
Outline • Fuzzy Set and Crisp Set • Expanding concepts • Standard operation of fuzzy set • Fuzzy relations • Operations on fuzzy relations
Set • Membership Function • Universal Set • user specify (crisp set)
Crisp set and Fuzzy set • Crisp set • membership function • membership degree: {0,1} • Fuzzy set • membership function: user specify • membership degree: [0,1]
Crisp set and Fuzzy set • Example • Universal set • Crisp Set • Fuzzy Set
Support • Support of A • example
Height • The maximum value of the membership degree
Normalized fuzzy set • Normalized fuzzy set • height is 1 • young, adult, and senior are normalized fuzzy sets
-cut • -cut set • Example • If , 5
Level set • Level set • Example
Relation of fuzzy sets • A and B are equivalent • A is a subset of B • A is a proper subset of B
Standard operation of fuzzy set • Complement • Union • Intersection
Complement • Example
Union • Example
Intersection • Example
Other operations • Disjunctive sum (exclusive OR)
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 • Afuzzy relation is a fuzzy setdefined on the Cartesian product of crisp sets A1, A2, ..., Anwhere tuples (x1, x2, ..., xn)may have varying degrees of membership within the relation. • The membership gradeindicates the strength of the relation present between the elements of the tuple.
(Crisp) (Fuzzy) Representation methods • Matrix
(Crisp) (Fuzzy) Representation methods • Digraph
α-cut of fuzzy relation • Example
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 • Example
Composition of fuzzy relations • Example