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Chapter 3 FUZZY RELATION AND COMPOSITION

Chapter 3 FUZZY RELATION AND COMPOSITION. G.Anuradha. Outline. Product set Crisp / fuzzy relations Composition / decomposition Projection / cylindrical extension Extension of fuzzy set / fuzzy relation. Product set. Product set. Product set. A={a1,a2} B={b1,b2} C={c1,c2}

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Chapter 3 FUZZY RELATION AND COMPOSITION

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  1. Chapter 3 FUZZY RELATION AND COMPOSITION G.Anuradha

  2. Outline • Product set • Crisp / fuzzy relations • Composition / decomposition • Projection / cylindrical extension • Extension of fuzzy set / fuzzy relation

  3. Product set

  4. Product set

  5. 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)}

  6. 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 :

  7. 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.

  8. (Crisp) (Fuzzy) Representation methods • Bipartigraph

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

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

  11. Domain and range of fuzzy relation domain range Domain: Range :

  12. Domain and range of fuzzy relation Fuzzy matrix

  13. Operations on fuzzy matrices Sum: Example

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

  15. Max product Example

  16. Max product Example

  17. Max product Example

  18. Operations on fuzzy matrices Scalar product: Example

  19. Operations on fuzzy relations Union relation For n relations

  20. Union relation Example

  21. Operations on fuzzy relations Intersection relation For n relations

  22. Intersection relation Example

  23. Operations on fuzzy relations Complement relation: Example

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

  25. Composition of fuzzy relations

  26. Composition of fuzzy relations • Example

  27. Composition of fuzzy relations • Example

  28. Composition of fuzzy relations

  29. α-cut of fuzzy relation • Example

  30. α-cut of fuzzy relation

  31. Decomposition of relation

  32. Decomposition of relation 0

  33. Decomposition of relation

  34. Projection / cylindrical extension

  35. Projection / cylindrical extension

  36. Projection in n dimension

  37. Projection

  38. Projection

  39. Projection

  40. Projection

  41. Projection / cylindrical extension

  42. Cylindrical extension

  43. Functions with Fuzzy Arguments • A crisp function maps its crisp input argument to its image. • Fuzzy arguments have membership degrees. • When computing a fuzzy mapping it is necessary to compute the image and its membership value.

  44. Crisp Mappings

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