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Chapter 1 Fuzzy Sets

Chapter 1 Fuzzy Sets. 1.1 Sets. 1.1.1 Elements of sets X : an universal set A : a set A in the universal set X ( A  X ) x : an element x is included in the set A ( x  A ) the elements of set A A = { a 1 , a 2 ,  , a n }

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Chapter 1 Fuzzy Sets

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  1. Chapter 1Fuzzy Sets

  2. 1.1 Sets • 1.1.1 Elements of sets • X : an universal set • A : a set A in the universal set X ( A  X ) • x : an element x is included in the set A ( x  A ) • the elements of set AA = {a1, a2, , an} • elements of set B should satisfy the conditions P1, P2, , Pn B = {b | b satisfies p1, p 2, , pn }

  3. 1.1 Sets • 1.1.2 Relation between sets • Family of sets {Ai | i I } • A  B iff (if and only if) x  A  x  B • If A  B and B  A then A = B • A  B and A  B then A  B (A is called a proper subset of B) • empty set : a set that has no element

  4. 1.1 Sets • 1.1.3 Membership Definition(Membership Function A) For a set A, we define a membership function A such as A(x) =1 if and only if x  A 0if and only if x  A We can say that the function A maps the elements in the universal set X to the set {0,1}. A: X {0,1}

  5. 1.2 Operation of Sets • ComplementB – A = {x | x  B, xA} • Union AB = {x | x A or x B} • Intersection A  B = {x | x  A and x  B • Partition (A)= {Ai | i  I, Ai A} (1) (2) (3)

  6. (1) Involution (2) Commutativity A  B = B  A A B = B A (3) Associativity (A  B) C = A  (B  C) (A  B) C = A  (B  C) (4) distributivity A  (B  C) = (A  B)  (A  C) A  (B  C) = (A  B)  (A  C) (5) Idempotency A  A = A A  A = A (6) Absorption A  (A  B) = A A  (A  B) = A (7) Absorption by X and  A  X = X A =  (8) Identity A  = A A  X = A (9) De Morgan’s law (10) Absorption of complement (11) Law of contradiction (12) Law of excluded middle 1.3 Characteristics of Crisp Set • 1.3.1 Ordinary Characteristics

  7. 1.3 Characteristics of Crisp Set • 1.3.2 Convex set Definition The term convex is applicable to a set A in Rn(n-dimensional Euclidian vector space) if the followings are satisfied. 1) Two arbitrary points s and r are defined in A. (N is a set of positive integers) 2) For arbitrary real number between 0 and 1, point t is involved in A where t is

  8. A1 y A2 A3 A6 A4 A5 x 1.3 Characteristics of Crisp Set Fig 1.1 convex sets A1, A2, A3 and non-convex sets A4, A5, A6 in R2 • 1.3.2 Convex set(example)

  9. 1.4 Definition of fuzzy set • 1.4.1 Concept for fuzzy set • Definition (Membership function of fuzzy set) In fuzzy sets, each elements is mapped to [0,1] by membership function. A : X [0, 1] Where [0,1] means real numbers between 0 and 1 (including 0 and 1).

  10. A X a b c d A a b c d A 1 A 1 x a a c c d d b b 0.5 x 1.4 Definition of fuzzy set • Example 1.2 Fig 1.2 Graphical representation of crisp set Fig 1.3 Graphical representation of fuzzy set

  11. 1.4 Definition of fuzzy set • Example 1.3 Consider fuzzy set ‘two or so’. In this instance, universal set X are the positive real numbers. X = {1, 2, 3, 4, 5, 6, } Membership function for A =‘two or so’ in this universal set X is given as follows: A(1) = 0, A(2) = 1,A(3) = 0.5, A(4) = 0… A 1 0.5 2 3 4 1

  12. 1.4.2 Examples of fuzzy set (1) • A= "young" , B="very young" Fig 1.5 Fuzzy sets representing “young” and “very young”

  13. (x) 1 0.5 x -3 -2 0 2 3 -1 1 1.4.2 Examples of fuzzy set (2) • A ={real number near 0} where A(x) = Fig 1.6 membership function of fuzzy set “real number near 0”

  14. (x) 1 0.5 x -3 -2 0 2 3 -1 1 0.25 1.4.2 Examples of fuzzy set (3) • B ={real number very near 0}, B(x) = Fig 1.7 membership function for “real number very near to 0”

  15. 1.4.3 Expansion of fuzzy set • Type-n Fuzzy Set The value of membership degree might include uncertainty. If the value of membership function is given by a fuzzy set, it is a type-2 fuzzy set. This conceptcan be extended up to Type-n fuzzy set. • Level-k fuzzy set The term “level-2 set” indicates fuzzy sets whose elements are fuzzy sets. The term “level-1 set” is applicable to fuzzy sets whose elements are no fuzzy sets ordinary elements. In the same way, we can derive upto level-k fuzzy set.

