\abstract{ This chapter defines what is meant by the terms
components, component fault modess and atomic component fault modes.
Mathematical constraints and definitions are made using set theory.
}

 
\section{Introduction}
When building a system the components used, will have known failure modes.
For most common electrical and mechanical components, the failure modes
for a given type of part can be obtained from standard literature\cite{mil1991}
\cite{mech}.

We can define a function $FM()$ to represent thiss, where K is the component
and F is the set of failure modes and A represents the set of atomic failure mode sets.

$$ FM : K \mapsto F | F \exits A $$


\subsection{Component failure modes : Atomic definition}

Electrical resistors can fail by going OPEN or SHORTED for instance.
However they cannot fail with both conditions active. The conditions
OPEN and SHORT are mutually exlusive.
Because of this these failure modes can be considered `atomic'.
A more complex component, say a micro controller could have several
faults active. It could for instance have a broken I/O output
and an unstable ADC input. Here the faults cannot be considered atomic.

A set of failure modes, where only one or no failure modes
are active is termed an atomic failure mode set. This
will be donoted as set $A$.

The function $active(f)$ deontes that the failure mode f is currently active.

Thus for the set $F$ to exist in $A$ the following condition must be true.

\begin{equation}
\label{atomic_def}
 active(f) | f \in F \wedge f1 \in F | f1 \neq f  \wedge \neg active(f1) 
\end{equation}

As an example the resistor $R$ 
has two failure modes $_{open}$ and $R_{shorted}$. 

$$ F = FM(R) = \{ R_{open}, R_{shorted} \} $$
 
Applying equation \ref{atomic_definition} to a resistor
for both fault modes

 $$ active(R_{short}) | R_{short} \in F \wedge R_{open} \in F | R_{open} \neq R_{short}  \wedge \neg active(R_{open})  $$
 $$ active(R_{open}) | R_{open} \in F \wedge R_{short} \in F | R_{short} \neq R_{open}  \wedge \neg active(R_{short})  $$

For the case of the resistor with only two failure modes the results above, being true,
show that the failure modes for a resistor of $ F = \{ R_{open}, R_{shorted} \} $ are atomic
component failure modes.

A general case can be stated by taking equation \ref{atomnic_def} and making it a function thus.


\begin{equation}
\label{atomic_def}
 Atomic(F) =  \forall f \in F | active(f) \wedge  f1 \in F \wedge f1 \neq f  \wedge \neg active(f1) 
\end{equation}