Crisp set theory is a certain element. Fuzzy sets

2.4. Description of uncertainties using fuzzy theory

Hello, citizens and citizens. At the behest of the left heel, I decided to start a series of popular science articles, where I will explain the basics of artificial intelligence. Therefore, in the future I will try on the role of a visiting lecturer, talking about how spaceships roam the expanses of the Bolshoi Theater.

I won’t be able to publish one article a day, so I won’t promise anything, so as not to embarrass myself with these obligations. The only thing: I will not torment others with an abundance of mathematics, I will try to present everything as clearly as possible, but without profanity. I’ll start the cycle with the apparatus of fuzzy logic, where I will explain what its intelligence is.

To start short excursion into set theory. A set is a collection of several objects that have a certain property. For example, the set of all the people on our planet. Lots of Audi cars with RGB color coordinates (255, 165, 0). A set of all male cockatoos sitting on a branch on one leg at exactly 1539 GMT. The essence of clear sets is their absolute categoricality. That is, in order to determine whether an object belongs to a certain set, it is necessary to answer the question whether it has a property that defines this set. Not really. No more, no less. Is one greater than zero? Yes. This means that it belongs to the set of positive numbers.

Let's move closer to the body, to the theory of fuzzy sets. It was created by an American scientist of Azerbaijani origin, Lotfi Zadeh, in order to adapt set theory to the way of human thinking. After all, how does a little man think? If, while on the beach, you ask a swimmer: “Tell me, dear man, what temperature is the water on the Fahrenheit scale, accurate to tenths of a degree?”, he will look at you as if you were mentally ill. And if you ask the question: “How is the water today?”, he will say: “Cold/hot/warm”, or he will mutter “wet” if he is not in the mood today. The whole point is that “ cold water" - this is a rather vague formulation. One will bask in bliss where the second will run to the shore to bask in two minutes. This is how humans are designed, subjectivity and the lack of clear boundaries - this is about us.

Some have already been able to figure out why fuzzy sets. It is extremely difficult to determine how many people have the trait “tall.” For me, a two-meter handsome man, slanting fathoms at the shoulders, tall is at least not lower than the level of my ear. And a short one and a half meters will look at a person 170 cm tall with his head raised - for him high growth starts much earlier. This is about subjectivity.

The second difficulty is the blurring of boundaries. Is it possible to accurately determine the number of centimeters that separates a person of average height from a short one? 170 and a half? 172 and three quarters? The division is very, very arbitrary. So, we have come close to the difference between fuzzy sets and crisp sets.

Drum roll, Moscow Art Theater pause... So, fuzzy sets differ from crisp sets in that objects belonging to fuzzy sets can have the property that defines them to varying degrees. We agreed to consider this degree of belonging to be in the range from zero to one, but if it is more convenient for someone, he can multiply by 100, and you will get the percentages.

Let's say you drink scalding coffee and the cup is smoking. With 0.99 confidence (99 percent – this is the first and last time I do the work for you), we can say that coffee has the “hot” property. If it (coffee, in the sense) has a temperature of 50 degrees Celsius, then the degree of possession of the “hot” property will be much lower, say, 0.76 (now do the math for yourself). At the same time, there are objects that belong to the “hot” set with zero or one degree. For example, half-frozen coffee can only be called hot by a madman or someone who doesn’t know the Russian language, but boiling coffee is a hundred poods hot. An endless number of examples can be given, fortunately, almost any human category that is used in everyday life is fuzzy. Relying on your rich imagination, I leave the task of finding other examples for you to solve on your own.

Why was the creation of such a theory so important, why did they pay such close attention to it? The answer is simple: there is a hidden goldmine here. Enormous breadth of application. Let's say you are an engineer and you are faced with the task of designing a microwave oven. To what temperature will a person heat food? Up to 40.2°C? Fuck it. To the hot one, which is a fuzzy set. And the task of the microwave is to give the snack a temperature that, with a single degree of certainty, would belong to the set of “hot”.

Then the fun begins, truants from math lessons can run away screaming. A? What? Did I promise to do without this? As old Arnie said in famous film- "I lied". The degree of membership is usually denoted by the Greek letter “mu” - μ. In order not to get bored, let's introduce the concept of a linguistic variable - this is a variable that can take on a value in the form of words of a human language. That is, the linguistic variable “height” can take the following values: “high”, “medium”, “low”. We will call the values of a linguistic variable term sets; please note that they are fuzzy. And finally, there is the concept of a universal set - an ordinary, clear set containing all the values that an ordinary variable can take. The usual variable “human height” can take values from zero to “I don’t remember how many Guinness records there are.”

The purpose of the membership function (MF) is to determine the degree to which an ordinary variable belongs to the value of a linguistic variable. Since I’ve started to emphasize the topic of height, I’ll elaborate: FP determines to what degree a person with a height of 184 cm belongs to the term set “average”. So, let's match the grandmothers. We have a linguistic variable. We have several of its values, each of which is a fuzzy set. Finally, we have a universal set - the set of numeric values of an ordinary variable. We have the following goal: to determine for each of the fuzzy sets its own membership function, i.e. for each element of the universal set, set the degree of membership in the corresponding fuzzy set. Then we can point to a specific value of a variable and see to what extent it belongs to any fuzzy set. That's it, the storm has passed, you can wipe off the sweat and relax for a while. Next will be funny pictures, after which we will continue to have fun for a while. In the pictures I will illustrate the meaning of the membership function, show what types of these animals there are, what they are eaten with, and explain how to build these animals. Let's return to your favorite topic of human growth. Let’s take the “average” set as an example and plot the membership function.

Now you can, armed with a sharpened pencil, select any value of “x” and see to what extent this x satisfies the condition of average height. The fact that meter eighty is ironclad. Meter seventy-two - with a degree of 0.5. A height of one meter and fifty is by no means average, so the degree of membership is zero. And so on. Note that the given function is called triangular. It's hard to believe, and yet.

