1. Fuzzy sets I
Prof. Dr. Jaroslav Ramík
Fuzzy sets I

2. Content Basic definitions Examples Operations with fuzzy sets (FS)
t-norms and t-conorms
Aggregation operators
Extended operations with FS
Fuzzy numbers: Convex fuzzy set, fuzzy interval, fuzzy number (FN), triangular FN, trapezoidal FN, L-R fuzzy numbers
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3. Basic definitions
Set - a collection well understood and distinguishable objects of our concept or our thinking about the collection.
Fuzzy set - a collection of objects in connection with expression of uncertainty of the property characterizing the objects by grades from interval between 0 and 1.
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4. Fuzzy set X - universe (of discourse) = set of objects
A : X  [0,1] - membership function
= {(x, A(x))| x X} - fuzzy set of X (FS)
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5. Examples Feasible daily car production Young man age Number around 8
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6. Example1. “Feasible car production per day”
X = {3, 4, 5, 6, 7, 8, 9} - universe
= {(3; 0), (4; 0), (5; 0,1), (6; 0,5), (7; 1), (8; 0,8), (9; 0)}
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7. Example 2. “Young man age”
X = [0, 100] - universe (interval)
Approximation of empirical evaluations (points):
20 respondents have been asked to evaluate the membership grade
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8. Example 3.“Number around eight”
X = (0, +) - universe (interval)
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9. Crisp set
Crisp set A of X = fuzzy set with a special membership function: A : X  {0,1} - characteristic function
Crisp set can be uniquely identified with a set:
(non-fuzzy) set A is in fact a (fuzzy) crisp set
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10. Support, height, normal fuzzy set
Support of fuzzy set , supp( ) = {xX| A(x) > 0}
support is a set (crisp set)!
Height of fuzzy set , hgt( ) = Sup{A(x) | xX }
Fuzzy set is normal (normalized), if there exists
x0X with A(x0) = 1
Ex.: Support of from Example 1: supp( ) = {5, 6, 7, 8}
hgt( ) = A(8) = 1  is normal!
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11. -cut (- level set)
 [0,1], fuzzy set, A = {x X|A(x)} - -cut of
- convex FS, if A is convex set (interval) for all  [0,1] !!!
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12. Operations with fuzzy sets
(X) -Fuzzy power set = set of all fuzzy sets of X (X)
 A(x) = B(x) for all x X - identity
 A(x)  B(x) for all x X - inclusion
- transitivity
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13. Union and Intersection of fuzzy sets
(X)
 AB(x) =Max{A(x), B(x)} - union
 AB(x) =Min{A(x), B(x)}- intersection
Properties:
Commutativity, Associativity, Distributivity,…
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14. Example 4.
= {(3; 0), (4; 0), (5; 0,1), (6; 0,5), (7; 1), (8; 0,8), (9; 0)}
= {(3; 1), (4; 1), (5; 0,9), (6; 0,8), (7; 0,4), (8; 0,1), (9; 0)}
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15. Complement, Cartesian product
(X)
 CA(x) =1 - A(x) - complement of
(X) ,
(Y)
 AB(x,y) =Min{A(x), B(y)} Cartesian product (CP)
CP is a fuzzy set of XY !
Extension to more parts possible e.g. X, Y, Z,…
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16. Complementarity conditions
(X)
 = 
 = X
Min and Max do not satisfy 1., 2. ! (only for crisp sets)
…later on …”bold” intersection and union will satisfy the complementarity…
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17. Examples
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18. Extended operations with FS
Intersection  and Union  = operations on
(X)
Realization by Min and Max operators
generalized by t-norms and t-conorms
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19. t-norms A function T: [0,1]  [0,1]  [0,1] is called t-norm
if it satisfies the following properties (axioms):
T1: T(a,1) = a a  [0,1] - “1” is a neutral element
T2: T(a,b) = T(b,a) a,b  [0,1] commutativity
T3: T(a,T(b,c)) = T(T(a,b),c) a,b,c  [0,1] - associativity
T4: T(a,b)  T(c,d) whenever a  c , b  d - monotnicity
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20. t-conorms A function S: [0,1]  [0,1]  [0,1] is called t-conorm
if it satisfies the following axioms:
S1: S(a,0) = a a  [0,1] - “0” is a neutral element
S2: S(a,b) = S(b,a) a,b  [0,1] commutativity
S3: S(a,S(b,c)) = S(S(a,b),c) a,b,c  [0,1] - associativity
S4: S(a,b)  S(c,d) whenever a  c , b  d - monotnicity
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21. Examples of t-norms and t-conorms #1
1. TM = Min, SM = Max - minimum and maximum 2.
- drastic product, drastic sum
Property:
TW(a,b)  T(a,b)  TM(a,b) , SM(a,b)  S(a,b)  SW(a,b)
for every t-norm T, resp. t-conorm S, and a,b  [0,1]
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22. Examples of t-norms and t-conorms #2
3. TP(a,b) = a.b SP (a,b) = a+b - a.b - product and probabilistic sum
4. TL(a,b) = Max{0,a+b - 1} SL (a,b) = Min{1,a+b}
- Lukasiewicz t-norm and t-conorm (satisfies complematarity!)
