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

9. Tautology

10. Tautology

11. Predicate logic

12. Fuzzy Propositions
Assuming that truthand falsity 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].

13. Fuzzy Propositions

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

15. 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 onV that 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)).

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

17. Representation of Fuzzy Rule

18. Representation of Fuzzy Rule

19. Fuzzy rule as a relation

20. Fuzzy implications

21. Example of Fuzzy implications

22. Example of Fuzzy implications

23. Example of Fuzzy implications

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

25. Representation of Fuzzy Rule
Multiple rules

26. 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:

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

28. One singleton input and one fuzzy output
Mamdani

29. One singleton input and one fuzzy output
Mamdani

30. One singleton input and one fuzzy output
Larsen

31. One singleton input and one fuzzy output
Larsen

32. One fuzzy input and one fuzzy output
Mamdani

33. One fuzzy input and one fuzzy output
Mamdani

34. Ri consists of R1 and R2

35. Example

36. Two singleton inputs and one fuzzy output
Mamdani

37. Two singleton inputs and one fuzzy output
Mamdani

38. Example

39. Two fuzzy inputs and one fuzzy output
Mamdani

40. Two fuzzy inputs and one fuzzy output
Mamdani

41. Two fuzzy inputs and one fuzzy output
Mamdani

42. Example

43. Multiple rules

44. Multiple rules

45. Multiple rules

46. Example

47. Mamdani method

48. Mamdani method

49. Mamdani method

50. Mamdani method

51. Larsen method

52. Larsen method

53. Larsen method

54. Larsen method

55. Fuzzy Logic Controller

56. Inference

57. Inference

58. Inference

59. Inference

60. Defuzzification
Mean of Maximum Method (MOM)

61. Defuzzification
Center of Area Method (COA)

62. Defuzzification
Bisector of Area (BOA)