What function is a fuzzy set characterized by? Examples of fuzzy sets

Let X be a universal (basic) set, x be an element of X, and R be some property. An ordinary (crisp) subset A of a universal set X, whose elements satisfy the property R, is defined as the set of ordered pairs
A = μ A x / x, where μ A x is a characteristic function that takes the value 1 if x satisfies property R, and 0 otherwise.

Fuzzy subset is different from regular topics that for elements x of X there is no clear yes-no answer regarding the property R. In this regard, a fuzzy subset A of a universal set X is defined as a set of ordered pairs A = μ A x / x , where μ A x – characteristic membership function(or simply membership function), taking values ​​in some completely ordered set M = 0 ;

1 . The membership function indicates the degree (or level) of membership of an element x to a subset A. The set M is called the membership set. If M = 0 ; 1, then the fuzzy subset A can be considered as an ordinary or crisp set. The degree of membership μ A x is a subjective measure of how much an element x ∈ X corresponds to the concept, the meaning of which is formalized by the fuzzy set A. Carrier fuzzy set A is a clear subset S A of the universal set X with the property μ A x > 0, i.e.

If the carrier of a fuzzy set A is a discrete subset S A , then the fuzzy subset A of a universal set X consisting of n elements can be represented as a union finite number single-point sets μ A x / x using the symbol ∑ : A = ∑ i = 1 n μ A x i / x i . This implies that the elements x i are ordered in ascending order in accordance with their indices, i.e.< x 2 < x 3 < … < x n .

x 1

If the carrier of a fuzzy set A is a continuous subset S A, then the fuzzy subset A of the universal set X, considering the symbol ∫ as a continuous analogue of the union symbol introduced above for discrete fuzzy sets ∑, can be represented as a union of an infinite number of single-point sets μ A x / x:

A = ∫ X μ A x / x . Example.

Let the universal set X correspond to the set of possible values ​​of product thickness from 10 mm to 40 mm with a discrete step of 1 mm. The fuzzy set A, corresponding to the fuzzy concept of “small thickness of the product,” can be represented in the following form:

A = 1 / 10 ;

0.9/11; 0.8/12; 0.7/13;

0.5 / 14 ;

0.3 / 15 ; 0.1 / 16 ; 0 / 17 ;

... ; 0 / 40 A = 1 / 10 + 0.9 / 11 + 0.8 / 12 + 0.7 / 13 + 0.5 / 14 + 0.3 / 15 + 0.1 / 16 + 0 / 17 + … + 0 / 40,

where the summation sign denotes a non-operation

A = ∫ X μ A x / x . arithmetic addition

, but combining elements into one set. The carrier of the fuzzy set A will be a finite subset (discrete carrier): Graphical representation fuzzy set small

The fuzzy set A is called final, if its support S A is a finite crisp set. In this case, by analogy with ordinary sets, we can say that such a fuzzy set has a finite card A = card S A . The fuzzy set A is called endless, if its support S A is not a finite crisp set. Wherein countable a fuzzy set will be called a fuzzy set with a countable medium having counting power in the usual sense in terms of crisp set theory, i.e. if S A contains infinite number elements, which, however, can be numbered with natural numbers 1,2,3. .. , and it is fundamentally impossible to reach the last element when numbering. Uncountable

A = ∫ X μ A x / x . a fuzzy set will be called a fuzzy set with an uncountable carrier having uncountable power of the continuum , i.e. if S A contains an infinite number of elements that cannot be numbered by natural numbers 1,2,3.. . Fuzzy concept

A = ∫ X μ A x / x ."a very small number of details" can be represented as a finite fuzzy set A = 1 / 0 + 0.9 / 1 + 0.8 / 2 + 0.7 / 3 + 0.5 / 4 + 0.1 / 5 + 0 / 6 + ... with card power (A) = 6 and carrier S A = 0 ;

1 ; 2 ; 3; 4 ; 5, which is a finite crisp set. The fuzzy concept of “very

a large number of details" can be represented as A = 0 / 0 + … + 0.1 / 1 0 + 0.4 / 11 + 0.7 / 12 + 0.9 / 13 + 1 / 14 + 1 / 15 + … + 1 / n + … , n ∈ N – fuzzy set with infinite countable support S A ≡ N (set natural numbers ), which has countable power in the usual sense. An uncountable fuzzy set A corresponding to the fuzzy concept “very hot” is defined on the universal set of temperature values ​​(in Kelvin) by temperature x ∈ [ 0 ;< 1 ∞) and the membership function μ A = 1 − e − x , with support S A ≡ R + (the set of non-negative real numbers), which has an uncountable continuum power.

