1. Introduction to Fuzzy Set Theory
主講人: 虞台文

2. Content Fuzzy Sets Set-Theoretic Operations MF Formulation
Extension Principle
Fuzzy Relations
Linguistic Variables
Fuzzy Rules
Fuzzy Reasoning

3. Introduction to Fuzzy Set Theory
Fuzzy Sets

4. Types of Uncertainty Stochastic uncertainty Linguistic uncertainty
E.g., rolling a dice
Linguistic uncertainty
E.g., low price, tall people, young age
Informational uncertainty
E.g., credit worthiness, honesty

5. Crisp or Fuzzy Logic Crisp Logic Fuzzy Logic
A proposition can be true or false only.
Bob is a student (true)
Smoking is healthy (false)
The degree of truth is 0 or 1.
Fuzzy Logic
The degree of truth is between 0 and 1.
William is young (0.3 truth)
Ariel is smart (0.9 truth)

6. Crisp Sets Classical sets are called crisp sets
either an element belongs to a set or not, i.e.,
Member Function of crisp set or

7. Crisp Sets P Y P : the set of all people.
Y : the set of all young people. Y 1 y 25

8. Crisp sets
Fuzzy Sets
Example 1 y

9. Fuzzy Sets L. A. Zadeh, “Fuzzy sets,” Information and Control,
Lotfi A. Zadeh, The founder of fuzzy logic.
Fuzzy Sets
L. A. Zadeh, “Fuzzy sets,” Information and Control,
vol. 8, pp , 1965.

10. Definition: Fuzzy Sets and Membership Functions
U : universe of discourse.
Definition: Fuzzy Sets and Membership Functions
If U is a collection of objects denoted generically by x, then a fuzzy set A in U is defined as a set of ordered pairs:
membership
function

11. Example (Discrete Universe)
# courses a student may take in a semester.
appropriate
# courses taken
0.5 1 2 4 6 8
x : # courses

12. Example (Discrete Universe)
# courses a student may take in a semester.
appropriate
# courses taken
Alternative Representation:

13. Example (Continuous Universe)
U : the set of positive real numbers
possible ages
about 50 years old
x : age
Alternative Representation:

14. Alternative Notation U : discrete universe U : continuous universe
Note that  and integral signs stand for the union of membership grades; “ / ” stands for a marker and does not imply division.

15. Membership Functions (MF’s)
A fuzzy set is completely characterized by a membership function.
a subjective measure.
not a probability measure.
Membership
value
height 1
“tall” in Asia
5’10”
“tall” in USA
“tall” in NBA

16. Fuzzy Partition
Fuzzy partitions formed by the linguistic values “young”, “middle aged”, and “old”:

17. MF Terminology
cross points x 1
0.5 MF
core
width 
-cut
support

18. More Terminologies Normality Fuzzy singleton Fuzzy numbers
core non-empty
Fuzzy singleton
support one single point
Fuzzy numbers
fuzzy set on real line R that satisfies convexity and normality
Symmetricity
Open left or right, closed

19. Convexity of Fuzzy Sets
A fuzzy set A is convex if for any  in [0, 1].

20. Introduction to Fuzzy Set Theory
Set-Theoretic Operations

21. Set-Theoretic Operations
Subset
Complement
Union
Intersection

22. Set-Theoretic Operations

23. Properties Involution De Morgan’s laws Commutativity Associativity
Distributivity
Idempotence
Absorption

24. Properties The following properties are invalid for fuzzy sets:
The laws of contradiction
The laws of exclude middle

25. Other Definitions for Set Operations
Union
Intersection

26. Other Definitions for Set Operations
Union
Intersection

27. Generalized Union/Intersection
Generalized Intersection
Generalized Union
t-norm
t-conorm

28. T-Norm Or called triangular norm. Symmetry Associativity Monotonicity
Border Condition

29. T-Conorm Or called s-norm. Symmetry Associativity Monotonicity
Border Condition

30. Examples: T-Norm & T-Conorm
Minimum/Maximum:
Lukasiewicz:
Probabilistic:

31. Introduction to Fuzzy Set Theory
MF Formulation

32. MF Formulation Triangular MF Trapezoidal MF Gaussian MF
Generalized bell MF

33. MF Formulation

34. Manipulating Parameter of the Generalized Bell Function

35. Sigmoid MF Extensions: Abs. difference of two sig. MF Product

36. L-R MF
Example:
c=65
=60
=10
c=25
=10
=40

37. Introduction to Fuzzy Set Theory
Extension Principle

38. Functions Applied to Crisp Setsy
B(y) x
y = f(x) B A x
A(x)

39. Functions Applied to Fuzzy Setsx
y = f(x) B
B(y)
A(x) A x

40. Functions Applied to Fuzzy Setsx
y = f(x) B
B(y)
A(x) A x

41. The Extension Principle
Assume a fuzzy set A and a function f.
How does the fuzzy set f(A) look like?
The Extension Principle y x
y = f(x) B
B(y)
A(x) A x

42. The Extension Principle
Assume a fuzzy set A and a function f.
How does the fuzzy set f(A) look like?
The Extension Principle y x
y = f(x) B
B(y)
A(x) A x

43. The Extension Principle
fuzzy sets
defined on
The extension of f operating on A1, …, An gives a fuzzy set F with membership function

44. Introduction to Fuzzy Set Theory
Fuzzy Relations

45. Binary Relation (R) A a1 a2 a3 a4 B b1 b2 b3 b4 b5

46. Binary Relation (R) A a1 a2 a3 a4 B b1 b2 b3 b4 b5

47. The Real-Life Relation
x is close to y
x and y are numbers
x depends on y
x and y are events
x and y look alike
x and y are persons or objects
If x is large, then y is small
x is an observed reading and y is a corresponding action

48. Fuzzy Relations
A fuzzy relation R is a 2D MF:

49. Example (Approximate Equal)

50. Max-Min Composition X Y Z R: fuzzy relation defined on X and Y.
S: fuzzy relation defined on Y and Z.
R。S: the composition of R and S.
A fuzzy relation defined on X an Z.

