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altered formulas for defining pure~intersection and enclosure

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Robin 2010-06-06 20:49:53 +01:00
parent 2365604723
commit 31b4422e9d
1 changed files with 36 additions and 11 deletions
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47
fzd/fzd.tex
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@ -374,10 +374,13 @@ i.e. for a Venn N diagram $Z\# = N^{2}$
\endequation
But for any diagram less
But for any diagram less
complicated than Venn N, where $ Z\# $
is small in comparison with $2^{N}$ the algorithm becomes far more efficient.
For FMMD failure analysis Venn 2 combinations are rare
and Venn 3 or 4 would only be required for special cases
(such as common components with a serial safety dependency; 747 engines;
relays contacts in series etc)
Examples of complexity savings are shown in section \ref{complexity}.
@ -477,17 +480,29 @@ contour against all others a collection of pure intersection relationships
and enclosure relationships can be determined.
This forms a list of relationship pairs from the cross product of all the contours.
%\equation
%%\label{crossprodsingle}
%\begin{array}{l}
% pi(a,b) \; \Rightarrow
%%\stackrel{\Delta}{=}
% \; \forall \; C \; \bullet \; a \; X \; \forall C \; \bullet \; b \\
% \; \bullet (a \cap b) \; \wedge \; \neg (a \supseteq b) \; \wedge \; \neg (b \supseteq a) \\
%\end{array}
%\endequation
%
\equation
%\label{crossprodsingle}
\begin{array}{l}
pi(a,b) \; \Rightarrow
%\stackrel{\Delta}{=}
\; \forall \; C \; \bullet \; a \; X \; \forall C \; \bullet \; b \\
\; \bullet (a \cap b) \; \wedge \; \neg (a \supseteq b) \; \wedge \; \neg (b \supseteq a) \\
pi(a,b) \; \implies
a \in D \wedge b \in D \wedge a \neq b \wedge
\; (a \cap b) \; \wedge \; \neg (a \supseteq b) \; \wedge \; \neg (b \supseteq a) \\
\end{array}
\endequation
Or in other words, a pure intersection is where the contours $a$ and $b$ in the diagram $D$
where $a$ intersects $b$ but $a$ does not enclode $b$ and $b$ does not enclose $a$.
\begin{figure}[h]
\centering
\includegraphics[width=200pt,bb=0 0 452 290,keepaspectratio=true]{fzd/pice.jpg}
@ -517,13 +532,23 @@ This again, forms a list of relationship pairs from cross product of all the co
\equation
%\label{crossprodsingle}
\begin{array}{l}
enc(a,b) \;
%\stackrel{\Delta}{=}
\Rightarrow \; \forall \; C \; \; \bullet a \; X \; \forall \; C \; \bullet \; b \\
\; \bullet (a \supset b) \\
enc(a,b) \; \implies
a \in D \wedge b \in D \wedge a \neq b \wedge
(a \supset b)
\end{array}
\endequation
%
%\equation
%%\label{crossprodsingle}
%\begin{array}{l}
% enc(a,b) \;
%%\stackrel{\Delta}{=}
%\Rightarrow \; \forall \; C \; \; \bullet a \; X \; \forall \; C \; \bullet \; b \\
% \; \bullet (a \supset b) \\
%\end{array}
%\endequation
%
\section{The Pure Intersection Chain}