The quantity sup x ∈ X μ A x is called height fuzzy set.

A non-empty subnormal set can always be normalized by dividing all values ​​of the membership function by its maximum value μ A x sup x ∈ X μ A x .

The quantity sup x ∈ X μ A x is called unimodal, if μ A x = 1 for only one point x ( fashion) of the universal set X.

The quantity sup x ∈ X μ A x is called point, if μ A x > 0 only for one point x of the universal set X .

Many α -level fuzzy set A defined on a universal set X is called a clear subset A α of the universal set X, defined as:

A α = x ∈ X ∣ μ A x ≥ α, where α ∈ 0;

A = ∫ X μ A x / x . 1 .

A = 0.8 / 1 + 0.6 / 2 + 0.2 / 3 + 1 / 4 , A 0.5 = 1 ; 2 ; 4, where A 0.5 is a clear set, including those elements x of ordered pairs μ A x / x that make up the fuzzy set A, for which the value of the membership function satisfies the condition μ A x ≥ α.

For α-level sets the following holds: next property: if α 1 ≥ α 2, then the power of the subset A α 1 is not greater than the power of the subset A α 2.

Elements x ∈ X for which μ A x = 0.5 are called transition points

fuzzy set A. Core< μ A x < 1 .

A = ∫ X μ A x / x . a fuzzy set A defined on a universal set X is called a clear set core A, the elements of which satisfy the condition core A = x ∈ X ∣ μ A x = 1.

Border of a fuzzy set A defined on a universal set X is called a clear set front A whose elements satisfy the condition front A = x ∈ X ∣ 0 Let X = 0;< x < b ; x , a , b ∈ X (рис.2.3).

1 ; 2 ;

By tradition, clear sets are usually illustrated by circles with sharply outlined boundaries. Fuzzy sets are circles formed by individual points: in the center of the circle there are many points, and closer to the periphery their density decreases to zero; the circle seems to be shaded at the edges. Such “fuzzy sets” can be seen... at a shooting range - on the wall where targets are hung. Bullet marks form random sets whose mathematics are known. It turned out that the long-developed apparatus of random sets is suitable for operating with fuzzy sets...

The concept of a fuzzy set - an attempt mathematical formalization fuzzy information for the purpose of using it in constructing mathematical models complex systems. This concept is based on the idea that the elements that make up a given set and have common property, may have this property to varying degrees and, therefore, belong to a given set with varying degrees.

One of the simplest ways mathematical description fuzzy set – characterization of the degree of membership of an element to a set by a number, for example, from the interval. Let X– a certain set of elements. In what follows we will consider subsets of this set.

Fuzzy set A in X is called a collection of pairs of the form ( x, m A(x)), Where xÎX, and m A– function x®, called membership function fuzzy set A. m value A(x) this function for a specific x is called the degree of membership of this element in the fuzzy set A.

As can be seen from this definition, a fuzzy set is fully described by its membership function, so we will often use this function as a designation for a fuzzy set.

Ordinary sets constitute a subclass of the class of fuzzy sets. Indeed, the membership function of an ordinary set BÌ X is its characteristic function: m B(x)=1 if xÎ B and m B(x)=0 if xÏ B. Then, in accordance with the definition of a fuzzy set, the ordinary set IN can also be defined as a set of pairs of the form ( x, m B(x)). Thus, a fuzzy set is more broad concept than an ordinary set, in the sense that the function belonging fuzzy a set can, generally speaking, be an arbitrary function or even an arbitrary mapping.