But we took a ready-made function that someone (someone!) kindly provided to us. How can we build a similar function ourselves? There are two ways: simple and complicated. For obvious reasons, I will only describe a simple one. First, you need to assemble a group of experts. Well, that is, those slackers who believe that they understand everything and know how the world really works. Give each expert a pencil and notepad. Then list the values of the variable and ask to put “1” (stick, cross - optional) opposite this value if the expert believes that the value of the variable belongs to a fuzzy set. Zero - in otherwise. Then, for each value of the variable, sum up the zeros and ones and take the average - that is, divide the resulting sum by the number of slackers. The resulting value will be in the range from zero to one (both values inclusive). Some might guess that we obtained the value of the membership function for a specific value of the variable. Having obtained the PT values for all values of the x variable, you can build a graph. Or don't build if you're lazy.

The theory of fuzzy sets allows the use of fuzzy linguistically defined variables when synthesizing a control algorithm.

The theory of fuzzy sets has gone from developing formal means of representing poorly defined concepts used by humans and an apparatus for processing them to modeling approximate reasoning that humans resort to in everyday life. professional activity and even before the creation of fuzzy logic computers.

The theory of fuzzy sets allows one to replace the strict membership of an object in a certain set with a continuous degree of membership. To become familiar with the theory of fuzzy sets and their application for research in the field of catalytic processes, the reader can refer to Section.

Fuzzy set theory is often confused with probability theory. Indeed, its critics have argued that fuzzy set theory is incapable of solving problems that are not formulated in terms probability theory. Apart from these quantities, the two measures are completely different, although both can be described as measures of uncertainty. Each of them measures a different aspect of uncertainty.

In the theory of fuzzy sets, as is known, membership functions are used, interpreted as characteristic functions for fuzzy sets. Its value equal to 0 corresponds to the statement that this element x does not belong to A, and its value equal to 1 indicates its unconditional membership in this set. Intermediate values / id (g) should not be interpreted in probabilistic sense, since the degree of membership of an element in a fuzzy set does not have to be of a statistical nature.

In fuzzy set theory important role plays the concept of a combination of two fuzzy relations.

In the theory of fuzzy sets, a number of operations on sets are introduced, which must correspond to combinations of fuzzy terms and their semantic loads when solving applied problems. The paper notes that in a particular case, operations on fuzzy sets should correspond to operations in the theory of ordinary sets. When deciding specific tasks: Each researcher uses his knowledge of the object of study and the role of each operation.

In fuzzy set theory, most arithmetic operations are defined for continuous areas. Operations for discrete areas are usually isolated as a special case.

In the theory of fuzzy sets, depending on the methods of specifying the operation (T), which satisfy axioms (2.1) - (2.5), there is an infinite number of fuzzy operations I. The following types are used in the theory of fuzzy control.

Elements of fuzzy set theory can be successfully applied to decision making under conditions of uncertainty. Fuzzy logic arose as the most convenient way to build control systems for complex technological processes, and has also found application in diagnostic and other expert systems. Despite the fact that the mathematical apparatus of fuzzy logic was first developed in the USA, active development this method began in Japan, Research in the field of fuzzy logic received widespread financial support, In Europe and the USA, efforts were aimed at closing the huge gap with the Japanese.

However, the axiomatics of the theory of fuzzy sets differs significantly from the axiomatics of probability theory and allows the use of simpler computational procedures. To verify this, it is enough to consider the operations of union and intersection of fuzzy sets.

Let us also mention the theory of fuzzy sets, in which initial concepts are described by fuzzy sets and variables and, accordingly, the resulting solution is interpreted in terms of fuzzy sets. As shown specific examples, these methods are in many ways similar to statistical ones. When using them, it is assumed given functions membership of the observation results, and on their basis the corresponding membership functions for the final results are obtained.

Using fuzzy sets, it is possible to formally define imprecise and ambiguous concepts such as “high temperature”, “young man”, “average height” or “ Big city" Before formulating the definition of a fuzzy set, it is necessary to define the so-called universe of discourse. In the case of the ambiguous concept of “a lot of money”, one amount will be considered large if we limit ourselves to the range and a completely different amount - in the range. The area of reasoning, called henceforth space or set, will most often be denoted by the symbol. It must be remembered that this is a clear set.

Definition 3.1

A fuzzy set in some (non-empty) space, which is denoted as , is a set of pairs

, (3.1)

Fuzzy set membership function. This function assigns to each element the degree of its membership in a fuzzy set, and three cases can be distinguished:

1) means the complete membership of an element in a fuzzy set, i.e. ;

2) means that the element does not belong to a fuzzy set, i.e.;

3) means the element partially belongs to a fuzzy set.

In the literature, a symbolic description of fuzzy sets is used. If is a space with a finite number of elements, i.e. , then the fuzzy set is written in the form

The above entry is symbolic. The “–” sign does not mean division, but means assigning degrees of membership to specific elements . In other words, the record

means a couple

Similarly, the “+” sign in expression (3.3) does not mean an addition operation, but is interpreted as multiple summation of elements (3.5). It should be noted that In a similar way One can also write clear sets. For example, many school grades can be symbolically represented as

, (3.6)

which is equivalent to writing

If is a space with an infinite number of elements, then the fuzzy set is symbolically written in the form

. (3.8)

Example 3.1

Let us assume that there is a set natural numbers. Let us define the concept of the set of natural numbers “close to the number 7”. This can be done by defining the following fuzzy set:

Example 3.2

If , where is the set of real numbers, then the set real numbers, “close to the number 7,” can be defined by a membership function of the form

. (3.10)

Therefore, the fuzzy set of real numbers “close to the number 7” is described by the expression

. (3.11)

Remark 3.1

Fuzzy sets of natural or real numbers “close to the number 7” can be written in various ways. For example, the membership function (3.10) can be replaced by the expression

(3.12)

In Fig. 3.1a and 3.1b present two membership functions for the fuzzy set of real numbers “close to the number 7”.

