% %%%% FORMAL DEFINITIONS %%%% THESE MIGHT BE MOVED TO AN APPENDIX 
% 
% 
% 
% \chapter{Formal Definitions}
% \label{sec:formalfmmd}
% \section{An algebraic notation for identifying FMMD enitities}
% Consider all `components' to exist as
% members of a set $\mathcal{C}$.
% %
% Each component $c$ has an associated  set of failure modes.
% We can define a function $fm$ that returns a 
% set of failure modes $F$, for the component $c$.
% 
% Let the set of all possible components  be $\mathcal{C}$
% and let the set of all possible failure modes be $\mathcal{F}$.
% 
% We now define the function $fm$
% as
% \begin{equation}
% \label{eqn:fm}
% fm : \mathcal{C} \rightarrow \mathcal{P}\mathcal{F}.
% \end{equation}
% This is defined by, where $c$ is a component and $F$ is a set of failure modes,
% $  fm ( c ) = F. $
% 
% We can use the variable name $\FG$ to represent a {\fg}. A {\fg} is a collection
% of components. 
% %We thus define $FG$ as a set of chosen components defining 
% %a {\fg}; all functional groups 
% We can state that
% {\FG} is a member of the power set of all components, $ \FG \in \mathcal{P} \mathcal{C}. $
% 
% We can overload the $fm$ function for a functional group {\FG}
% where it will return all the failure modes of the components in {\FG}
% 
% 
% given by
% 
% $$ fm ({\FG}) = F. $$
% 
% Generally, where $\mathcal{{\FG}}$ is the set of all functional groups,
% 
% \begin{equation}
% fm : \mathcal{{\FG}} \rightarrow \mathcal{P}\mathcal{F}.
% \end{equation}
% \section{Relationships between functional~groups and failure modes}
% 
% Let the set of all possible components  be $\mathcal{C}$
% and let the set of all possible failure modes be $\mathcal{F}$, and $\mathcal{PF}$
% is the power-set of $\mathcal{F}$.
% 
% In order to analyse failure mode effects we need to be able to determine the 
% failure modes of a component. We define a function $fm$ to perform this (see equation~\ref{eqn:fmset}).
% \label{fmdef}
% 
% \begin{equation}
% fm : \mathcal{C} \rightarrow \mathcal{P}\mathcal{F}
%   \label{eqn:fmset}
% \end{equation}
% 
% %% 
% % Above def gives below anyway
% %
% %The is defined by equation \ref{eqn:fminstance}, where C is a component and F is a set of failure modes.
% %
% %\begin{equation}
% %  fm ( C ) = F
% %  \label{eqn:fminstance}
% %\end{equation}
% 
% \paragraph{Finding all failure modes within the functional group.}
% 
% For FMMD failure  mode analysis %we need to consider the failure modes
% from all the components in a functional~group.
% In a functional group we have a collection of Components
% which have associated failure mode sets.
% we need to collect failure mode sets from the components and place them all
% %modes 
% into a single set; this can be termed flattening the set of sets.
% %%Consider the components in a functional group to be $C_1...C_N$.
% The flat set of failure modes $FSF$ we are after can be found by applying function $fm$ to all the components
% in the functional~group and taking the union of them thus:
% 
% %%$$ FSF = \bigcup_{j=1}^{N} fm(C_j) $$
% $$ FSF = \bigcup_{c \in FG} fm(c) \; .$$
% 
% We can actually overload the notation for the function $fm$ % FM
% and define it for the set components within a functional group $\mathcal{FG}$ (i.e. where $\mathcal{FG} \subset \mathcal{C} $) 
% in equation \ref{eqn:fmoverload}.
% 
% \begin{equation}
% fm : \mathcal{FG} \rightarrow \mathcal{F}
% \label{eqn:fmoverload}
% \end{equation}