C++ Neural Networks and Fuzzy Logic: Preface

Similarity Relations

A reflexive, symmetric, and transitive fuzzy relation is said to be a fuzzy equivalence relation. Such a relation

is also called a similarity relation. When you have a similarity relation s, you can define the similarity class of

an element x of the domain as the fuzzy set in which the degree of membership of y in the domain is m

s

(x, y).

The similarity class of x with the relation s can be denoted by [x]

s

.

Resemblance Relations

Do you think similarity and resemblance are one and the same? If x is similar to y, does it mean that x

resembles y? Or does the answer depend on what sense is used to talk of similarity or of resemblance? In

everyday jargon, Bill may be similar to George in the sense of holding high office, but does Bill resemble

George in financial terms? Does this prompt us to look at a ‘resemblance relation’ and distinguish it from the

‘similarity relation’? Of course.

Recall that a fuzzy relation that is reflexive, symmetric, and also transitive is called similarity relation. It helps

you to create similarity classes. If the relation lacks any one of the three properties, it is not a similarity

relation. But if it is only not transitive, meaning it is both reflexive and symmetric, it is still not a similarity

relation, but it is a resemblance relation. An example of a resemblance relation, call it t, is given by the

following matrix.

Let the domain have elements a, b, and c:

               1     0.4   0.8

t =  0.4   1     0.5

               0.8   0.5   1

This fuzzy relation is clearly reflexive, and symmetric, but it is not transitive. For example:

      min (m

t

(a, c) , m

t

(c, b) ) = min (0.8, 0.5) = 0.5 ,

but the following:

      m

t

(a, b) = 0.4 < 0.5 ,

is a violation of the condition for transitivity. Therefore, t is not a similarity relation, but it certainly is a

resemblance relation.

C++ Neural Networks and Fuzzy Logic:Preface

Similarity Relations

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