We are speaking fuzzy set. And many what? If we are consistent, we have to state that an element of a fuzzy set turns out to be... a new fuzzy set of new fuzzy sets, etc. Let's turn to classic example- To pile of grain. An element of this fuzzy set will be million grains, For example. But a million grains is not clear at all element, and new fuzzy set. After all, when counting grains (manually or automatically), it’s not surprising to make a mistake - taking 999,997 grains as a million, for example. Here we can say that element 999,997 has a membership function value for the set “million” equal to 0.999997. In addition, the grain itself is again not an element, but a new fuzzy set: there is a full-fledged grain, and there are two fused grains, an underdeveloped grain or just a husk. When counting grains, a person must reject some, take two grains as one, and in another case, one grain as two. A fuzzy set is not so easy to stuff into a digital computer with classical languages: the elements of an array (vector) must be new arrays of arrays (nested vectors and matrices, if we talk about Mathcad). Classical crisp set mathematics (number theory, arithmetic, etc.) is the hook by which reasonable man fixes (determines) himself in the slippery and unclear world around him. And a hook, as you know, is a rather crude tool, often spoiling what it clings to. Terms representing fuzzy sets – “a lot”, “slightly”, “a little”, etc. etc. - it’s difficult to “stuff” it into a computer also because they context dependent. It’s one thing to say “Give me some seeds” to a person who has a glass of seeds, and another thing to say to a person sitting behind the wheel of a truck with seeds.

Fuzzy subset A sets X characterized by membership function m A:X→, which assigns each element xÎ X number m A(x) from the interval characterizing the degree of membership of the element X subset A. Moreover, 0 and 1 represent, respectively, the lowest and highest degree belonging of an element to a specific subset.

Let us give basic definitions.

· Value sup m A(x) called 3; fuzzy set A. Fuzzy set A details" can be represented as A = 0 / 0 + … + 0.1 / 1 0 + 0.4 / 11 + 0.7 / 12 + 0.9 / 13 + 1 / 14 + 1 / 15 + … + 1 / n + … , n ∈ N – fuzzy set with infinite countable support S A ≡ N (set , if its height is 1 , i.e. the upper bound of its membership function is 1. When sup mA(x)<1 fuzzy set is called ∞) and the membership function μ A = 1 − e − x , with support S A ≡ R + (the set of non-negative real numbers), which has an uncountable continuum power.

A fuzzy set is called height, if its membership function is equal to zero on the entire set X, i.e. m 0 (x)= 0 " xÎ X.

Fuzzy set empty , If " xÎ E m A ( x)=0 . A non-empty subnormal set can be normalized by the formula

(Fig. 1).

Fig.1. Normalization of a fuzzy set with a membership function. .

1 . The membership function indicates the degree (or level) of membership of an element x to a subset A. The set M is called the membership set. If M = 0 ; fuzzy set A(designation supp A) with membership function m A(x) called a set of the form suppA={x|xÎ X, m A(x)> 0). For practical applications carriers of fuzzy sets are always limited. Thus, the carrier of a fuzzy set of admissible modes for a system can be a clear subset (interval), for which the degree of admissibility is not equal to zero (Fig. 2).

Rice. 3. Core, carrier and α- section of a fuzzy set

Meaning α called α -level. The carrier (kernel) can be considered as a section of a fuzzy set on zero (unit) α -level.

Rice. 3 illustrates definitions carrier, core,α - sections andα - level fuzzy set.

Annotation: The lecture presents modeling methods economic tasks using fuzzy sets in the Mathcad environment. The basic concepts of the theory of fuzzy sets are introduced. Examples show operations on sets and calculation of properties. Original problems in which a fuzzy-set approach is used in the decision-making process are considered. The modeling technique is implemented using matrices from the Mathcad program.

Purpose of the lecture. Introduce fuzzy sets. Learn to pose a problem for constructing a fuzzy set model. Show how to construct fuzzy sets and perform operations on them in Mathcad. Present methods for solving a fuzzy set model in the process of solving problems.

6.1 Fuzzy-set modeling

When modeling a wide class of real objects, it becomes necessary to make decisions under conditions of incomplete fuzzy information. Modern promising direction of modeling various types uncertainty is the theory of fuzzy sets. Within the framework of fuzzy set theory, methods have been developed for formalizing and modeling human reasoning, such concepts as “more or less high level inflation", "stable position in the market", "more valuable", etc.

