting any contour in the pure instersection +chain, does not belong to it. 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. + + +\vspace{40pt} + diff --git a/eulerg/eulerg.tex.backup b/eulerg/eulerg.tex.backup new file mode 100644 index 0000000..5c80bf3 --- /dev/null +++ b/eulerg/eulerg.tex.backup @@ -0,0 +1,119 @@ + + + +\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 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