Spell checked

eulerg/eulerg.tex

@@ -6,9 +6,9 @@
 \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
+algorithms to investigate properties of the diagrams, are possible, without
 having to resort
-to extra unecessary CPU expensive area operations on the concrete diagrams. 
+to extra unnecessary CPU expensive area operations on the concrete diagrams. 
 
 The graph representations presented here form the basis for several algorithms
 and time saving procedures, implemented in the FMMD analysis tool.
@@ -19,9 +19,9 @@ and time saving procedures, implemented in the FMMD analysis tool.
 \section{Introduction}
 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
+algorithms to investigate properties of the diagrams, are possible, without
 having to resort
-to extra unecessary CPU expensive area operations on the concrete diagrams. 
+to extra unnecessary CPU expensive area operations on the concrete diagrams. 
 
 The graph representations presented here form the basis for several algorithms
 and time saving procedures, implemented in the FMMD analysis tool.
@@ -31,7 +31,7 @@ and time saving procedures, implemented in the FMMD analysis tool.
 \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.
+The spatial 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
@@ -41,7 +41,7 @@ The spaitial relationship between the curves defines the set theoretic relations
 The definitions above allow us to read an Euler diagram
 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.
+processing and parsing of the diagram without resorting to extra area operations in the concrete plane.
 
 \section{Defining `pair-wise intersection' \\ and `enclosure'}
 
@@ -140,7 +140,7 @@ $$ B {\enc} A \wedge A {\enc} C \implies B {\enc} C $$
 
 \begin{definition}
 \label{def:enctrans}
-Enlcosure relationships are transitive.
+Enclosure relationships are transitive.
 \end{definition}
 
 
@@ -158,7 +158,7 @@ The diagram in figure \ref{fig:eulerg_enc} can be represented by the following r
 $$ B {\enc} A $$
 $$ A {\enc} C $$
 
-\section{Represeting Euler diagrams as graphs}
+\section{Representing Euler diagrams as graphs}
 
 As the relationships {\em enclosure} and {\pic} are mutually exclusive
 and {\em enclosure} is transitive and {\pic} is not, we can represent
@@ -187,7 +187,7 @@ Figure \ref{fig:eulerg_pic} shows a {\pic} consisting of contours $M,N,O,P$ and
 \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 
+all the contours within the 
 {\pic} will be enclosed by it. 
 \end{definition}
 
@@ -218,17 +218,17 @@ 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 pair-wise instersection relationships,$M,N,O,P,Q$
+Thus by the definition of being reachable by pair-wise intersection 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 pair-wise instersection
+Contour $A$, by virtue of not bisecting any contour in the pair-wise intersection
 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.
+Knowing this can save on unnecessary area operations on the concrete diagram.
 
 \begin{figure}[h]
  \centering
@@ -259,7 +259,7 @@ we can represent them both on the same graph, see figure \ref{fig:eulerg_pic_g_a
  \centering
  \includegraphics[width=200pt,bb=0 0 330 162,keepaspectratio=true]{./eulerg/eulerg_pic_g_a.jpg}
  % eulerg_pic_g_a.jpg: 330x162 pixel, 72dpi, 11.64x5.72 cm, bb=0 0 330 162
- \caption{Graph of euler diagram in figure \ref{fig:eulerg_pic}.}
+ \caption{Graph of Euler diagram in figure \ref{fig:eulerg_pic}.}
  \label{fig:eulerg_pic_g_a}
 \end{figure}
 
@@ -267,7 +267,7 @@ we can represent them both on the same graph, see figure \ref{fig:eulerg_pic_g_a
 \subsection{Reducing clutter in the graph}
 Contour A encloses the pure intersection chain $PIC1$;
 using definition \ref{def:encpic}, we can draw this in a less cluttered way,
-by only drawing one enclosure ralationship to the {\pic}
+by only drawing one enclosure relationship to the {\pic}
 (see figure \ref{fig:eulerg_pic_g_a_unc}).
 We only need to show contour A enclosing one member of the {\pic} $PIC1$
 in order to show that contour A encloses all contours in $PIC1$.
@@ -278,7 +278,7 @@ in order to show that contour A encloses all contours in $PIC1$.
  \centering
  \includegraphics[width=200pt,bb=0 0 330 162]{./eulerg/eulerg_pic_g_a_unc.jpg}
  % eulerg_pic_g_a_unc.jpg: 330x162 pixel, 72dpi, 11.64x5.72 cm, bb=0 0 330 162
- \caption{Uncluttered graph of euler diagram in figure \ref{fig:eulerg_pic}.}
+ \caption{Uncluttered graph of Euler diagram in figure \ref{fig:eulerg_pic}.}
  \label{fig:eulerg_pic_g_a_unc}
 \end{figure}
 
@@ -287,7 +287,7 @@ in order to show that contour A encloses all contours in $PIC1$.
 \section{Reduction of searches \\ for available zones}
 
 Another property of any {\pic} $P$, is that
-the maximum number of euler zones within it is 
+the maximum number of Euler zones within it is 
 
 $$ MaxZones =  2^{|P|}  $$