Rice. 3.1. Illustration for example 3.2: membership functions of a fuzzy set of real numbers “close to the number 7”.

Example 3.3

Let’s formalize the imprecise definition of “suitable temperature for swimming in the Baltic Sea.” Let us define the area of reasoning in the form of a set . Vacationer I, who feels best at a temperature of 21°, would define for himself a fuzzy set

Vacationer II, who prefers a temperature of 20°, would suggest a different definition of this set:

Using fuzzy sets, we formalized the imprecise definition of the concept of “suitable temperature for swimming in the Baltic Sea.” Some applications use standard forms membership functions. Let us specify these functions and consider their graphical interpretations.

1. The class membership function (Fig. 3.2) is defined as

(3.15)

Where . The membership function belonging to this class has a graphical representation (Fig. 3.2), reminiscent of the letter “”, and its shape depends on the selection of parameters , and . At the point the class membership function takes a value of 0.5.

2. The class membership function (Fig. 3.3) is determined through the class membership function:

(3.16)

Rice. 3.2. Class membership function.

Rice. 3.3. Class membership function.

The class membership function takes zero values for and . At points its value is 0.5.

3. The class membership function (Fig. 3.4) is given by the expression

(3.17)

The reader will easily notice the analogy between the forms of the class membership functions and .

4. The class membership function (Fig. 3.5) is defined as

(3.18)

Rice. 3.4. Class membership function.

Rice. 3.5. Class membership function.

In some applications, the class membership function may be an alternative to the class function.

5. The class membership function (Fig. 3.6) is determined by the expression

(3.19)

Example 3.4

Let's consider three imprecise formulations:

1) “low vehicle speed”;

2) " average speed car";

3) “high vehicle speed.”

As the area of reasoning, we will take the range , where is the maximum speed. In Fig. 3.7 presents the fuzzy sets , and , corresponding to the above formulations. Note that the membership function of a set has type , sets have type , and sets have type . At a fixed point km/h, the membership function of the fuzzy set “low car speed” takes on the value 0.5, i.e. . The membership function of the fuzzy set “average car speed” takes on the same value, i.e. , whereas .

Example 3.5

In Fig. Figure 3.8 shows the membership function of the fuzzy set “big money”. This is a class function, and , , .

Rice. 3.6. Class membership function.

Rice. 3.7. Illustration for example 3.4: membership functions of fuzzy sets “small”, “medium”, “high” car speed.

Rice. 3.8. Illustration for example 3.5: Membership function of the fuzzy set “big money”.

Consequently, amounts exceeding 10,000 rubles can definitely be considered “large”, since the values of the membership function become equal to 1. Amounts less than 1000 rubles are not considered “large”, since the corresponding values of the membership function are equal to 0. Of course, such a definition of the fuzzy set “big money” is subjective. The reader may have his own understanding of the ambiguous concept of “big money”. This representation will be reflected by other values of the parameters and functions of the class.

Definition 3.2

The set of space elements for which , is called the support of a fuzzy set and is denoted by (support). Its formal notation has the form

. (3.20)

Definition 3.3

The height of a fuzzy set is denoted and defined as

. (3.21)

Example 3.6

If And

, (3.22)

That .

, (3.23)

Definition 3.4

A fuzzy set is called normal if and only if . If the fuzzy set is not normal, then it can be normalized using the transformation

, (3.24)

where is the height of this set.

Example 3.7

Fuzzy set

(3.25)

after normalization it takes the form

. (3.26)

Definition 3.5

A fuzzy set is called empty and is denoted if and only if for each .

Definition 3.6

A fuzzy set is contained in a fuzzy set, which is written as , if and only if

(3.27)

for each .

An example of the inclusion (content) of a fuzzy set in a fuzzy set is illustrated in Fig. 3.9. The concept of degree of inclusion of fuzzy sets is also found in the literature. The degree of inclusion of a fuzzy set in a fuzzy set in Fig. 3.9 is equal to 1 (full inclusion). Fuzzy sets presented in Fig. 3.10 do not satisfy dependence (3.27); therefore, there is no inclusion in the sense of definition (3.6). However, a fuzzy set is contained in a fuzzy set to the degree

, (3.28)

, the condition is satisfied

Rice. 3.12. Fuzzy convex set.

Rice. 3.13. Fuzzy concave set.

Rice. Figure 3.13 illustrates a fuzzy concave set. It is easy to check that a fuzzy set is convex (concave) if and only if all its -cuts are convex (concave).

The foundations of the theory of fuzzy sets and fuzzy logic were laid in the late 1960s in the works of the famous American mathematician Lotfi Zadeh. His work "Fuzzy Sets", published in 1965 in the journal "Information and Control", became the impetus for the development of a new mathematical theory. He gave the name to a new branch of science - “fuzzy sets” (fuzzy - fuzzy, blurry, soft). The main reason for the appearance new theory became fuzzy and approximate reasoning that was used to describe processes, systems, and objects by humans. Mathematical theory fuzzy sets and fuzzy logic are generalizations classical theory sets and classical formal logic.

Before fuzzy modeling approach complex systems received recognition all over the world, more than a decade has passed. What did L. Zadeh propose? First, he expanded the classical Cantorian concept of a set, admitting that the characteristic function (the function of an element’s membership in a set) can take any value in the interval [ 0 , 1 ], and not just the values 0 or 1. He called such sets fuzzy [21]. L. Zadeh also defined a number of operations with fuzzy sets and proposed a generalization of logical inference methods.

