Crisp and Fuzzy Relations

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Crisp and Fuzzy Relations

Crisp Relation

A crisp relation is used to represents the presence or absence of interaction, association, or interconnectedness between the elements of more than a set. This crisp relational concept can be generalized to allow for various degrees or strengths of relation or interaction between elements.

Operations on Crisp Relations

Let A and B be two relations defined on X x Y and are represented by relational matrices. The following operations can be performed on these relations A and B
Union
A ∪ B (x,y) = max [ A (x,y) , B (x,y) ]

Intersection
A ∩ B (x,y) = min [ A(x,y) , B (x,y) ]

Fuzzy relation

Degrees of association can be represented by grades of the membership in a fuzzy relation in the same way as degrees of set membership are represented in the fuzzy set. In fact, just as the crisp set can be viewed as a restricted case of the more general fuzzy set concept, the crisp relation can be considered to be a restricted case of the fuzzy relations.

Cartesian product

The Cartesian product of two crisp sets X and Y, denoted by

Cartesian

, is the crisp set of all ordered pairs such that the first element in each pair is a member of X and the second element is a member of Y. Formally,

Cartesian

Relation among sets

A relation among crisp sets

relations

is a subset of the Cartesian product

is a subset of the Cartesian product

It is denoted either by

relations

or by the abbreviated form

relations

Thus,

relations

so for relations among sets

so for relations among sets

, the Cartesian product represents

product

the universal set. Because a relation is itself a set, the basic set concepts such as containment or subset, union, intersection, and complement can be applied without modification to relations.

Each element of the first dimension i1 of this array corresponds to exactly one member of X1, each element of the first dimension i2 to exactly one member of X2, and so on. If the n-tuple

dimension i1

,then

membership

Crisp and Fuzzy Relations

Thus, a fuzzy relation is a fuzzy set defined on the Cartesian product of crisp sets

Cartesian product of crisp sets

, may have varying degrees of membership within the relation. The membership grade is usually represented by a real number in the closed interval,

0,1

and indicates the strenght of the relation present between the elements of the tuple.

A fuzzy relation can also conveniently be represented by an n-dimensional membership array whose entries correspond to n-tuples in the universal set. These entries take values representing the membership grades of the corresponding n-tuples.

Examples

Let Q be a crisp relation among the two sets A={dollar, pound, franc, mark} and Y={United States, France, Canada, Britain, Germany}, which associates a country with a currency as follows:

(pound,Britain),(mark,Germany)}

This relation can also be represented by the following two dimensional membership array:

 U.S. France Canada Britain Germany dollar 1 0 1 0 0 pound 0 0 0 1 0 franc 0 1 0 0 0 mark 0 0 0 0 1

Let R be a fuzzy relation among the two sets the distance to the target X={far, close, very close} and the speed of the car Y={very slow, slow, normal, quick, very quick}, which represents the relational concept “the break must be pressed very strong”.

This relation can be written in list notation as

R(X,Y) = {0/(far, very slow) + .3/(close, very slow) + .8/(very close, very slow)

+ 0/(far, slow) + .4/(close, slow) + .9/(very close, slow)

+ 0/(far, normal) + .5/(close, normal) + 1/(very close, normal)

+ .1/(far, quick) + .6/(close, quick) + 1/(very close, quick)

+ .2/(far,very quick)+ .7/(close,very quick)+ 1/(very close,very quick)}

This relation can also be represented by the following two dimensional membership array:

 very slow slow normal quick very quick far 0 0 0 .1 .2 close .3 .4 .5 .6 .7 very close .8 .9 1 1 1

binary