expanded final equation and gave reason why it is useful

component_failure_modes_definition/component_failure_modes_definition.tex

@@ -358,7 +358,7 @@ from $1$ to $cc$ thus
 %
 
 \begin{equation}
- \#\mathcal{P}_{cc} S = \sum^{k}_{1..cc} \frac{\#S!}{k!(\#S-k)!}    
+ |{\mathcal{P}_{cc}S}| = \sum^{k}_{1..cc} \frac{|{S}|!}{ k! ( |{S}| - k)!}    
  \label{eqn:ccps}
 \end{equation}
 
@@ -377,7 +377,11 @@ from the cardinality constrain powerset number.
 
 Thus were we to have a simple functional group with two components R and T, of which
 $$FM(R) = \{R_o, R_s\}$$ and $$FM(T) = \{T_o, T_s, T_h\}$$.
-For a cardinality constrained powerset of 2, because there are 5 error modes
+
+This means that a functional~group $FG=\{R,T\}$ will have a component failure modes set % $FM_{cfg} $
+of $FM_{cfg} = \{R_o, R_s, T_o, T_s, T_h\}$
+
+For a cardinality constrained powerset of 2, because there are 5 error modes ( $|{FG_{cfg}}|=5$),
 applying equation \ref{eqn:ccps} gives :-
 
 $$\frac{5!}{1!(5-1)!} + \frac{5!}{2!(5-2)!}  = 15$$
@@ -392,41 +396,60 @@ $R_o \wedge R_s$. As a combination ${2 \choose 2} = 1$ . For $T$ the component w
 Thus for $cc == 2$ we must subtract $(3+1)$.
 The number of combinations to check is thus 11 for this example and this can be verified
 by listing all the required combinations:
+%
+%\vbox{
+%\subsubsection{All Eleven Cardinality Constrained \\ Powerset of 2 Elements Listed}
+%%\tiny
+%\begin{enumerate}
+%\item $\{R_o T_o\}$
+%\item $\{R_o T_s\}$
+%\item $\{R_o T_h\}$
+%\item $\{R_s T_o\}$
+%\item $\{R_s T_s\}$
+%\item $\{R_s T_h\}$
+%\item $\{R_o \}$
+%\item $\{R_s \}$
+%\item $\{T_o \}$
+%\item $\{T_s \}$
+%\item $\{T_h \}$
+%\end{enumerate}
+%%\normalsize
+%}
+%
 
-\vbox{
+$$ \mathcal{P}_{2}(FG_cfg) =  \{
-\subsubsection{All Eleven Cardinality Constrained \\ Powerset of 2 Elements Listed}
+ \{R_o T_o\}, \{R_o T_s\}, \{R_o T_h\}, \{R_s T_o\}, \{R_s T_s\}, \{R_s T_h\}, \{R_o \}, \{R_s \}, \{T_o \}, \{T_s \}, \{T_h \}
-%\tiny
+ \}
-\begin{enumerate}
+$$
-\item $\{R_o T_o\}$
-\item $\{R_o T_s\}$
-\item $\{R_o T_h\}$
-\item $\{R_s T_o\}$
-\item $\{R_s T_s\}$
-\item $\{R_s T_h\}$
-\item $\{R_o \}$
-\item $\{R_s \}$
-\item $\{T_o \}$
-\item $\{T_s \}$
-\item $\{T_h \}$
-\end{enumerate}
-%\normalsize
-}
 
+And by inspection
+
+$$ |\mathcal{P}_{2}(FG_cfg)| = 11 $$
 
 The cardinality constrained powerset equation \ref{eqn:ccps} corrected for 
 unitary state failure modes can be 
-written as a general formula, where C is a set of the components (indexed by j where J 
+written as a general formula (see equation \ref{eqn:correctedccps}), where C is a set of the components (indexed by j where J 
-is the set of components  in the functional~group under analyis) and $\#C$
+is the set of components  in the functional~group under analyis) and $|{C}|$
 indicates the number of mutually exclusive fault modes each component has:-
 
 %$$ \#\mathcal{P}_{cc} S = \sum^{k}_{1..cc} \frac{\#S!}{k!(\#S-k)!}    $$
 \begin{equation}
- \#\mathcal{P}_{cc} S = {\sum^{k}_{1..cc}  \frac{\#S!}{k!(\#S-k)!}}  - {\sum^{j}_{j \in J} {\#C_{j} \choose cc}} 
+ |{\mathcal{P}_{cc}S}| = {\sum^{k}_{1..cc}  \frac{|{S}|!}{k!(|{S}| - k)!}}  - {\sum^{j}_{j \in J} {|{C_{j}}| \choose cc}} 
  \label{eqn:correctedccps}
 \end{equation}
 
+Expanding the combination in equation \ref{eqn:correctedccps}
 
 
+\begin{equation}
+ |{\mathcal{P}_{cc}S}| = {\sum^{k}_{1..cc}  \frac{|{S}|!}{k!(|{S}| - k)!}}   - {\sum^{j}_{j \in J}   \frac{|{C_j}|!}{cc!(|{C_j}| - cc)!}}  
+ \label{eqn:correctedccps2}
+\end{equation}
+
+The equation \ref{eqn:correctedccps2} is now useful for an automated tool that
+would verify that a `N' simultaneous failures model had been completly covered
+by knowing how many test case should be covered.
+
 %$$ \#\mathcal{P}_{cc} S = \sum^{k}_{1..cc}  \big[ \frac{\#S!}{k!(\#S-k)!}  - \sum_{j} (\#C_{j} \choose cc \big]  $$
 
 

component_failure_modes_definition/style.tex

@@ -57,6 +57,7 @@ depth -0.5ex\hfill}\newcommand{\innerhead}[1]{\def\lp@innerhead{#1}}
 \newcommand{\mins}[1]{$#1^{\scriptsize\prime}$}         % Minutes symbol
 \newcommand{\secs}[1]{$#1^{\scriptsize\prime\prime}$}   % Seconds symbol
 \newcommand{\key}[1]{\fbox{\sc#1}}                      % Box for keys
+\newcommand{\modulus}[1]{\ensuremathmode{|#1|}}
 \newcommand{\?}{\_\hspace{0.115em}}                     % Proper spacing for 
                                                         %   underscore
 \newcommand{\rev}{PA5}