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1. Course Name: Fuzzy Logic and Neural Networks Faculty Name: Prof. Dilip Kumar Pratihar Department : Mechanical Engineering Week 1
2. Course Name: Fuzzy Logic and Neural Networks Faculty Name: Prof. Dilip Kumar Pratihar Department : Mechanical Engineering Topic Lecture 01: Introduction to Fuzzy Sets
3. Concepts Covered:  Classical Set/Crisp Set  Properties of Classical Set/Crisp Set  Fuzzy Set  Representation of Fuzzy Set
4. • Universal Set/Universe of Discourse (X): A set consisting of all possible elements Ex: All technical universities in the world • Classical or Crisp Set is a set with fixed and well-defined boundary • Example: A set of technical universities having at least five departments each Classical Set/Crisp Set (A)
5. Representation of Crisp Sets • A={a1,a2,……,an} • A={x|P(x)}, P: property • Using characteristic function μA(x)= 1, if x belongs to A, 0, if x does not belong to A.
6. Notations Used in Set Theory • : Empty/Null set • : Element x of the Universal set X belongs to set A • : x does not belong to set A • : set A is a subset of set B • : set A is a superset of set B • : A and B are equal • : A and B are not equal Φ A x ∈ x A ∉ A B ⊆ A B ⊇ A B = A B ≠
7. • : A is a proper subset of B • : A is a proper superset of B • : Cardinality of set A is defined as the total number of elements present in that set • p(A) : Power set of A is the maximum number of subsets including the null that can be constructed from a set A Note: ( ) p A A = 2 A B ⊂ A B ⊃ A
8. Crisp Set Operations
9. Properties of Crisp Sets
10. Fuzzy Sets • Sets with imprecise/vague boundaries • Introduced by Prof. L.A. Zadeh, University of California, USA, in 1965 • Potential tool for handling imprecision and uncertainties • Fuzzy set is a more general concept of the classical set
11. Representation of a Fuzzy Set ( ) ( ) ( ) { } X x x x x A A ∈ µ = , , Note: Probability: Frequency of likelihood that an element is in a class Membership: Similarity of an element to a class
12. Types of Fuzzy sets 1. Discrete Fuzzy set n: Number of elements present in the set ( ) ( ) , / 1 ∑ = µ = n i i i A x x x A ( ) ( ) ∫µ = X A x x x A / 2. Continuous Fuzzy set
13. Convex vs. Non-Convex Membership Function Distribution A fuzzy set A(x) will be convex, if Where 0.0 ≤ λ ≤ 1.0 ( ) { } ( ) ( ) { } 2 1 2 1 , min 1 x x x x A A A µ µ ≥ λ − + λ µ
14. Various Types of Membership Function Distributions 1. Triangular Membership               − − − − = µ 0 , b c x c , a b a x min max triangle
15. 2. Trapezoidal Membership max min ,1, ,0 trapezoidal x a d x b a d c µ   − −   =     − −    
16. 3. Gaussian Membership 2 2 1 1       σ − = µ m x Gaussian e
17. 4. Bell-shaped Membership Function b shaped Bell a c x 2 1 1 − + = µ −
18. 5. Sigmoid Membership ( ) b x a Sigmoid e − − + = µ 1 1
19. Reference: Pratihar D.K.: Soft Computing: Fundamentals and Applications, Narosa Publishing House, New-Delhi, 2014
20. Conclusion: Classical Set/Crisp Set has been defined Properties of Classical Set/Crisp Set has been explained Fuzzy Set has been defined Deals with representation of Fuzzy Set
21. Course Name: FUZZY LOGIC AND NUERAL NETWORKS Faculty Name: Prof. Dilip Kumar Pratihar Department: Mechanical Engineering, IIT Kharagpur Topic Lecture 02: Introduction to Fuzzy Sets (contd.)
