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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Properties of Fuzzy Relations

A relation on a set, that is a subset of a Cartesian product of some set with itself, may have some interesting

properties. It may be reflexive. For this you need to have 1 for the degree of membership of each main

diagonal entry. Our example here is evidently not reflexive.

A relation may be symmetric. For this you need the degrees of membership of each pair of entries

symmetrically situated to the main diagonal to be the same value. For example (Jeff, Mike) and (Mike, Jeff)

should have the same degree of membership. Here they do not, so our example of a relation is not symmetric.

A relation may be antisymmetric. This requires that if a is different from b and the degree of membership of

the ordered pair (a, b) is not 0, then its mirror image, the ordered pair (b, a), should have 0 for degree of

membership. In our example, both (Steve, Mike) and (Mike, Steve) have positive values for degree of

membership; therefore, the relation much_more_educated over the set {Jeff, Steve, Mike} is not

antisymmetric also.

A relation may be transitive. For transitivity of a relation, you need the following condition, illustrated with

our set {Jeff, Steve, Mike}. For brevity, let us use r in place of much_more_educated, the name of the

relation:

     min (m

r

(Jeff, Steve) , m

r

(Steve, Mike) )[le]m

r

(Jeff, Mike)

     min (m

r

(Jeff, Mike) , m

r

(Mike, Steve) )[le]m

r

(Jeff, Steve)

     min (m

r

(Steve, Jeff) , m

r

(Jeff, Mike) )[le]m

r

(Steve, Mike)

     min (m

r

(Steve, Mike) , m

r

(Mike, Jeff) )[le]m

r

(Steve, Jeff)

     min (m

r

(Mike, Jeff) , m

r

(Jeff, Steve) )[le]m

r

(Mike, Steve)

     min (m

r

(Mike, Steve) , m

r

(Steve, Jeff) )[le]m

r

(Mike, Jeff)

In the above listings, the ordered pairs on the left−hand side of an occurrence of [le] are such that the second

member of the first ordered pair matches the first member of the second ordered pair, and also the right−hand

side ordered pair is made up of the two nonmatching elements, in the same order.

In our example,

     min (m

r

(Jeff, Steve) , m

r

(Steve, Mike) ) = min (0.2, 0.3) = 0.2

     m

r

(Jeff, Mike) = 0.7 > 0.2

For this instance, the required condition is met. But in the following:

     min (m

r

(Jeff, Mike), m

r

(Mike, Steve) ) = min (0.7, 0.6) = 0.6

     m

r

(Jeff, Steve) = 0.2 < 0.6

The required condition is violated, so the relation much_more_educated is not transitive.

C++ Neural Networks and Fuzzy Logic:Preface

Properties of Fuzzy Relations

385

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NOTE:  If a condition defining a property of a relation is not met even in one instance, the

relation does not possess that property. Therefore, the relation in our example is not reflexive,

not symmetric, not even antisymmetric, and not transitive.

If you think about it, it should be clear that when a relation on a set of more than one element is symmetric, it

cannot be antisymmetric also, and vice versa. But a relation can be both not symmetric and not antisymmetric

at the same time, as in our example.

An example of reflexive, symmetric, and transitive relation is given by the following matrix:

          1     0.4   0.8

          0.4   1     0.4

          0.8   0.4   1