Mo Jamshidi and Aly El-Osery
8.7 PROPERTIES OF FUZZY SETS
8.7.2 Extension Principle
complement does not give the universe of discourse (see Figure 8.12).
0
x
1 ~
A A~
~
~ B
µAU
Figure 12: Law of Excluded Middle.
8.7.1 Alpha-Cut Fuzzy Sets
It is the crisp domain in which we perform all computations with today s computers. The conversion from fuzzy to crisp sets can be done by two means, one of which is
alpha-cut sets.
Given a fuzzy set
A~ , the alpha-cut (or lambda cut) set of
A~ is defined by Aα =xµA x ≥α
~
( ) (8.39)
Note that by virtue of the condition on ( )
~
A x
µ in Equation 8.39, i.e., a common property, the set Aα in Equation 8.39 is now a crisp set. In fact, any fuzzy set can be converted to an infinite number of cut sets.
8.7.2 Extension Principle
In fuzzy sets, just as in crisp sets, one needs to find means to extend the domain of a function, i.e., given a fuzzy set
A~ and a function f(⋅), then what is the value of function f(
~
A)? This notion is called the extension principle which was first proposed by Zadeh.
Let the function f be defined by
f U: →V (8.40)
where U and V are domain and range sets, respectively. Define a fuzzy set
Then the extension principle asserts that the function f is a fuzzy set, as well, which is defined below:
The complexity of the extension principle would increase when more than one member of u1 x u2 is mapped to only one member of v; one would take the maximum membership grades of these members in the fuzzy set
~
A.
Example 8.1
Given two universes of discourse U1=U2={1,2, ,10} and two fuzzy sets (numbers) defined by
It is desired to find approximately 10
SOLUTION:
The function f = ×u1 u2:→v represents the arithmetic product of these two fuzzy numbers and is given by
" approximately 10"= + +
membership function 1 and the other 8 pairs are spread around the point (1, 10).
Example 8.2
Consider two fuzzy sets (numbers) defined by Approximately 2 =
It is desired to find approximately 8 SOLUTION:
The function f = ×u1 u2:→v represents the arithmetic product of these two fuzzy numbers and is given by
" approximately 8"= + +
Classical relations are structures that represent the presence or absence of correlation or interaction among elements of various sets. There are only two degrees of relationship between elements of the sets in a crisp relation, namely, the relationships completely related or not related . Fuzzy relations, on the other hand, are developed by allowing the relationship between elements of two or more sets to take an infinite number of degrees of relationship between the extremes of completely related and not related [6,7].
The classical relation of two universes U and V is defined as
U× =V
{
( , )u v u U v V∈ , ∈}
(8.43)which combines ∀u∈U and ∀v∈V in an ordered pair and forms unconstrained matches between u and v. That is, every element in universe U is related completely to every element in universe V. The strength of this relationship
between ordered pairs of elements in each universe is measured by the characteristic function, where a value of unity is associated with complete relationship and a value of zero is associated with no relationship, i.e., the binary values 1 and 0.
As an example, if U={1,2} and V={a,b,c}, then UxV={(1,a), (1,b), (1,c), (2,a), (2,b), (2,c)}. The above product is said to be crisp relation, which can be expressed by either a matrix expression
a b c R= × = U V
1 2
1 1 1
1 1 1 (8.44)
Or in a so-called Sagittal diagram (see Figure 8.13)
U V
1 2
a b c
Figure 8.13: Sagittal Diagram.
Fuzzy relations map elements of one universe to those of another universe, through Cartesian product of the two universes. Unlike crisp relations, the strength of the relation between ordered pairs of the two universes is not measured with the characteristic function, but rather with a membership function expressing various degrees of the strength of the relation on the unit interval [0,1]. In other words, a fuzzy relation
~
Ris a mapping:
R U V
~: × →[ , ]0 1 (8.45)
The following example illustrates this relationship, i.e., µR u v µA B u v µA u µB v
~ ~ ~ ~ ~
( , )= × ( , )=min( ( ), ( )) (8.46)
Example 8.3
Consider two fuzzy sets
2
Determine the fuzzy relation between these sets.
SOLUTION:
R~ be a relation that relates elements from universe U to universe V, and let
~
Sbe a relation that relates elements from universe V to universe W. Is it possible to find the relation
~
Tthat relates the same elements in universe U that R~ contains to elements in universe W that
~
Scontains? The answer is yes, using an operation known as composition.
In crisp or fuzzy relations, the composition of two relations, using the max-min rule, is given below. Given two fuzzy relations ( , ) the composition of these is
T R S u v v w
or using the max-product rule, the characteristic function is given by
µT µ µ
The same composition rules hold for crisp relations.