22. Concepts Covered:  A few terms of Fuzzy Sets  Standard Operations in Fuzzy Sets  Properties of Fuzzy Sets  Fuzziness and Inaccuracy of Fuzzy Sets
23. Numerical Example Triangular Membership: Determine μ, corresponding to x=8.0 1.0 μ 0.0 a=2 b=6 c=10 x 8
24. We put, x=8.0
25. Trapezoidal Membership •Determine μ corresponding to x = 3.5
26.             − − − − = 0 , 8 10 10 , 1 , 2 4 2 min max x x             − − = 0 , 2 10 , 1 , 2 2 min max x x             − − − − = 0 , , 1 , min max c d x d a b a x
27. •We put x = 3.5             = 0 , 2 5 . 6 , 1 , 2 5 . 1 min max [ ] 0 , 75 . 0 max = 75 . 0 =
28. 1.0 0.0
29. 1.0 μ 0.0 c x
30. Take c=10.0, a=2.0, b=3.0 We put x=8.0
31. Sigmoid Membership Function: Determine µ corresponding to x = 8.0 2 6 0 2 2 0 4 1 1 1 1 1 1 0 98 1 1 − − − − − × − + + = = + + µ = µ = µ = Take b = 6.0; a = 2 we put x = 8.0 sigmoid a(x b) sigmoid (x . ) sigmoid . e e . e e
32. Difference Between Crisp and Fuzzy Sets
33. A Few Definitions in Fuzzy Sets
34. • Strong α-cut of a Fuzzy Set
35. The membership function distribution of a fuzzy set is assumed to follow a Gaussian distribution with mean m = 100 and standard deviation σ =20 . Determine 0.6 – cut of this distribution. Solution: Gaussian distribution : where m : Mean ; σ : Standard deviation By substituting the values of µ = 0.6, m = 100, σ =20 and taking log (ln) on both sides, we get Numerical Example 2 1 2 1   −   σ   µ = x m e
36. ( ) 2 2 2 1 100 2 20 1 100 2 20 1 100 2 20 1 1 0 6 1 6667 79 7846 120 2153 0 6 By taking ln x x x e e . ln e ln . x ( . , . ) .   −       −       −     ⇒ =     =     ⇒ = = Figure : 0.6-cut of a fuzzy set.
37. • Support of a Fuzzy Set A(x) • Scalar Cardinality of a Fuzzy Set A(x)
38. ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) 1 2 3 4 0 1 0 2 0 3 0 4 0 1 0 2 0 3 0 4 1 0 = = + + + = Let us consider a fu S zzy set A x as follow calar Cardinality s: A x x , . , x , . , x , . , x , . A x . . . . . Numerical Example
39. • Core of a Fuzzy Set A(x) It is nothing but its 1-cut • Height of a Fuzzy Set A(x) It is defined as the largest of membership values of the elements contained in that set.
40. • Normal Fuzzy Set For a normal fuzzy set, h(A) = 1.0 • Sub-normal Fuzzy Set For a sub-normal fuzzy set, h(A) < 1.0
41. Some Standard Operations in Fuzzy Sets • Proper Subset of a Fuzzy Set
42. Numerical Example ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4 0 1 0 2 0 3 0 4 0 5 0 7 0 8 0 9 µ µ = = ∈ < ⊂ Let us consider the two fuzzy As for al sets, as l that is , is the prop follows: er subset of A B A x x , . , x , . , x , . , x , . B x x , . , x , . , x , . , x , . x X, x x , A x B x , A x B x
43. Some Standard Operations in Fuzzy Sets (contd.) • Equal fuzzy sets
44. Numerical Example ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) 1 2 3 4 1 2 3 4 0 1 0 2 0 3 0 4 0 5 0 7 0 8 0 9 µ µ = = ∈ ≠ ≠ As for a Let us consider the two fuzzy sets, as follows ll : A B A x x , . , x , . , x , . , x , . B x x , . , x , . , x , . , x , . x X, x x , A x B x
45. • Complement of a Fuzzy Set
46. Numerical Example ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) { } 1 2 3 4 1 2 3 4 0 1 0 2 0 3 0 4 0 9 0 8 0 7 0 6 = = Let us consider a fuzzy set A x as follo Complement ws: A x x , . , x , . , x , . , x , . A x x , . , x , . , x , . , x , .
47. • Intersection of Fuzzy Sets
48. Note: Intersection is analogous to logical AND operation x
49. Numerical Example ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) { } { } ( ) ( ) { } 1 2 3 4 1 2 3 4 1 1 1 2 0 1 0 2 0 3 0 4 0 5 0 7 0 8 0 9 0 1 0 5 0 1 0 2 0 7 0 2 µ µ µ µ µ = = = = = = =   Now, Let us consider the two fuzzy s Similarly, ets as follo s w : A B A B A B A A x x , . , x , . , x , . , x , . B x x , . , x , . , x , . , x , . x min x , x min . , . . x min . , . . ( ) ( ) { } ( ) ( ) { } 3 4 0 3 0 8 0 3 0 4 0 9 0 4 µ = = = =   B A B x min . , . . x min . , . .