Having subsequently introduced the concept of a linguistic variable and assumed that its values (terms) are fuzzy sets, L. Zadeh created an apparatus for describing processes intellectual activity, including vagueness and vagueness of expressions (for example, high, medium, low risks).

The task of fuzzy sets is to determine the membership of some object or element in given set. Let E - some set, and A- subset E, that is A Ì E. The fact that element x of the set E belongs to many A in set theory it is denoted as follows: x Ì A. To express this affiliation, you can use another concept - characteristic functionμA ( x), the value of which indicates whether (yes or no) X element A:

According to the theory of fuzzy sets, the characteristic membership function can take any value in the interval, and not just two - 0 and 1. In accordance with this, the element X i sets E may not belong A (μ Α ( X) = 0), be an element A small degree (μA value ( x) close to zero), be an element A to a large extent (μA ( x) close to 1) or be an element A(μA ( x) = 1). So, the concept of belonging is generalized. Fuzzy under set A universal set E denote A n and are determined by ordered pairs [ 22 ]:

Characteristic membership function (or simply membership function) μA ( x) takes values in some ordered set M(For example, M =). This membership function indicates the degree (or level) of membership of an element x to a subset A. A bunch of M called the reliability set. If M= (0, 1), then the fuzzy subset A can be considered as an ordinary or crisp set.

For fuzzy sets, as for ordinary sets, the basic logical operations are defined.

Equality. Two fuzzy sets A And IN are called equal if for all x Î E their characteristic functions are equal: μA ( x) = μB ( x). Designations: A = B.

Dominance. It is believed that a fuzzy set A belongs to the fuzzy set IN, if for everyone X Î E the following relation is valid: μA ( x) £ μB ( x) stand for: A Ì IN. The term "dominance" is sometimes used, that is, when A Ì IN, they say that IN dominates A.

Addition. Let M = , A And IN - fuzzy sets defined on E. A And IN complement each other if ∀x is Εμ /, (x) = 1 - μB (χ). Designations: A = A

Intersection two fuzzy sets (fuzzy “and”), denoting A ∩ IN - the largest fuzzy subset that is simultaneously in A And IN. Define like this:

An association two fuzzy sets (fuzzy "OR"), denoting A ∪ IN- the smallest fuzzy subset, which includes both A, so IN, with membership function

Difference two fuzzy sets A - IN = A ∩ B with membership function

Let M= and A - fuzzy set with elements X from the universal set E and a set of membership function values M. The quantity is called height fuzzy set A. Fuzzy set A is normal, if its height is 1, that is upper limit its membership function is 1(). A fuzzy set is called subnormal.

A fuzzy set is empty, If . A non-empty subnormal set can be normalized using the formula

Visual representation of operations on fuzzy sets. Let's consider rectangular system coordinates, on the ordinate axis of which the values of μA ( x), on the x-axis - elements are placed in random order E. If the set E is ordered in nature, then it is desirable to preserve this order in the placement of elements on the x-axis. This representation clearly shows simple operations on fuzzy sets.

Let A - fuzzy interval between 5 and 8, and IN- fuzzy number close to 4 (Fig. 4.4, A , b) .

The fuzzy set between 5 and 8 I (AND) about 4 (dark line) is illustrated in Fig. 4.4, V, fuzzy set between 5 and 8 OR (OR) about 4 - fig. 4.4, G(dark line).

Rice. 4.4. Examples of fuzzy sets ( A , b), them intersections (V) and associations ( G)

To describe fuzzy sets, the concepts of fuzzy and linguistic variables are introduced. A fuzzy variable is described by a set<β, X, A>, where β is the name of the variable, X- universal set (domain β), A- fuzzy set on X, describing restrictions on the values of the fuzzy variable β.

The values of a linguistic variable can be fuzzy variables, that is, the linguistic variable is on high level than a fuzzy variable. Each linguistic variable consists of: name; set of its values, also called the basic TERM set T. The elements of the basic term set are the names of the fuzzy variables of the universal set X syntactic rule G, by which new terms are generated using words of natural or formal language; semantic rule R, which each value of a linguistic variable is associated with a fuzzy subset of the set X.

A linguistic variable is described by a set<β, Τ , X , G , M>, where

β - name of the linguistic variable;

T - the set of its values (term set), which are the names of fuzzy variables, the domain of definition of each of which is the set X; a bunch of T called the basic term set of a linguistic variable;

G- a syntactic procedure that allows you to operate with elements of a term set T, in particular, generate new terms (values);

M - a semantic procedure that allows you to turn each new value into a linguistic variable formed by the procedure G, to a fuzzy variable, that is, to form a corresponding fuzzy set.

2. DESCRIPTION OF UNCERTAINTIES IN DECISION-MAKING THEORY

Interval number

Interval, million rubles per month

Number of responses in interval

Proportion of responses in the interval

Cumulative number of responses

Cumulative response rate

2.4.1. Fuzzy sets

Let A- some set. Subset B sets A characterized by its characteristic function

What is a fuzzy set? It is usually said that a fuzzy subset C sets A characterized by its membership function The value of the membership function at a point X shows the degree to which this point belongs to the fuzzy set. A fuzzy set describes the uncertainty corresponding to a point X– it is both included and not included in the fuzzy set WITH. For entry - chances, for second - (1-) chances.

If the membership function has the form (1) for some B, That C there is a regular (crisp) subset A. Thus, fuzzy set theory is no less a general mathematical discipline than ordinary set theory, since ordinary sets are a special case of fuzzy ones. Accordingly, we can expect that the theory of fuzziness as a whole generalizes classical mathematics. However, later we will see that the theory of fuzziness in a certain sense reduces to the theory random sets and thus is part classical mathematics. In other words, in terms of generality, ordinary mathematics and fuzzy mathematics are equivalent. However for practical application in decision theory, the description and analysis of uncertainties using the theory of fuzzy sets is very fruitful.

