C++ Neural Networks and Fuzzy Logic: Preface

C++ Neural Networks and Fuzzy Logic

by Valluru B. Rao

MTBooks, IDG Books Worldwide, Inc.

ISBN: 1558515526   Pub Date: 06/01/95

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Now let us determine the fuzzy sets that have John Smith as an element, besides possibly others. We need the

values of degrees of membership for John Smith (actually his attribute values) in various fuzzy sets. Let us

pick them as follows:

——————————————————————————————————————————————————————————————————————————

 Age:   m

very young

(35) = 0 (degree of membership of 35 in very young is 0.

                          We will employ this notation from now on).

        m

young

(35) = 0.75

        m

somewhat old

(35) = 0.3

        m

old

(35) = 0

——————————————————————————————————————————————————————————————————————————

Assume that similar values are assigned to the degrees of membership of values of John Smith’s attributes in

other fuzzy sets. Just as John Smith’s age does not belong in the fuzzy sets young and old, some of his other

attribute values do not belong in some of the other fuzzy sets. The following is a list of fuzzy sets in which

John Smith appears:

age_young = {0.75/35, ...}

age_somewhat old = {0.3/35, ... }

A similar statement attempted for the number of visits may prompt you to list nov_rarely = {0.7/1, 0.2/2},

and nov_quite a few = {0.3/2, .6/3, ...}. But you readily realize that the number of visits by itself does not

mean much unless it is referenced with the country of visit. A person may visit one country very often, but

another only rarely. This suggests the notion of a fuzzy relation, which is also a fuzzy set.

NOTE:  What follows is an explanation of relations and discussion of fuzzy relations. If

you want to skip this part for now, you may go to the “Fuzzy Queries” section a few pages

later in this chapter.

Fuzzy Relations

A standard relation from set A to set B is a subset of the Cartesian product of A and B, written as A×B. The

elements of A×B are ordered pairs (a, b) where a is an element of A and b is an element of B. For example, the

ordered pair (Joe, Paul) is an element of the Cartesian product of the set of fathers, which includes Joe and the

set of sons which includes Paul. Or, you can consider it as an element of the Cartesian product of the set of

men with itself. In this case, the ordered pair (Joe, Paul) is in the subset which contains (a, b, if a is the father

of b. This subset is a relation on the set of men. You can call this relation “father.”

A fuzzy relation is similar to a standard relation, except that the resulting sets are fuzzy sets. An example of

such a relation is ‘much_more_educated’. This fuzzy set may look something like,

much_more_educated = { ..., 0.2/(Jeff, Steve), 0.7/(Jeff, Mike), ... }

C++ Neural Networks and Fuzzy Logic:Preface

Fuzzy Relations

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