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Fuzzy Set and Opertion

Fuzzy Set and Opertion. 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

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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

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