A regular subset could be identified with its characteristic function. Mathematicians do not do this, since to specify a function (in the currently accepted approach) it is necessary to first define a set. From a formal point of view, a fuzzy subset can be identified with its membership function. However, the term "fuzzy subset" is preferable when constructing mathematical models real phenomena.

Fuzzy theory is a generalization of interval mathematics. Indeed, the membership function

specifies interval uncertainty - all that is known about the value under consideration is that it lies in a given interval [ a,b]. Thus, the description of uncertainties using fuzzy sets is more general than using intervals.

The beginning of the modern theory of fuzziness was laid by the work of 1965 by the American scientist of Azerbaijani origin L.A. Zadeh. To date, thousands of books and articles have been published on this theory, and several international journals, quite a lot of both theoretical and applied work has been done. First book Russian author on fuzzy theory was published in 1980.

L.A. Zadeh considered the theory of fuzzy sets as an apparatus for analyzing and modeling humanistic systems, i.e. systems in which humans participate. His approach is based on the premise that the elements of human thinking are not numbers, but elements of some fuzzy sets or classes of objects for which the transition from “belonging” to “not belonging” is not abrupt, but continuous. Currently, fuzzy theory methods are used in almost all applied areas, including in enterprise management, product quality and technological processes.

L.A. Zadeh used the term "fuzzy set" (fuzzy set). The term “fuzzy” has been translated into Russian as fuzzy, fuzzy, indistinct, and even fluffy and foggy.

The apparatus of fuzzy theory is cumbersome. As an example, we give definitions of set-theoretic operations on fuzzy sets. Let C And D- two fuzzy subsets A with membership functions and respectively. By intersection, product CD, union , negation , sum C+ D fuzzy subsets are called A with membership functions

respectively.

How already noted, the theory of fuzzy sets in a certain sense reduces to the theory of probability, namely, to the theory of random sets. The corresponding cycle of theorems is given below. However, when solving applied problems, probabilistic-statistical methods and methods of fuzzy theory are usually considered different.

To get acquainted with the specifics of fuzzy sets, let's consider some of their properties.

In what follows, we assume that all fuzzy sets under consideration are subsets of the same set Y.

De Morgan's laws for fuzzy sets. As is known, the following identities of the algebra of sets are called Morgan’s laws

Theorem 1. For fuzzy sets the following identities hold:

(4)

The proof of Theorem 1 consists of directly checking the validity of relations (3) and (4) by calculating the values of the membership functions of the fuzzy sets involved in these relations based on the definitions given above.

Let us call identities (3) and (4) De Morgan's laws for fuzzy sets. Unlike the classical case of relations (2), they consist of four identities, one pair of which relates to the operations of union and intersection, and the second to the operations of product and sum. Like relation (2) in set algebra, de Morgan’s laws in fuzzy set algebra allow one to transform expressions and formulas that include negation operations.

Distributive law for fuzzy sets. Some properties of set operations do not hold for fuzzy sets. Yes, except when A- a “crisp” set (i.e. the membership function takes only the values 0 and 1).

Is the distributive law true for fuzzy sets? The literature sometimes vaguely states that “not always.” Let's be completely clear.

Theorem 2. For any fuzzy sets A, B And WITH

At the same time equality

fair if and only if, for all

Proof. We fix arbitrary element. To shorten the notation, we denote To prove identity (5), it is necessary to show that

Consider different orderings of three numbers a, b, c. Let first Then left side relationship (7) is and the right one, i.e. equality (7) is true.

Let Then in relation (7) on the left is a on the right, i.e. relation (7) is again an equality.

If then in relation (7) on the left is and on the right, i.e. both parts match again.

Three remaining number orderings a, b, c there is no need to disassemble, since in relation (6) the numbers b And c enter symmetrically. Identity (5) is proven.

The second statement of Theorem 2 follows from the fact that, in accordance with the definitions of operations on fuzzy sets

These two expressions coincide if and only if, when, what was required to be proved.

Definition 1. Carrier of a fuzzy set A is called the set of all points , for which

Corollary of Theorem 2. If the carriers of fuzzy sets IN And WITH coincide with Uh, then equality (6) takes place if and only if A -"crisp" (i.e. ordinary, classical, not fuzzy) set .

Proof. By condition in front of everyone. Then from Theorem 2 it follows that those. or , which means that A- clear set.

2.4.2. An example of describing uncertainty using

fuzzy set

The concept of “rich” is often used when discussing socio-economic problems, including in connection with preparation and decision-making. However, it is obvious that different faces put into this concept different content. In 1996, employees of the Institute of High Statistical Technologies and Econometrics conducted a sociological study of the perceptions of various segments of the population about the concept of a “rich person.”

The mini-survey looked like this:

1. At what monthly income (in million rubles per person) would you consider yourself a rich person?

2. Having assessed your current income, which category do you place yourself in:

a) rich;

b) above average income;

c) income below average;

d) poor;

d) below the poverty line?

(In the future, instead of the full names of categories, we will use letters, for example “c” - category, “b” - category, etc.)

3. Your profession, specialty.

A total of 74 people were interviewed, of which 40 were scientists and teachers, 34 people not employed in the field of science and education, including 5 workers and 5 retirees. Of all the respondents, only one (!) considers himself rich. Several typical answers from researchers and teachers are given in Table 1, and similar information for commercial workers is given in Table 2.

Table 1.

Typical answers from researchers and teachers

Answers to question 3		Answers to question 1, million rubles/person.	Answers to question 2
PhD
Teacher


	Senior. Researcher
	Physicist engineer
	Programmer
	scientist

table 2

Typical answers from commercial workers.