50. • Union of Fuzzy Sets
51. Note: Union is analogous to logical OR operation
52. Numerical Example ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) { } { } ( ) ( ) 1 2 3 4 1 2 3 4 1 1 1 2 0 1 0 2 0 3 0 4 0 5 0 7 0 8 0 9 0 1 0 5 0 5 µ µ µ µ = = = = = =   Let us consider the following two fuzzy s Now, Similarly, ets: A B A B A B A x x , . , x , . , x , . , x , . B x x , . , x , . , x , . , x , . x max x , x max . , . . x max{ } ( ) ( ) { } ( ) ( ) { } 3 4 0 2 0 7 0 7 0 3 0 8 0 8 0 4 0 9 0 9 µ µ = = = = =   A B A B . , . . x max . , . . x max . , . .
53. • Algebraic product of Fuzzy Sets
54. Numerical Example ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) ( ) { } 1 2 3 4 1 2 3 4 1 2 3 4 0 1 0 2 0 3 0 4 0 5 0 7 0 8 0 9 0 05 0 14 0 24 0 36 = = = Let us consider the following two fuzzy sets: A x x , . , x , . , x , . , x , . B x x , . , x , . , x , . , x , . A x B x x , . , x , . , x , . , x , . .
55. • Multiplication of a Fuzzy Set by a Crisp Number
56. Numerical Example ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) { } 1 2 3 4 1 2 3 4 0 1 0 2 0 3 0 4 0 2 0 02 0 04 0 06 0 08 = = and a crisp number Let us consider a fuzzy set A x x , . , x , . , x , . , x , . d . d A x x , . , x , . , x , . , x , . .
57. • Power of a Fuzzy Set AP(x): p-th power of a fuzzy set A(x) such that Concentration: p=2 Dilation: p=1/2
58. Numerical Example ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) { } 1 2 3 4 2 1 2 3 4 0 1 0 2 0 3 0 4 2 0 01 0 04 0 09 0 16 and power Let us consider a fuzzy set A x x , . , x , . , x , . , x , . p A x x , . , x , . , x , . , x , . = =
59. • Algebraic Sum of two Fuzzy Sets A(x) and B(x) where
60. Numerical Example ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) ( ) { } 1 2 3 4 1 2 3 4 1 2 3 4 0 1 0 2 0 3 0 4 0 5 0 7 0 8 0 9 0 55 0 76 0 86 0 94 = = ∴ + = Let us consider the following two fuzzy sets: A x x , . , x , . , x , . , x , . B x x , . , x , . , x , . , x , . A x B x x , . , x , . , x , . , x , .
61. • Bounded Sum of two Fuzzy Sets where
62. Numerical Example ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) ( ) { } 1 2 3 4 1 2 3 4 1 2 3 4 0 1 0 2 0 3 0 4 0 5 0 7 0 8 0 9 0 6 0 9 1 0 1 0 = = ∴ ⊕ = Let us consider the following two fuzzy sets: A x x , . , x , . , x , . , x , . B x x , . , x , . , x , . , x , . A x B x x , . , x , . , x , . , x , .
