Chapter 3 FUZZY RELATION AND COMPOSITION Chi-Yuan Yeh

Outline Product set Crisp / fuzzy relations Composition / decomposition Projection / cylindrical extension Extension of fuzzy set / fuzzy relation Fuzzy distance between fuzzy sets

Product set

Product set

Product set A={a1,a2} B={b1,b2} C={c1,c2} AxBxC = {(a1,b1,c1),(a1,b1,c2),(a1,b2,c1),(a1,b2,c2),(a2,b1,c1),(a2,b1,c2),(a2,b2,c1), (a2,b2,c2)}

Crisp relation A relation among crisp sets is a subset of the Cartesian product. It is denoted by . Using the membership function defines the crisp relation R :

Fuzzy relation A fuzzy relation is a fuzzy set defined on the Cartesian product of crisp sets A1, A2, ..., An where tuples (x1, x2, ..., xn) may have varying degrees of membership within the relation. The membership grade indicates the strength of the relation present between the elements of the tuple.

Representation methods Bipartigraph (Crisp) (Fuzzy)

Representation methods Matrix (Crisp) (Fuzzy)

Representation methods Digraph (Crisp) (Fuzzy)

Domain and range of fuzzy relation

Domain and range of fuzzy relation Fuzzy matrix

Operations on fuzzy matrices Sum: Example

Operations on fuzzy matrices Max product: C = A・B=AB= Example

Max product Example

Max product Example

Max product Example

Operations on fuzzy matrices Scalar product: Example

Operations on fuzzy relations Union relation For n relations

Union relation Example

Operations on fuzzy relations Intersection relation For n relations

Intersection relation Example

Operations on fuzzy relations Complement relation: Example

Composition of fuzzy relations Max-min composition Example

Composition of fuzzy relations

Composition of fuzzy relations Example

Composition of fuzzy relations Example

Composition of fuzzy relations

α-cut of fuzzy relation Example

α-cut of fuzzy relation

Decomposition of relation

Decomposition of relation

Decomposition of relation

Projection / cylindrical extension

Projection / cylindrical extension

Projection in n dimension

Projection

Projection

Projection

Projection

Projection / cylindrical extension

Cylindrical extension

Cylindrical extension

Cylindrical extension x1 = 0 : x，x1 = 1 : y x2 = 0 : a， x2 = 1 : b x3 = 0 : α， x3 = 1 : β

Cylindrical extension Join(R123’，R123’’) = C(R123’)∩C(R123’’) = Min(R123’，R123’’) = R123’’’

Extension of fuzzy set A crisp function Let then

Extension of fuzzy set There are two universal sets And We can obtain B by A and R, use

Extension of fuzzy set By

Extension of fuzzy set If A is a fuzzy set and R is We can also get B by A an R, use

Extension of fuzzy set By use

Extension of fuzzy set If A is a fuzzy set and R is a fuzzy relation We can get B by using

Extension of fuzzy set By

Extension of fuzzy set Extension of a crisp relation

Extension of fuzzy set

Extension by fuzzy relation

Extension by fuzzy relation

Extension by fuzzy relation

Extension by fuzzy relation

Extension by fuzzy relation

Extension by fuzzy relation

Fuzzy distance between fuzzy sets nonnegative

Fuzzy distance between fuzzy sets

Fuzzy distance between fuzzy sets

Fuzzy distance between fuzzy sets

Fuzzy distance between fuzzy sets

Thanks for your attention!