Answers to question 3	Answers to question 1	Answers to question 2
Vice President of the Bank
Deputy bank director
Boss. credit department
Department head valuable papers
Chief Accountant
Accountant
Bank manager
Head of Design Department

The range of answers to the first question is from 1 to 100 million rubles. per month per person. The survey results show that the wealth criterion for financial workers in general is slightly higher than for scientific workers (see histograms in Fig. 1 and Fig. 2 below).

The survey showed that to identify any specific meaning the amount that is necessary “for complete happiness”, even with a small variation, is impossible, which is quite natural. As can be seen from tables 1 and 2, the monetary equivalent of wealth ranges from 1 to 100 million rubles per month. The opinion was confirmed that the overwhelming majority of educators classify their income as category “c” and below (81% of respondents), including 57% who classified their income as category “d”.

With employees of commercial structures and budgetary organizations a different picture: “d” - category 1 person (4%), “d” - category 4 people (17%), “b” - category - 46% and 1 person “a” - category.

Pensioners, which is not surprising, classified their income as category “d” (4 people), and only one person indicated the “g” category. The workers answered like this: 4 people - “c”, and one person - “b”.

To present the general picture, Table 3 shows data on the responses of workers in other professions.

Table 3.

Typical answers from workers in various professions.

Answers to question 3	Answers to question 1	Answers to question 2
Trade worker

Driver
Serviceman
Gas station owner
Pensioner
Factory Manager

Housewife
Mechanic

Computer's operator
Social Security Worker
Architect

Traceable interesting phenomenon: the higher the level of wealth for a person, the lower the category relative to this level he considers himself.

A natural way to summarize data is to use histograms. To do this, you need to group the answers. 7 classes (intervals) were used:

1 – up to 5 million rubles per month per person (inclusive);

2 – from 5 to 10 million;

3- from 10 to 15 million;

4 – from 15 to 20 million;

5 – from 20 to 25 million;

6 – from 25 to 30 million;

7 – more than 30 million.

(In all intervals, the left boundary is excluded, and the right, on the contrary, is included.)

Summary information is presented in Fig. 1 (for researchers and teachers) and Fig. 2 (for all others, i.e. for persons not engaged in the field of science and education - employees of other budgetary organizations, commercial structures, workers, pensioners) .

Fig.1. Histogram of answers to question 1 for researchers and teachers (40 people).

Fig.2. Histogram of answers to question 1 for people not employed in the field of science and education (34 people).

For the two selected groups, as well as for some subgroups of the second group, summary average characteristics were calculated - sample arithmetic averages, medians, modes. At the same time, the median of the group is the number of million rubles, called central by serial number respondents in an increasing series of answers to question 1, and the mode of the group is the interval at which the histogram bar is the highest, i.e. it got hit maximum amount respondents. The results are shown in table. 4.

Table 4.

Summary average characteristics of responses to question 1

for various groups (in million rubles per month per person).

Group of respondents	arithmetic
Scientists and teachers
Persons not engaged in the field of science and education
Employees of commercial structures and budgetary organizations

Pensioners

Let’s construct a fuzzy set that describes the concept of “rich person” in accordance with the views of the respondents. To do this, we will compile Table 5 based on Fig. 1 and Fig. 2, taking into account the range of answers to the first question.

Table 5.

Number of responses falling within the intervals

Continuation of Table 5.

	Interval number
	Interval, million rubles per month
	Number of responses in interval
	Proportion of responses in the interval
	Cumulative number of responses
	Cumulative response rate

The fifth line of Table 5 specifies the membership function of a fuzzy set expressing the concept of a “rich person” in terms of his monthly income. This fuzzy set is a subset of the set of 9 intervals specified in line 2 of Table 5. Or a set of 9 conditional numbers (0, 1, 2, ..., 8). Empirical function The distribution, constructed from a sample of the responses of 74 respondents to the first question of the mini-questionnaire, describes the concept of “rich person” as a fuzzy subset of the positive semi-axis.

2.4.3. On the development of pricing methodology

based on fuzzy set theory

To assess the values of indicators that do not have quantification, you can use fuzzy set methods. For example, in the dissertation of P.V. Bityukov fuzzy sets were used to model pricing problems for electronic training courses used in distance learning. He conducted a study of the values of the factor “Level of course quality” using fuzzy sets. During practical use proposed by P.V. Bityukov pricing methods, the values of a number of other factors can also be determined using the theory of fuzzy sets. For example, it can be used to determine the forecast of the rating of a specialty at a university with the help of experts, as well as the values of other factors related to the “Course Features” group. Let us describe the approach of P.V. Bityukov as an example of the practical use of fuzzy set theory.

The value of the score assigned to each interval for the factor “Course quality level” is determined on a universal scale, where it is necessary to place the values of the linguistic variable “Course quality level”: LOW, MEDIUM, HIGH. The degree of belonging of a certain value is calculated as the ratio of the number of answers in which it occurred in a certain interval of the scale to the maximum (for this value) number of answers over all intervals.

During the work on the dissertation, a survey of experts was conducted about the degree of influence of the quality level electronic courses on their use value. During the survey, each expert was asked to evaluate from the consumer’s perspective the value of a particular class of courses depending on the level of quality. The experts gave their assessment for each course class on a 10-point scale (where 1 is min, 10 is max). To move to a universal scale, all values of the 10-point value rating scale were divided by a maximum score of 10.

Using the properties of the membership function, it is necessary to pre-process the data in order to reduce the distortions introduced by the survey. Natural properties membership functions are the presence of one maximum and smooth fronts that decay to zero. To process statistical data, you can use the so-called hint matrix. Obviously erroneous elements are first removed. The deletion criterion is the presence of several zeros in the line around this element.

