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eulerg/eulerg.tex

@@ -253,21 +253,22 @@ shows contour A enclosing all elements in $PIC1$
 
 \pagebreak[1]
 \subsection{Enclosure and pair-wise \\ intersection in the graph}
-Because enclose is a directed relationship and {\em pair-wise intersection} is non-directed
+Because enclosure is a directed relationship and {\em pair-wise intersection} is non-directed
 we can represent them both on the same graph, see figure \ref{fig:eulerg_pic_g_a}.
 \begin{figure}[h]
  \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{PIC1 including enclosing contour A as a graph}
+ \caption{Graph of euler diagram in figure \ref{fig:eulerg_pic}.}
  \label{fig:eulerg_pic_g_a}
 \end{figure}
 
 \pagebreak[1]
 \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
-see figure \ref{fig:eulerg_pic_g_a_unc}.
+using definition \ref{def:encpic}, we can draw this in a less cluttered way,
+by only drawing one enclosure ralationship 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 $PIC1$.
 
@@ -277,7 +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}