63. • Algebraic Difference of two Fuzzy Sets where
64. Numerical Example •Let us consider the following two fuzzy sets: 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 ( ) {( ,0.1),( ,0.2),( ,0.3),( ,0.4)} ( ) {( ,0.5),( ,0.7),( ,0.8),( ,0.9)} , ( ) {( ,0.5),( ,0.3),( ,0.2),( ,0.1)} ( ) ( ) {( ,0.1),( ,0.2),( ,0.2),( ,0.1)} A x x x x x B x x x x x Now B x x x x x A x B x x x x x = = = ∴ − =
65. • Bounded Difference of two Fuzzy Sets where
66. Numerical Example •Let us consider the following two fuzzy sets: 1 2 3 4 1 2 3 4 1 2 3 4 ( ) {( ,0.1),( ,0.2),( ,0.3),( ,0.4)} ( ) {( ,0.5),( ,0.7),( ,0.8),( ,0.9)} ( ) ( ) {( ,0.0),( ,0.0),( ,0.1),( ,0.3)} A x x x x x B x x x x x A x B x x x x x = = Θ =
67. • Cartesian product of two Fuzzy Sets Two fuzzy sets A(x) defined in X and B(y) defined in Y Cartesian product of two fuzzy sets is denoted by A(x)×B(y), such that
68. Numerical Example •Let us consider the following two fuzzy sets: 1 2 3 4 1 2 3 1 1 1 2 ( ) {( ,0.2),( ,0.3),( ,0.5),( ,0.6)} ( ) {( ,0.8),( ,0.6),( ,0.3)} min( ( ), ( )) min(0.2,0.8) 0.2 min( ( ), ( )) min(0.2,0.6) 0.2 A B A B A x x x x x B y y y y x y x y µ µ µ µ = = = = = =
69. 1 3 2 1 2 2 2 3 min( ( ), ( )) min(0.2,0.3) 0.2 min( ( ), ( )) min(0.3,0.8) 0.3 min( ( ), ( )) min(0.3,0.6) 0.3 min( ( ), ( )) min(0.3,0.3) 0.3 A B A B A B A B x y x y x y x y µ µ µ µ µ µ µ µ = = = = = = = =
70. 3 1 3 2 3 3 4 1 min( ( ), ( )) min(0.5,0.8) 0.5 min( ( ), ( )) min(0.5,0.6) 0.5 min( ( ), ( )) min(0.5,0.3) 0.3 min( ( ), ( )) min(0.6,0.8) 0.6 A B A B A B A B x y x y x y x y µ µ µ µ µ µ µ µ = = = = = = = =
71. 4 2 4 3 min( ( ), ( )) min(0.6,0.6) 0.6 min( ( ), ( )) min(0.6,0.3) 0.3 0.2 0.2 0.2 0.3 0.3 0.3 0.5 0.5 0.3 0.6 0.6 0.3 A B A B x y x y A B µ µ µ µ = = = =       ∴ × =      
72. Composition of fuzzy relations Let A = [aij] and B = [bjk] be two fuzzy relations expressed in the matrix form. Composition of these two fuzzy relations, that is, C is represented as follows: C=A о B In matrix form [cik] = [aij] о [bjk] Where cik =max[min(aij, bjk)]
73. Numerical Example •Let us consider the following two Fuzzy relations: [ ] [ ] [ ] ik jk ij c b B a A       = =       = = 6 . 0 7 . 0 8 . 0 6 . 0 1 . 0 3 . 0 7 . 0 5 . 0 3 . 0 2 . 0 •Elements of matrix can be determined as follows:
74. [ ] [ ] [ ] 2 . 0 1 . 0 , 2 . 0 max ) 1 . 0 , 3 . 0 min( ), 3 . 0 , 2 . 0 min( max ) , min( ), , min( max 21 12 11 11 11 = = = = b a b a c
75. [ ] [ ] [ ] 3 . 0 3 . 0 , 2 . 0 max ) 8 . 0 , 3 . 0 min( ), 6 . 0 , 2 . 0 min( max ) , min( ), , min( max 22 12 12 11 12 = = = = b a b a c
76. [ ] [ ] [ ] 3 . 0 3 . 0 , 2 . 0 max ) 6 . 0 , 3 . 0 min( ), 7 . 0 , 2 . 0 min( max ) , min( ), , min( max 23 12 13 11 13 = = = = b a b a c
77. [ ] [ ] [ ] 3 . 0 1 . 0 , 3 . 0 max ) 1 . 0 , 7 . 0 min( ), 3 . 0 , 5 . 0 min( max ) , min( ), , min( max 21 22 11 21 21 = = = = b a b a c
78. [ ] [ ] [ ] 7 . 0 7 . 0 , 5 . 0 max ) 8 . 0 , 7 . 0 min( ), 6 . 0 , 5 . 0 min( max ) , min( ), , min( max 22 22 12 21 22 = = = = b a b a c
79. [ ] [ ] [ ] 6 . 0 6 . 0 , 5 . 0 max ) 6 . 0 , 7 . 0 min( ), 7 . 0 , 5 . 0 min( max ) , min( ), , min( max 23 22 13 21 23 = = = = b a b a c
80.       = ∴ 6 . 0 3 . 0 7 . 0 3 . 0 3 . 0 2 . 0 C
81. Properties of Fuzzy Set Fuzzy sets follow the properties of crisp sets except the following two: • Law of excluded middle