The concept of fuzzy sets was first proposed by the American scientist L.A. Zade (1965). His ideas contributed to the development of fuzzy logic. Unlike standard logic with two binary states (1/0, Yes/No, True/False), fuzzy logic allows you to determine intermediate values ​​between standard estimates. Examples of such assessments are: “more likely yes than no”, “probably yes”, “a little to the right”, “sharply to the left”, in contrast to the standard ones: “to the right” or “to the left”, “yes”. In the theory of fuzzy sets, fuzzy numbers are introduced as fuzzy subsets specialized type, corresponding to statements like “the value of the variable is approximately equal to a.” As an example, consider a triangular fuzzy number, where three points are distinguished: the minimum possible, the most expected and the maximum possible meaning factor a. Triangular numbers are the most commonly used type of fuzzy numbers in practice, and most often they are used as predictive parameter values. For example, the expected value of inflation at next year. Let the most probable value be 10%, the minimum possible value be 5%, and the maximum possible value be 20%, then all these values ​​can be reduced to the form of a fuzzy subset or fuzzy number A: A: (5, 10, 20)

With the introduction of fuzzy numbers, it has become possible to predict future values ​​of parameters that vary within a specified design range. A set of operations on fuzzy numbers is introduced, which are reduced to algebraic operations with ordinary numbers when specifying a certain confidence interval (membership level). The use of fuzzy numbers allows you to set a calculated corridor for the values ​​of the predicted parameters. Then the expected effect is also assessed by the expert as a fuzzy number with its own calculated spread (degree of fuzzyness).

Fuzzy logic as a model of human thought processes, built into systems artificial intelligence and to automated support tools decision making(in particular, in control systems technological processes).

6.2 Basic concepts of fuzzy set theory

Set is an undefined concept in mathematics. Georg Cantor (1845 – 1918) – German mathematician whose work forms the basis modern theory sets, gives the following concept: “... a set is a lot, conceived as one.”

The set that includes all the objects considered in the problem is called the universal set. Universal set usually denoted by the letter . Universal set is maximum set in the sense that all objects are its elements, i.e. the statement within the problem is always true. The minimal set is empty set– , which does not contain any elements. All other sets in the problem under consideration are subsets of the set. Recall that a set is called a subset of a set if all elements are also elements of . Specifying a set is a rule that allows you to unambiguously establish, relative to any element of a universal set, whether it belongs to the set or not. In other words, it is a rule to determine which of two statements, or , is true and which is false. One way to define sets is to specify them using characteristic function.

The characteristic function of a set is a function defined on a universal set and taking the value one on those elements of the set that belong to and the value zero on those elements that do not belong to:

(6.1)

As an example, consider universal set and its two subsets: - the set of numbers less than 7, and - the set of numbers slightly less than 7. The characteristic function of the set has the form

(6.2)

Set in in this example is an ordinary set.

It is impossible to write down the characteristic function of the set using only 0 and 1. For example, should the numbers include 1 and 2? Is the number 3 “much” or “not much” less than 7? Answers to these and similar questions can be obtained depending on the conditions of the problem in which the sets and are used, as well as on the subjective view of the person solving this problem. The set is called a fuzzy set. When compiling the characteristic function of a fuzzy set problem solver(an expert) can express his opinion as to the extent to which each of the numbers in the set belongs to the set. You can choose any number from the segment as the degree of membership. At the same time, it means the expert’s complete confidence that - equally complete confidence, which indicates that the expert finds it difficult to answer the question whether it belongs to the set or does not belong. If , then the expert is inclined to classify it as a set, but if , then I am not inclined.

The membership function of a fuzzy set is a function that

This function is called membership function fuzzy set. - Maximum value membership function present in the set - top edge- called supremum. Membership function reflects a specialist’s subjective view of the problem and brings individuality to its solution.

The characteristic function of an ordinary set can be considered as a membership function of this set, but unlike a fuzzy set, it takes only two values: 0 or 1.

A fuzzy set is a pair , Where - universal set, - membership function fuzzy set.

The carrier set or carrier of a fuzzy set is a subset of the set, consisting of elements on which .

The transition point of a fuzzy set is called set element, on which .

