A density lemma

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Order reduction diagram. + Y which did not depend on inputs explicitly, the reduced SDS will probably exhibit explicit input dependence through X Ix U + Y x X 2 . In fact, in many applications, such reductions can explain the original presence of u in g ( x , u). 5-1. 5-2. Show that the action (2-71) makes (2-70) a bijection. Let S and T be sets. Show that 1, x 1, = l,, T. Notice that to establish an equality of functions requires showing that they have the same domains, codomains, and actions.

The elements in the set S have come to be called the states of the system. As will already be apparent, the creation of a state set S provides a new mechanism SXl-+O (2-38) 34 2 Set-Dynamical Systems for producing the system output. Moreover, provision is traditionally made for tracking any changes in the state set which may be dependent upon the input selected. This leads to another mechanism SxI+S. (2-39) The domain and codomain structure made explicit in (2-38) and (2-39) may be used to specify pairs of functions.

This turns out to be a very restrictive way to look at general systems, a way which is really not flexible enough to encompass the accepted applications. What is needed is a sort of parameterization of the input-output function. To see this, consider the following example. 2-1 A student in a course on psychokinesis is given the task of manipulating a set of levers so as to cause certain colored lights to be activated. The situation is sketched in Fig. 1. 2 Local Dynamical Functions 33 red 0 I blue II ,I green down yellow Fig.

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