(bounded difference, bounded sum)
Also: b - bold intersection, b - bold union
Property:
T*(a,b) = 1 - T(1-a,1-b) , S*(a,b) = 1 - S(1-a,1-b)
If T is a t-norm then T* is a t-conorm ( T and T* are dual )
If S is a t-conorm then S* is a t-norm ( S and S* are dual )
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23. Examples of t-norms and t-conorms #3
5. q [1,+)
a,b  [0,1]
Yager’s t-norm and t-conorm
6. Einstein, Hamacher, Dubois-Prade product and sum etc.
Property:
If q =1, then Tq, (Sq) is Lukasiewicz t-norm (t-conorm)
If q = +, then Tq, (Sq) is Min (Max)
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24. Extended Union and Intersection of fuzzy sets
(X), T - t-norm, S - t-conorm
 AsB(x) =S(A(x), B(x)) - S-union
 ATB(x) =T(A(x), B(x)) -T-intersection
Properties:
Commutativity, Associativity?,…
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25. Aggregation operators
A function G: [0,1]  [0,1]  [0,1] is called
aggregation operator
if it satisfies the following properties (axioms):
A1: G(0,0) = 0 - boundary condition 1
A2: G(1,1) = 1 - boundary condition 2
A3: G(a,b)  G(c,d) whenever a  c , b  d - monotnicity
NO commutativity or associativity conditions!
All t-norms and t-conorms are aggregation operators!
May be extended to more parts, e.g. a,b,c,…
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26. Compensative operators (CO) #1
CO = Aggregation operator G satisfying
Min(a,b)  G(a,b)  Max(a,b)
Example 1. Averages:
1: G(a,b) = (a +b)/2 - arithmetic mean (average)
2: G(a,b) = geometric mean
3: G(a,b) = harmonic mean S
Max G
Min T
Extension to more elements possible!
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27. Compensative operators #2
Examples. Compensatory operators:
1: TW(a,b) = .Min(a,b) + (1- ) fuzzy „and“
SW(a,b) = .Max(a,b) + (1- ) fuzzy „or“ (by Werners)
2: ATS(a,b) = .T(a,b) + (1 - ).S(a,b) COs by PTS(a,b) =T(a,b) . S(a,b)1- Zimmermann and Zysno
T - t-norm, S - t-conorm,   [0,1] - compensative parameter
CO compensate trade-offs between conflicting evaluations
extension to more elements possible
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28. Fuzzy numbers - fuzzy set of R (real numbers) - convex
- normal (there exists x0  R with A(x0) = 1)
- A is closed interval for all  [0,1]
Then is called fuzzy interval
Moreover if there exists only one x0  R with A(x0) = 1
then is called fuzzy number
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29. Positive and negative fuzzy numbers
- fuzzy number is
- positive if A(x) = 0 for all x  0
- negative if A(x) = 0 for all x  0
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30. Example 5. Fuzzy number “About three”
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31. Example 6. Triangular fuzzy number “About three”
spread
mean value
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32. L-R fuzzy intervals
L, R : [0,+)  [0,1] - non-increasing, non-constant functions - shape functions
L(0) = R(0) = 1, m, n,  > 0,  > 0 - real numbers
- fuzzy interval of L-R-type if
fuzzy number of L-R-type if m = n, L, R - decreasing functions
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33. Example 7. L-R fuzzy number “Around eight”
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34. Example 8. L-R fuzzy number “About eight”
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35. Example 9. L-R fuzzy interval
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36. Example 10. Fuzzy intervals
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37. Summary
Basic definitions: set, fuzzy set, membership function, crisp set, support, height, normal fuzzy set, -level set
Examples: daily production, young man age, around 8
Operations with fuzzy sets: fuzzy power set, union, intersection, complement, cartesian product
Extended operations with fuzzy sets: t-norms and t-conorms, compensative operators
Fuzzy numbers: Convex fuzzy set, fuzzy interval, fuzzy number (FN), triangular FN, trapezoidal FN, L-R fuzzy numbers
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38. References
[1] J. Ramík, M. Vlach: Generalized concavity in fuzzy optimization and decision analysis. Kluwer Academic Publ. Boston, Dordrecht, London, 2001.
[2] H.-J. Zimmermann: Fuzzy set theory and its applications. Kluwer Academic Publ. Boston, Dordrecht, London, 1996.
[3] H. Rommelfanger: Fuzzy Decision Support - Systeme. Springer - Verlag, Berlin Heidelberg, New York, 1994.
[4] H. Rommelfanger, S. Eickemeier: Entscheidungstheorie - Klassische Konzepte und Fuzzy - Erweiterungen, Springer - Verlag, Berlin Heidelberg, New York, 2002.
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