A First Course in FUZZY and NEURAL CONTROL

170 CHAPTER 5. NEURAL NETWORKS FOR CONTROL

First, to see whether or not this problem is solvable, we look at the domain
of the functiong, coloring the points white where the function value is 1 ,and
black where the function value is 0.

The input domain is divided into the two subsetsB ={(0,0)}andW =
{(0,1),(1,1),(1,0)}corresponding to the two output values 0 and 1 , respec-
tively. A solution to the perceptron design is a straight linew 1 x 1 +w 2 x 2 −b=0,
in thex 1 - x 2 plane, and we can see that there are lines that can separate these
two subsets,

that is, this is a linearly separable problem. Of course, there are many such lines,
and any such line gives a solution. In view of the simplicity of this functiong,
no sophisticated mathematics is needed. Just observe that

g(0,0) = 0implies 0 <b

g(0,1) = 1impliesw 2 ≥b

g(1,0) = 1impliesw 1 ≥b

g(1,1) = 1impliesw 1 +w 2 ≥b

and choose any numbersw 1 ,w 2 ,andbsatisfying these inequalities. One solution
isw 1 =w 2 =b=2.

or

The basic logical functions are often written in set notation. For setsA
andB, the set notation for the logical function AND isA∩B, also calledA