The elements of the hint matrix are calculated using the formula: ,

where is a table element with survey results grouped by intervals. The hint matrix is a row in which the maximum element is selected: , and then all its elements are transformed according to the formula:

For columns where , linear approximation is applied:

The calculation results are summarized in a table, on the basis of which membership functions are constructed. For this purpose there are maximum elements line by line: . The membership function is calculated using the formula: . The calculation results are given in table. 6.

Table 6

Linguistic variable membership function values

	Interval on a universal scale

Rice. 3 . Graph of membership functions for the values of the linguistic variable “Course quality level”

In Fig.3 solid lines shows the membership functions of the values of the linguistic variable “Course quality level” after processing the table containing the survey results. As can be seen from the graph, the membership functions satisfy the properties described above. For comparison, the dotted line shows the membership function of the linguistic variable for the LOW value without data processing.

2.4.4. On the statistics of fuzzy sets

Fuzzy sets – private view objects of non-numerical nature. Statistical methods analysis of objects of non-numerical nature are described in. In particular, the average value of a fuzzy set can be determined by the formula:

As is known, methods of statistics of non-numerical data are based on the use of distances (or difference indicators) in the corresponding spaces of a non-numerical nature. Distance between fuzzy subsets A And IN sets X = {x 1 , x 2 , …, x k) can be defined as

where is the fuzzy set membership function A, a - membership function of a fuzzy set B. Other distances can be used:

(Let us take this distance to be 0 if the membership functions are identically equal to 0.)

In accordance with the axiomatic approach to the choice of distances (metrics) in spaces of non-numerical nature, an extensive set of axiom systems has been developed, from which one or another type of distances (metrics) in specific spaces is derived. Using probabilistic models the distance between random fuzzy sets is itself a random variable, which in a number of formulations has asymptotically normal distribution.

2.4.5. Fuzzy sets as projections of random sets

From the very beginning modern theory fuzzyness began to be discussed in the 1960s about its relationship with probability theory. The fact is that the membership function of a fuzzy set resembles a probability distribution. The only difference is that the sum of the probabilities over all possible values of the random variable (or the integral, if the set possible values uncountable) is always equal to 1, and the sum S values of the membership function (in the continuous case - the integral of the membership function) can be any non-negative number. There is a temptation to normalize the membership function, i.e. divide all its values by S(at S 0) to reduce it to a probability distribution (or probability density). However, fuzziness specialists rightly object to such a “primitive” reduction, since it is carried out separately for each fuzziness (fuzzy set), and the definitions of ordinary operations on fuzzy sets cannot be consistent with it. The last statement means the following. Let the membership functions of fuzzy sets be transformed in the indicated way sets A And IN. How are the membership functions transformed? Install this impossible in principle. The last statement becomes completely clear after considering several examples of pairs of fuzzy sets with the same sums of values of membership functions, but different results of set-theoretic operations on them, and the sums of values of the corresponding membership functions for these results of set-theoretic operations, for example, for intersections of sets are also different.

In works on fuzzy sets, it is quite often stated that the theory of fuzzyness is an independent branch of applied mathematics and is not related to probability theory (see, for example, a review of the literature in monographs). Authors who compared fuzzy theory and probability theory usually emphasized the difference between these areas of theoretical and applied research. Usually axiomatics are compared and application areas are compared. It should be immediately noted that the arguments for the second type of comparisons do not have evidentiary force, since regarding the limits of applicability of even such a long-standing scientific field, as probabilistic-statistical methods, there are different opinions. Let us recall that the result of the reasoning of one of the most famous French mathematicians, Henri Lebesgue, regarding the limits of applicability of arithmetic is as follows: “Arithmetic is applicable when it is applicable” (see his monograph).

When comparing various axiomatics of fuzzy theory and probability theory, it is easy to see that the lists of axioms differ. From this, however, it does not at all follow that a connection cannot be established between these theories, such as the well-known reduction of Euclidean geometry on the plane to arithmetic (more precisely, to the theory numerical system- see, for example, monograph). Let us recall that these two axiomatics - Euclidean geometry and arithmetic - at first glance are very different.

One can understand the desire of enthusiasts of the new direction to emphasize the fundamental novelty of their scientific apparatus. However, it is equally important to establish connections between the new approach and previously known ones.

As it turns out, the theory of fuzzy sets is closely related to the theory of random sets. Back in 1974, it was shown in the work that fuzzy sets can naturally be considered as “projections” of random sets. Let's consider this method of reducing the theory of fuzzy sets to the theory of random sets.

Definition 2. Let - a random subset of a finite set Y. Fuzzy set IN, defined on Uh, called projection A and is designated Proj A, If

(8)

in front of everyone

Obviously, every random set A can be correlated using formula (8) with a fuzzy set B = Proj A. It turns out the opposite is also true.

Theorem 3. For any fuzzy subset IN final sets U there is a random subset A sets U such that B = Proj A.

Proof. It is enough to set the distribution of the random set A. Let U 1- carrier IN(see definition 1 above). Without loss of generality we can assume that at some m and elements U 1 numbered in such an order that

Let us introduce sets

For all other subsets X sets U let's put P(A=X)=0. Since the element y t included in the set Y(1), Y(2),…, Y(t) and is not included in sets Y(t+1),…, Y(m), That From the above formulas it follows that If then, obviously, Theorem 3 is proven.

The distribution of a random set with independent elements, as follows from the considerations in Chapter 8 of the monograph, is completely determined by its projection. For a finite random set general view this is wrong. To clarify the above, we need the following theorem.

Theorem 4. For a random subset A sets U from finite number elements sets of numbers And expressed one through the other .

Proof. The second set is expressed in terms of the first in the following way:

The elements of the first set can be expressed through the second using the formula of inclusions and exclusions from formal logic, according to which

In this formula in the first sum at runs through all elements of the set Y\X, in the second sum the summation variables at 1 And at 2 do not coincide and also run through this set, etc. A reference to the inclusion and exclusion formula completes the proof of Theorem 4.

