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

 
\section{Introduction}
When building a system from components, 
we should be able to find all known failure modes for each component.
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}. %The failure modes for a given component $K$ form a set $F$.

An important factor in defining a set of failure modes is that they
should be as clearly defined as possible.
%
It should not be possible for instance for
a component to have two or more failure modes active at once.

Having a set of failure modes whhere $N$ modes could be active simultaneously
would mean having to consider $2^N$ failure mode scenarios.
% 
Should a component be analysed and simultaneous failure mode cases exit,
the combinations could be represented by a new failure modes, or
the component should be considered from a fresh perspective,
perhaps considering it as several smaller components
within one package.

\begin{definition}
A set of failure modes where only one fault mode
can be active at a time is termed a `unitary~state' failure mode set.
\end{definition}

We can define a function $FM()$ to 
take a given component $K$ and return its set of failure modes $F$.

$$  FM : K \mapsto F $$

We can further define a set $U$ which is a set of sets of failure modes, where
the component failure modes in each of its members are unitary~state.
Thus if the failure modes of $F$ are unitary~state, we can say $F \in U$.


\subsection{Component failure modes : Unitary State example}

A component with simple ``unitary~state'' failure modes is the electrical resistor.

Electrical resistors can fail by going OPEN or SHORTED.
However they cannot fail with both conditions active. The conditions
OPEN and SHORT are mutually exlusive.
Because of this the failure mode set $F=FM(R)$ is `unitary~state'.
%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 `unitary~state'.

% A set of failure modes, where only one or no failure modes
% are active is termed an `unitary~state' failure mode set. This
% will be donoted as set $A$.
% 
To define `unitary~state' using set theory we can define a function
`active'.
The function $active(f)$ deontes that the failure mode $f$ (where $f$ is an element of $F$) is currently active.

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

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

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

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

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

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 `unitary~state'
component failure modes.

Thus 
     $$ FM(R) =  \{ R_{open}, R_{shorted} \} \in U $$


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


\begin{equation}
\label{`unitarystate'_def}
 UnitaryState(F) =  \forall f \in F | active(f) \wedge  f1 \in F \wedge f1 \neq f  \wedge \neg active(f1) 
\end{equation}

%Which can be written

%$$  UnitaryState(FM(K)) $$



% should this be a paragraph in Symptom Abstraction ????



