1. (Fuzzy Set Operations)
Lecture 4
(Fuzzy Set Operations)
“We need a radically different kind of mathematics, the mathematics of fuzzy or cloudy quantities which are not describable in terms of probability distributions”
L.A.Zadeh, 1962
Spring 2002
Lecture 04

2. Some points of the previous lecture
Fuzzy Logic is some kind of multi-valued logic. Unlike crisp two valued logic, the truth value in fuzzy logic can be any number between 0 and 1 and hence is an extension to classical logic
Consequently, many of the Natural Language propositions can be represented with fuzzy logic
Spring 2002
Lecture 04

3. A ={ (x,A(x)) | xU , 0 A(x)  1}
A fuzzy set (A generalized concept of the conventional crisp set) is specified with a membership function A(x) which represents degree of membership in the set.
A ={ (x,A(x)) | xU , 0 A(x)  1}
Spring 2002
Lecture 04

4. Basic Set-Theoretic Operations
Equality:
Subset:
Complement:
Union:
Intersection:
Slides for fuzzy sets, J.-s. Roger Jang
Spring 2002
Lecture 04

5. De Morgan’s Laws
Set operations on fuzzy sets are supposed to be defined such that the more previous known laws and equalities remain true in fuzzy sets as well.
Having the previous definitions for fuzzy set operations, we can verify De Morgan laws in fuzzy logic as well.
Exercise: Prove the above equalities
Spring 2002
Lecture 04

6. Fuzzy Set operations in details
- Fuzzy Complement
Fuzzy complement is actually a function say c that maps the membership function to the membership function of the complement set
Definition: Any function c :[0,1][0,1] that satisfies the following Axioms c1 and c2 is called a fuzzy complement
Spring 2002
Lecture 04

7. Axiom c1. (boundary conditions) Axiom c2. (non-increasing condition)
Requirements
Axiom c1. (boundary conditions)
Axiom c2. (non-increasing condition)
Axiom c1 requires that if an element belongs to a fuzzy set to degree zero (one), then it should belong to the complement of this fuzzy set to degree one (zero).
Axiom c2 means that an increase in membership value of a fuzzy set must result in a decrease or no change in membership value of the complement set
Spring 2002
Lecture 04

8. Examples of fuzzy complements 1. Basic Fuzzy Complement
Clearly, in classical crisp logic (where domain of definition of the complement function is {0,1}) there is only one complement function which satisfies the above axioms whereas in fuzzy logic, there are many functions with a domain [0,1] which satisfy the above conditions.
Examples of fuzzy complements
1. Basic Fuzzy Complement
Spring 2002
Lecture 04

9. 2. Sugeno class of fuzzy complements
For any value of the parameter , a particular fuzzy complement function is obtained
3. Yager class of fuzzy complements
For any value of the parameter , a particular fuzzy complement function is obtained
Spring 2002
Lecture 04

10. Graphical Representation of the Sugeno Class Complement
Spring 2002
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11. Graphical Representation of the Yager Class Complement
Spring 2002
Lecture 04

12. - Fuzzy Union s-norm (t-conorm)
Intuitively, the union of two sets, AB means a fuzzy set (in particular the smallest one) containing both A and B.
The union of two fuzzy sets can be defined with a function named s-norm s:[0,1]x[0,1][0,1] which maps the membership functions of fuzzy sets A and B into the membership function of the union of A and B
(called AB)
The requirements for a function to be an s-norm are as follow:
Spring 2002
Lecture 04

13. Axiom s1. (boundary conditions) Axiom s2. (commutative condition)
Axiom s3. (non-decreasing condition)
Axiom s4. (associative condition)
Spring 2002
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14. Examples of fuzzy s-norms 1. Dombi calss
Definition: Any function s:[0,1]x[0,1][0,1] that satisfies the above 4 axioms is called an s-norm
Examples of fuzzy s-norms
1. Dombi calss
2. Dubois-Prade calss
Spring 2002
Lecture 04