In the example under consideration, where , is the set of numbers less than 7, is the set of numbers slightly less than 7, we subjectively select values ​​for the set that will form the membership function. Table 6.1 presents the membership functions for and and .

A more compact notation for finite or countable fuzzy sets is often used. So, instead of the above tabular representation of the subsets and , these subsets can be written as follows.

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 crisp sets can also be written in a similar way. 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 is a set of 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 of real numbers “close to the number 7” can be determined 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 of 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).

Modern science and technology cannot be imagined without the widespread use of mathematical modeling, since full-scale experiments cannot always be carried out, they are often too expensive and require considerable time, and in many cases they are associated with risk and large material or moral costs. The essence of mathematical modeling is to replace a real object with its “image” - a mathematical model - and further study of the model using computational and logical algorithms implemented on computers. The most important requirement for mathematical model, is the condition of its adequacy (correct correspondence) to the real object being studied relative to the selected system of its properties. This, first of all, means a correct quantitative description of the properties of the object under consideration. The construction of such quantitative models is possible for simple systems.

The situation is different with complex systems. To obtain significant conclusions about the behavior of complex systems, it is necessary to abandon high accuracy and rigor when building a model and use approaches that are approximate in nature when constructing it. One of these approaches is associated with the introduction of linguistic variables that describe a person’s unclear reflection of the surrounding world. In order for a linguistic variable to become a full-fledged mathematical object, it was introduced concept of fuzzy multitudes.

In the theory of crisp sets, the characteristic function of a crisp set was considered in universal space
, equal to 1 if element satisfies the property and therefore belongs to the set , and equal to 0 otherwise. Thus, we were talking about a clear world (Boolean algebra), in which the presence or absence of a given property is determined by the values ​​0 or 1 (“no” or “yes”).

However, everything in the world cannot be divided only into white and black, truth and lies. So, even the Buddha saw a world filled with contradictions, things could be true to some extent and, to some extent, false at the same time. Plato laid the foundation for what would become fuzzy logic by pointing out that there was a third realm (beyond Truth and Falsehood) where these contradictions are relative.

University of California professor Zadeh published the paper “Fuzzy Sets” in 1965, in which he extended the two-valued estimate of 0 or 1 to an unlimited multi-valued estimate above 0 and below 1 in a closed interval and first introduced the concept of a “fuzzy set.” Instead of the term “characteristic function,” Zadeh used the term “membership function.” Fuzzy set (the same notation is retained as for a crisp set) in the universal space
through the membership function
(same notation as for the characteristic function) is defined as follows

(3.1)

The membership function is most often interpreted as follows: the value
means subjective assessment degree of element membership fuzzy set , For example,
means that 80% owned . Therefore, there must be “my membership function”, “your membership function”, “specialist’s membership function”, etc. The graphical representation of a fuzzy set, a Venn diagram, is represented by concentric circles in Fig. 1. The membership function of a fuzzy set has a bell-shaped graph, in contrast to the rectangular characteristic function of a clear set, Fig. 1.

You should pay attention to the connection between the crisp and fuzzy sets. Two values ​​(0,1) of the characteristic function belong to a closed interval of values ​​of the membership function. Therefore, a crisp set is a special case of a fuzzy set, and the concept of a fuzzy set is an extended concept that also covers the concept of a crisp set. In other words, a crisp set is also a fuzzy set.

A fuzzy set is strictly defined using the membership function and does not contain any vagueness. The fact is that a fuzzy set is strictly defined using the estimated values ​​of a closed interval, and this is the membership function. If the universal set
consists of a discrete finite set of elements, then, based on practical considerations, indicate the value of the membership function and the corresponding element using the separation signs / and +. For example, let the universal set consist of integers less than 10, then the fuzzy set "small numbers" can be represented as

A=1/0 + 1/1 + 0.8/2 + 0.5/3 + 0.1/4

Here, for example, 0.8/2 means
. The + sign denotes a union. When writing a fuzzy set in the above form, the elements of the universal set are omitted
with membership function values ​​equal to zero. Usually all elements of the universal set are written down with the corresponding values ​​of the membership function. A fuzzy set notation is used, as in probability theory,

Definition. IN general case fuzzy subset universal set
is defined as the set of ordered pairs

. (3.2)