51. Example
min
max

52. Max-Product Composition
Max-min composition is not mathematically tractable, therefore other compositions such as max-product composition have been suggested.
Max-Product Composition X Y Z
R: fuzzy relation defined on X and Y.
S: fuzzy relation defined on Y and Z.
R。S: the composition of R and S.
A fuzzy relation defined on X an Z.

53. Dimension Reduction
Projection R

54. Dimension Reduction
Projection

55. Cylindrical Extension
Dimension Expansion
Cylindrical Extension
A : a fuzzy set in X.
C(A) = [AXY] : cylindrical extension of A.

56. Introduction to Fuzzy Set Theory
Linguistic Variables

57. Linguistic Variables
Linguistic variable is “a variable whose values are words or sentences in a natural or artificial language”.
Each linguistic variable may be assigned one or more linguistic values, which are in turn connected to a numeric value through the mechanism of membership functions.

58. Motivation
Conventional techniques for system analysis are intrinsically unsuited for dealing with systems based on human judgment, perception & emotion.

59. Example
if temperature is cold and oil is cheap then heating is high

60. Example if temperature is cold and oil is cheap then heating is high
Linguistic
Variable
cold
if temperature is cold and oil is cheap then heating is high
Linguistic
Value
Linguistic
Value
Linguistic
Variable
cheap
high
Linguistic
Variable
Linguistic
Value

61. Definition [Zadeh 1973]
A linguistic variable is characterized by a quintuple
Universe
Term Set
Name
Syntactic Rule
Semantic Rule

62. Example A linguistic variable is characterized by a quintuple age
[0, 100]
Example semantic rule:

63. Example (x) cold warm hot x Linguistic Variable : temperature
Linguistics Terms (Fuzzy Sets) : {cold, warm, hot}
(x)
cold
warm
hot 20 60 1 x

64. Introduction to Fuzzy Set Theory
Fuzzy Rules

65. Classical ImplicationB
A  B T F A B
A  B 1
A  B
A  B A B
A  B T F A B
A  B 1

66. Classical ImplicationB
A  B 1
A  B
A  B A B
A  B 1

67. Modus Ponens A  B A  B If A then B  A  A  A is true B B1
Modus Ponens
A  B
A  B
If A then B  A  A 
A is true B B
B is true

68. Fuzzy If-Than Rules A  B  If x is A then y is B. antecedent or
premise
consequence or
conclusion

69. Examples A  B  If x is A then y is B.
If pressure is high, then volume is small.
If the road is slippery, then driving is dangerous.
If a tomato is red, then it is ripe.
If the speed is high, then apply the brake a little.

70. Fuzzy Rules as Relations
A  B R 
If x is A then y is B.
Depends on how
to interpret A  B
A fuzzy rule can be defined as a binary relation with MF

71. Interpretations of A  B
A coupled with B A B x y A B
A entails B x y

72. Interpretations of A  B
A coupled with B A B x y
A coupled with B (A and B) A B
A entails B x y
t-norm

73. Interpretations of A  B
A coupled with B A B x y
A coupled with B (A and B) A B
A entails B x y
E.g.,

74. Interpretations of A  B
A entails B (not A or B)
Material implication
Propositional calculus
Extended propositional calculus
Generalization of modus ponens
A coupled with B A B x y A B
A entails B x y

75. Interpretations of A  B
A entails B (not A or B)
Material implication
Propositional calculus
Extended propositional calculus
Generalization of modus ponens

76. Introduction to Fuzzy Set Theory
Fuzzy Reasoning

77. Generalized Modus Ponens
Single rule with single antecedent
Rule:
if x is A then y is B
Fact:
x is A’
Conclusion:
y is B’

78. Fuzzy Reasoning Single Rule with Single Antecedentx A A’ B
B’ = ?

79. Fuzzy Reasoning Single Rule with Single Antecedent
Max-Min Composition
Firing Strength
Firing Strength x A A’ B

80. Fuzzy Reasoning Single Rule with Single Antecedent
Max-Min Composition x A A’ B

81. Fuzzy Reasoning Single Rule with Multiple Antecedents
if x is A and y is B then z is C
Fact:
x is A and y is B
Conclusion:
z is C

82. Fuzzy Reasoning Single Rule with Multiple Antecedents
if x is A and y is B then z is C
Fact:
x is A’ and y is B’
Conclusion:
z is C’ x A B C A’ B’
C’ = ?

83. Fuzzy Reasoning Single Rule with Multiple Antecedents
Max-Min Composition
Firing Strength B A A’ B’ C x

84. Fuzzy Reasoning Single Rule with Multiple Antecedents
Max-Min Composition
Firing Strength B A A’ B’ C x

85. Fuzzy Reasoning Multiple Rules with Multiple Antecedents
if x is A1 and y is B1 then z is C1
Rule2:
if x is A2 and y is B2 then z is C2
Fact:
x is A’ and y is B’
Conclusion:
z is C’

86. Fuzzy Reasoning Multiple Rules with Multiple Antecedentsx A1 y B1 A’ B’ x A2 y B2 z C2 A’ B’
C’ = ?

87. Fuzzy Reasoning Multiple Rules with Multiple Antecedents
Max-Min Composition z C1 x A1 y B1 A’ B’ x A2 y B2 z C2 A’ B’
Max z