  16. A(x) A A(x) x Example (Type-n Fuzzy Set ) • A = “adult” A(x) = “youth”A(y) = “manhood”A(z) =  Fig 1.8 Fuzzy Set of Type-2

  17. A2 A3 A1 1.0 1.0 0.5 A3 A1 A2 Example (Level-k fuzzy set ) • A(A1) = 0.5 A(A2) = 1.0 A(A3) = 0.5 (b) elements of level-2 fuzzy set, A1, A2, A3 (a) level-2 fuzzy set

  18. age(element) infant young adult senior 5 0 0 0 0 15 0 0.2 0.1 0 25 0 1 0.9 0 35 0 0.8 1 0 45 0 0.4 1 0.1 55 0 0.1 1 0.2 65 0 0 1 0.6 75 0 0 1 1 85 0 0 1 1 1.5 Expanding Concepts of Fuzzy Set • 1.5.1 Example of Fuzzy Set • X= {5, 15, 25, 35, 45, 55, 65, 75, 85} : age domain Table 1.2 example of fuzzy set

  19. 1.0 1.5 Expanding Concepts of Fuzzy Set • Support Support(A) = {x X | A(x) > 0} : a set that is made up of elements contained in A ex) Support(youth) = {15, 25, 35, 45, 55} • Height : maximum value of the membership Fig 1.9 Non-Normalized Fuzzy Set and Normalized Fuzzy Set

  20. 1.5.2 -Cut Set • -cut set • A= {x  X | A(x) },  is an arbitrary real number in [0,1] • -cut set is a crisp set • Level Set A = { | A(x) = ,  0, x  X} • Example 1.11 • Young0.2 = {12, 25, 35, 45} • If =0.4, Young0.4 = {25, 35, 45} • If =0.8, Young0.8 = {25, 35} • “young” = {0, 0.1, 0.2, 0.4, 0.8, 1.0}

  21. 1 A′ : -cut set x 1 A : -cut set A : fuzzy set 1 ’ x  1.5.2 -Cut Set Fig 1.10 -cut set ′ , A A

  22. y  = 0.4  = 0.2  = 0.8  = 1  = 0 A = {0, 0.2, 0.4, 0.8, 1} x 1.5.3 Convex Fuzzy Set • Convex fuzzy set(1) where r, s n,  [0,1]

  23. A(s) A(r) A(t) r t s 1.5.3 Convex Fuzzy Set • Convex fuzzy set(2) Convex Fuzzy Set A(t) A(r) Non-Convex Fuzzy Set A(t) A(r)

  24. 1.5.4 Fuzzy Number • (Crisp) number A = “positive number less than equal to 10 (including 0)” = {x | 0  x  10, x }, or A(x) = 1 if 0 x 10, x = 0 if x < 0 or x > 10 • Fuzzy number • convex • normalized • piecewise continuous on 

  25. 1 1 0 10 fuzzy set “number near 0” fuzzy set “number near 10” A(x) 1 1 0 0 10 x set “positive number” set “positive number not exceeding 10” 1.5.4 Fuzzy Number Fig 1.14 Sets denoting intervals and fuzzy numbers

  26. 1.5.5 The magnitude of fuzzy set • scalar cardinality |A|= • relative cardinality ||A|| = ex) |senior| = 0.1 + 0.4 + 1 = 1.5 , |X| = 6, ||senior|| = 1.5/6 = 0.25

  27. 1.5.5 The magnitude of fuzzy set • Fuzzy cardinality • ex) senior0.1={45, 55, 65, 75, 85}, |senior0.1| = 5, senior0.2={55, 65, 75, 85}, |senior0.2|= 4, senior0.6={65, 75, 85}, |senior0.6| = 3, senior1.0={75, 85}, |senior1.0| = 2. Fuzzy cardinality |senior| ={(5, 0.1), (4, 0.2), (3, 0.6), (2,1)}

  28. 1.0 A B B A 1.5.6 Subset of fuzzy set • equivalence : A = B iff A(x) = B(x), x  X • subset : A  B iff A(x)B(x), x  X Fig 1.15 Subset A  B

  29. 1.6 Standard Operation of Fuzzy Set • 1.6.1 Complement • 1.6.2 Union • 1.6.3 Intersection Ex)A = “adult” , B=“young” = {(5, 1), (15, 0.9), (25, 0.1)} A  B= {(15,0.2), (25,1), (35,1), (45,1), (55,1), (65,1), (75,1), (85,1)} A B = {(15, 0.1), (25, 0.9), (35, 0.8), (45, 0.4), (55, 0.1)}

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