C++ Neural Networks and Fuzzy Logic: Preface

4.  The maximum value for the function is 1, at x = 0. It has a sort of a bell−shaped curve.

If A is a fuzzy number with the ordered pair (a, b) with b > 0, and if the reference function is L, you do the

following:

You write the fuzzy number as A = (a, b)

L

. You get the membership of any element x, by taking the quantity

(x − a) / b (which reminds you of how you get a z−score), and evaluate the reference function L at this

argument. That is:

     m

A

(x) = L( (x−a) / b)

Examples of a reference function L are:

Example 1:

L(x) = max ( 0, 1− x

2

)

You obtain the following shown in Table 16.8.

Table 16.8 Reference Function L

xL(x)

−20

−10

−0.50.75

01

0.50.75

10

20

C++ Neural Networks and Fuzzy Logic:Preface

Fuzzy Numbers

397

---

This function is not strictly decreasing though. It remains a constant at 0 for x > 1.

Example 2:

L(x) = 1/ (1 + x

2

 )

You get the values shown in Table 16.9.