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Fuzzy Inference and Reasoning

Fuzzy Inference and Reasoning. Proposition. Logic variable. Basic connectives for logic variables. (1) Negation (2) Conjunction. Basic connectives for logic variables. (3) Disjunction (4) Implication. Logical function. Logic Formula . Tautology. Tautology. Predicate logic.

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Fuzzy Inference and Reasoning

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  1. Fuzzy Inference and Reasoning

  2. Proposition

  3. Logic variable

  4. Basic connectives for logic variables (1) Negation (2) Conjunction

  5. Basic connectives for logic variables (3) Disjunction (4) Implication

  6. Logical function

  7. Logic Formula

  8. Tautology

  9. Tautology

  10. Predicate logic

  11. Fuzzy Propositions • Assuming that truthandfalsity are expressed by values 1 and 0, respectively, the degree of truth of each fuzzy proposition is expressed by a number in the unit interval [0, 1].

  12. Fuzzy Propositions

  13. p : temperature (V) is high (F).

  14. Fuzzy Propositions p : V is F is S • V is a variable that takes values v from some universal set V • F is a fuzzy set onVthat represents a fuzzy predicate • S is a fuzzy truth qualifier • In general, the degree of truth, T(p), of any truth-qualified proposition p is given for each v e V by the equation T(p) = S(F(v)).

  15. p : Age (V) is very(S) young (F).

  16. Representation of Fuzzy Rule

  17. Representation of Fuzzy Rule

  18. Fuzzy rule as a relation

  19. Fuzzy implications

  20. Example of Fuzzy implications

  21. Example of Fuzzy implications

  22. Example of Fuzzy implications

  23. Representation of Fuzzy Rule Single input and single output Multiple inputs and single output Multiple inputs and Multiple outputs

  24. Representation of Fuzzy Rule Multiple rules

  25. Compositional rule of inference The inference procedure is called as the “compositional rule of inference”. The inference is determined by two factors : “implication operator” and “composition operator”. For the implication, the two operators are often used: For the composition, the two operators are often used:

  26. Representation of Fuzzy Rule Max-min composition operator Mamdani: min operator for the implication Larsen: product operator for the implication

  27. One singleton input and one fuzzy output Mamdani

  28. One singleton input and one fuzzy output Mamdani

  29. One singleton input and one fuzzy output Larsen

  30. One singleton input and one fuzzy output Larsen

  31. One fuzzy input and one fuzzy output Mamdani

  32. One fuzzy input and one fuzzy output Mamdani

  33. Ri consists of R1 and R2

  34. Example

  35. Two singleton inputs and one fuzzy output Mamdani

  36. Two singleton inputs and one fuzzy output Mamdani

  37. Example

  38. Two fuzzy inputs and one fuzzy output Mamdani

  39. Two fuzzy inputs and one fuzzy output Mamdani

  40. Two fuzzy inputs and one fuzzy output Mamdani

  41. Example

  42. Multiple rules

  43. Multiple rules

  44. Multiple rules

  45. Example

  46. Mamdani method

  47. Mamdani method

  48. Mamdani method

  49. Mamdani method

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