Example 8.4
It is desired to evaluate
Using the max-min composition for
~
where, for example, the element (1,1) is obtained by max{min(0.6,0.3), min(0.8,0.2)}=0.3.
For ~ ~
R
So we get the following result
S R R S
Using the max-product rule, we have R S~ ~
where, for example, the element (2,2) is obtained by max{(0.7)(0.1), (0.9)(0.8)}=0.72.
In this chapter a quick overview of classical and fuzzy sets was given. Main similarities and differences between classical and fuzzy sets were introduced.
In general, set operations are the same for classical and fuzzy sets. The exceptions were excluded middle laws. Alpha-cut sets and extension principle were presented followed by a brief introduction to classical vs. fuzzy relations.
This chapter presented issues that are important in understanding fuzzy sets and their advantages over classical sets. A set of problems at the end of the book will further enhance the reader s understanding of these concepts.
REFERENCES
1. Zadeh, L. A, Fuzzy sets, Information and Control, Vol. 8, 338−353, 1965.
2 . Mamdani, E. H., Applications of fuzzy algorithms for simple dynamic plant, Proc. IEE, 121, No. 12, 1585−1588, 1974.
Fuzzy Logic - Toward High Machine Intelligence Quotient Systems, Vol. 9, Prentice Hall series on Environmental and Intelligent Manufacturing Systems (M. Jamshidi, ed. ), Prentice Hall, Upper Saddle River, NJ, 1997.
4. Ross, T. J., Fuzzy Logic with Engineering Application, McGraw-Hill, New York, 1995.
5. Jamshidi, M., Vadiee, N. and Ross, T. J. (eds.), Fuzzy Logic and Control:
Software and Hardware Applications. Vol 2. Prentice Hall Series on Environmental and Intelligent Manufacturing Systems, (M. Jamshidi, ed.).
Prentice Hall, Englewood Cliffs, NJ, 1993.
6. Dubois, D. and Prade, H., Fuzzy Sets and Systems, Theory and Applications, Academic, New York, 1980.
7. Zimmermann, H., Fuzzy Set Theory and Its Applications, 2nd ed., Kluwer Academic Publishers, Dordrecht, Germany, 1991.
9
Mo Jamshidi, Aly El-Osery, and Timothy J. Ross
9.1 INTRODUCTION
The need and use of multilevel logic can be traced from the ancient works of Aristotle, who is quoted as saying, “There will be a sea battle tomorrow.” Such a statement is not yet true or false, but is potentially either. Much later, around AD 1285-1340, William of Occam supported two-valued logic but speculated on what the truth value of “if p then q” might be if one of the two components, p or q, as neither true nor false. During the time period of 1878-1956, Lukasiewicz proposed a three-level logic as a “true” (l), a “false” (0), and a “neuter” (1/2), which represented half true or half false. In subsequent times, logicians in China and other parts of the world continued on the notion of multi-level logic. Zadeh, in his seminal 1965 paper [1], finished the task by following through with the speculation of previous logicians and showing that what he called “fuzzy sets”
were the foundation of any logic, regardless of the number of truth levels assumed. He chose the innocent word “fuzz” for the continuum of logical values between 0 (completely false) and 1 (completely true). The theory of fuzzy logic deals with two problems 1) the fuzzy set theory, which deals with the vagueness found in semantics, and 2) the fuzzy measure theory, which deals with the ambiguous nature of judgments and evaluations.
The primary motivation and “banner” of fuzzy logic is the possibility of exploiting tolerance for some inexactness and imprecision. Precision is often very costly, so if a problem does not require precision, one should not have to pay for it. The traditional example of parking a car is a noteworthy illustration.
If the driver is not required to park the car within an exact distance from the curb, why spend any more time than necessary on the task as long as it is a legal parking operation? Fuzzy logic and classical logic differ in the sense that the former can handle both symbolic and numerical manipulation, while the latter can handle symbolic manipulation only. In a broad sense, fuzzy logic is a union of fuzzy (fuzzified) crisp logics [2]. To quote Zadeh, “Fuzzy logic’s primary aim is to provide a formal, computationally-oriented system of concepts and techniques for dealing with modes of reasoning which are approximate rather than exact.” Thus, in fuzzy logic, exact (crisp) reasoning is considered to be the limiting case of approximate reasoning. In fuzzy logic one can see that everything is a matter of degrees.
This chapter is organized as follows. In section 9.2, a brief introduction to predicate logic is given. In section 9.3, fuzzy logic is presented, followed by approximate reasoning in section 9.4.