15. 3. Yager calss
4. Drastic Sum:
5. Einstein Sum:
Spring 2002
Lecture 04

16. 7. Maximum (Basic fuzzy Union)
6. Algebraic Sum:
7. Maximum (Basic fuzzy Union)
Theorem S1: For any s-norm, s(a,b) the following inequality holds: (for any a,b  [0,1]
It means that the smallest s-norm (or smallest union of two fuzzy sets) is maximum while the largest s-norm is Drastic sum
Spring 2002
Lecture 04

17. Theorem S2: Dombi s-norm and Yager s-norm cover the whole spectrum of s-norms when their parameters change
In its extreme cases:
And Also
So it is possible to build any s-norm with choosing the right parameter in any of the yager or dombi s-norms
Spring 2002
Lecture 04

18. - Fuzzy Intersection t-norm
Intuitively, the intersection of two sets, AB means a fuzzy set (in particular the largest one) containing by both A and B.
The Intersection of two fuzzy sets can be defined with a function named t-norm t:[0,1]x[0,1][0,1] which maps the membership functions of fuzzy sets A and B into the membership function of the intersection of A and B
The requirements for a function to be a t-norm are as follow:
Spring 2002
Lecture 04

19. Axiom t1. (boundary conditions) Axiom t2. (commutative condition)
Axiom t3. (non-decreasing condition)
Axiom t4. (associative condition)
Spring 2002
Lecture 04

20. Examples of fuzzy t-norms 1. Dombi calss
Definition: Any function t:[0,1]x[0,1][0,1] that satisfies the above 4 axioms is called a t-norm
Examples of fuzzy t-norms
1. Dombi calss
2. Dubois-Prade calss
Spring 2002
Lecture 04

21. 3. Yager calss 4. Drastic Product: 5. Einstein Product: Lecture 04
Spring 2002
Lecture 04

22. 7. Minimum (Basic fuzzy Intersection)
6. Algebraic Product:
7. Minimum (Basic fuzzy Intersection)
Theorem T1: For any t-norm, t(a,b) the following inequality holds: (for any a,b  [0,1] )
It means that the largest t-norm (or largest intersection of two fuzzy sets) is minimum while the smallest t-norm is Drastic product
Spring 2002
Lecture 04

23. Theorem T2: Dombi t-norm and Yager t-norm cover the whole spectrum of t-norms when their parameters change
In its extreme cases:
And also:
So it is possible to build any t-norm with choosing the right parameter in any of the yager or dombi t-norms.
Spring 2002
Lecture 04

24. Graphical representation of theorem S1 and S2
Algebraic sum: S(a,b)=sas (a,b)
Yager: S(a,b)=sw (a,b)
W=3
S(a,b)=max(a,b)
Spring 2002
Lecture 04

25. Graphical representation of theorem T1 and T2
S(a,b)=min(a,b)
Yager: t(a,b)=tw (a,b)
W=3
Algebraic product: S(a,b)=sap (a,b)
Spring 2002
Lecture 04

26. Generalized De Morgan’s Law
Using the new definitions of s-norm and t-norm instead of the basic fuzzy union and basic fuzzy intersection respectively, the generalized De Morgan’s Law can be shown as follow:
c( t(a,b) ) = s( c(a), c(b) )
c( s(a,b) ) = t( c(a), c(b) )
where c(.) denotes for any fuzzy complement and s(.) and t(.) denote for fuzzy s-norm and fuzzy t-norm respectively.
Spring 2002
Lecture 04

27. c[s(a,b)]=t[c(a),c(b)]
Associated class
An s-norm s(a,b), a t-norm t(a,b) and a fuzzy complement c(a) form an associated class if they all together satisfy the Generalized De Morgan laws
c[s(a,b)]=t[c(a),c(b)]
It can be shown that there is a t-norm associated with each s-norm in the sense that there is a complement such that the De Morgan laws are satisfied. For example the Yager s-norm and t-norms are associated with each other through basic fuzzy complement
Spring 2002
Lecture 04

28. References 1. L.X. Wang, A course in Fuzzy Systems and control
2. Tutorial on Fuzzy Logic, Jan Jantzen, Technical University of Denmark, Technical report no 98-E 868, 1999
3. Slides for fuzzy sets, J.-s. Roger Jang
Spring 2002
Lecture 04