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Fuzzy Set and Opertion

Fuzzy Set and Opertion

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Fuzzy Set and Opertion

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  1. Fuzzy Set and Opertion

  2. Outline • Fuzzy Set and Crisp Set • Expanding concepts • Standard operation of fuzzy set • Fuzzy relations • Operations on fuzzy relations

  3. Set • Membership Function • Universal Set • user specify (crisp set)

  4. Crisp set and Fuzzy set • Crisp set • membership function • membership degree: {0,1} • Fuzzy set • membership function: user specify • membership degree: [0,1]

  5. Crisp set and Fuzzy set • Example • Universal set • Crisp Set • Fuzzy Set

  6. Expression of fuzzy set

  7. Example of fuzzy set

  8. Support • Support of A • example

  9. Height • The maximum value of the membership degree

  10. Normalized fuzzy set • Normalized fuzzy set • height is 1 • young, adult, and senior are normalized fuzzy sets

  11. -cut • -cut set • Example • If , 5

  12. -cut

  13. Level set • Level set • Example

  14. Convex fuzzy set

  15. Non-convex fuzzy set

  16. Relation of fuzzy sets • A and B are equivalent • A is a subset of B • A is a proper subset of B

  17. Subset of fuzzy set

  18. Standard operation of fuzzy set • Complement • Union • Intersection

  19. Complement • Example

  20. Union • Example

  21. Intersection • Example

  22. Other operations • Disjunctive sum (exclusive OR)

  23. Other operations

  24. Other operations

  25. Product set

  26. Product set

  27. 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)}

  28. 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 :

  29. 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.

  30. (Crisp) (Fuzzy) Representation methods • Matrix

  31. (Crisp) (Fuzzy) Representation methods • Digraph

  32. α-cut of fuzzy relation • Example

  33. α-cut of fuzzy relation

  34. Operations on fuzzy matrices Sum: Example

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

  36. Max product Example

  37. Max product Example

  38. Max product Example

  39. Operations on fuzzy matrices Scalar product: Example

  40. Operations on fuzzy relations Union relation For n relations

  41. Union relation Example

  42. Operations on fuzzy relations Intersection relation For n relations

  43. Intersection relation Example

  44. Operations on fuzzy relations Complement relation: Example

  45. Composition of fuzzy relations • Max-min composition • Example

  46. Composition of fuzzy relations

  47. Composition of fuzzy relations • Example

  48. Composition of fuzzy relations • Example

  49. Composition of fuzzy relations