In accordance with Theorem 4, a random set A can be characterized not only by a distribution, but also by a set of numbers There are no other relations of the equality type in this set. This set includes numbers; therefore, fixing the projection of a random set is equivalent to fixing k = Card(Y) parameters from (2k-1) parameters defining the distribution of a random set A in general.

The following theorem will be useful.

Theorem 5. If Proj A = B, That

To prove it, it is enough to use the identity from the theory of random sets, the formula for the probability of covering , the definition of the negation of a fuzzy set, and the fact that the sum of all P(A=X) is equal to 1. In this case, the formula for the probability of covering means the following statement: to find the probability of covering a fixed element q random subset S finite set Q, it is enough to calculate

where the summation is over all subsets A sets Q containing q.

2.4.6. Intersections and products of fuzzy

and random sets

Let us find out how operations on random sets relate to operations on their projections. By virtue of De Morgan's laws (Theorem 1) and Theorem 5, it is sufficient to consider the operation of intersection of random sets.

Theorem 6. If random subsets A 1 And A 2 finite set U are independent, then the fuzzy set is the product of fuzzy sets Proj A 1 And Proj A 2 .

Proof. It must be shown that for any

According to the formula for the probability of covering a point with a random set (see above)

It is easy to check that the distribution of the intersection of random sets can be expressed in terms of their joint distribution in the following way:

From relations (10) and (11) it follows that the covering probability for the intersection of random sets can be represented as a double sum

Note now that the right-hand side of formula (12) can be rewritten as follows:

(13)

Indeed, formula (12) differs from formula (13) only in that it groups terms in which the intersection of the summation variables takes a constant value. Using the definition of independence of random sets and the rule for multiplying sums, we obtain that from (12) and (13) the equality follows

Then equality (14) reduces to the condition

It is clear that relation (15) is satisfied if and only if p 2 p 3=0 for all i.e. there is not a single element such that at the same time And , and this is equivalent to the emptiness of the intersection of the supports of random sets and . Theorem 7 is proven.

24.7. Reduction of the sequence of operations

over fuzzy sets to a sequence of operations

over random sets

Above we obtained some connections between fuzzy and random sets. It is worth noting that the study of these connections in the work began with the introduction of random sets with the aim of developing and generalizing the apparatus of fuzzy sets by L. Zadeh. (To fix the priority at the global level, it is advisable to note that this work was carried out in 1974 and reported at the Central Economics and Mathematics Institute of the USSR Academy of Sciences at the all-Union scientific seminar "Multidimensional statistical analysis and probabilistic modeling real processes"December 18, 1974 - see.) The fact is that the mathematical apparatus of fuzzy sets does not allow adequate consideration various options dependencies between concepts (objects) modeled with its help are not flexible enough. Thus, to describe the “common part” of two fuzzy sets there are only two operations - product and intersection. If the first of them is applied, then it is actually assumed that the sets behave as projections of independent random sets (see Theorem 6 above). The operation of intersection also imposes well-defined restrictions on the type of dependence between sets (see Theorem 7 above), and in this case even the necessary and sufficient conditions. It is desirable to have broader capabilities for modeling dependencies between sets (concepts, objects). Usage mathematical apparatus random sets provides such capabilities.

The purpose of reducing the theory of fuzzy sets to the theory of random sets is to see behind any construction of fuzzy sets a construction of random sets that determines the properties of the first, similar to how we see behind the probability distribution density random variable. In this section we present results on reducing the algebra of fuzzy sets to the algebra of random sets.

Definition 4. Probability space (Ω ,G,P} we call it divisible if for any measurable set X G and any positive number , smaller P(X), you can specify measurable a bunch of such that

Example. Let be the unit cube of a finite-dimensional linear space, G is the sigma algebra of Borel sets, and P- Lebesgue measure. Then (Ω ,G,P} - divisible probability space.

Thus, divisible probability space is not exotic. An ordinary cube is an example of such a space.

The proof of the statement formulated in the example is carried out using standard mathematical techniques. They are based on the fact that a measurable set can be approximated as accurately as desired by open sets, the latter being represented as a sum of no more than a countable number of open balls, and for balls the divisibility is checked directly (a body of volume is separated from a ball X by a corresponding plane).

Theorem 8. Let a random set be given A on a divisible probability space {Ω,G,P) with values in the set of all subsets of the set U of a finite number of elements, and a fuzzy set D on U. Then there are random sets C 1, C 2, C 3, C 4 on same probability space such that

Where B = Proj A.

Proof. Due to the validity of De Morgan's laws for fuzzy (see Theorem 1 above) and for random sets, as well as Theorem 5 above (on negations), it is sufficient to prove the existence of random sets C 1 And C 2 .

Consider the probability distribution in the set of all subsets of the set U, corresponding to the random set WITH such that Proj C = D(it exists by virtue of Theorem 3). Let's build a random set C 2 with the specified distribution, independent of A. Then by Theorem 6.

So that for the resulting random set the random set does not change). After going through all the elements U, we get a random set , for which the required is fulfilled. Theorem 8 is proven.

The main result on reducing the theory of fuzzy sets to the theory of random sets is given by the following theorem.

Theorem 9. Let - some fuzzy subsets of the set U from a finite number of elements. Let's consider the results of sequential execution of set-theoretic operations

where is a symbol of one of the following set-theoretic operations on fuzzy sets: intersection, product, union, sum (on different places there may be different symbols). Then there are random subsets the same set U such that

and, in addition, the results of set-theoretic operations are related by similar relations

where the sign means that at the place in question there is a symbol for the intersection of random sets, if in the definition Bm there is a symbol of intersection or a symbol of the product of fuzzy sets, and, accordingly, a symbol of the union of random sets, if in Bm there is a union symbol or a sum symbol of fuzzy sets.