Basic concepts and definitions of the theory of fuzzy sets. Fuzzy sets in control systems

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

Function belonging fuzzy sets. 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

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

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

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

Remark 3.1

Fuzzy sets 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

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

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 a point, the class membership function takes a value equal to 0.5.

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

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

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

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

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 function of the class, 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

Definition 3.3

The height of a fuzzy set is denoted and defined as

Example 3.6

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

where is the height of this set.

Example 3.7

Fuzzy set

after normalization it takes the form

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

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

Condition is met

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).

By a clear set or simply a set, we usually understand a certain set of defined and distinguishable objects of our intuition and intellect, conceived as a single whole. IN this statement Let us note the following point: set A is a collection certain objects. This means that for any x one can unambiguously say whether it belongs to the set A or not.

The condition for an element x to belong to the set A can be written using the concept of the membership function m(x), namely

Consequently, the set can be specified as a set of pairs: an element and the value of its membership function

A = ((x|m(x)) (1)

Example 1. The department offers five elective courses x 1 , x 2 , x 3 , x 4 and x 5 . In accordance with the program, three courses are required. The student has chosen courses x 2, x 3 and x 5 to study. Let's write this fact using the membership function

where the first element of each pair means the name of the course, and the second describes the fact that it belongs to the subset chosen by the given student ("yes" or "no").

There are infinitely many examples of crisp sets: list of students study group, many houses on a given city street, many molecules in a drop of water, etc.

Meanwhile, a huge volume human knowledge and connections with outside world include such concepts that cannot be called sets in the sense of (1). They should rather be considered classes with fuzzy boundaries, when the transition from belonging to one class to belonging to another occurs gradually, not abruptly. Thus, it is assumed that the logic of human reasoning is based not on classical two-valued logic, but on logic with fuzzy truth values ​​- fuzzy connectives and fuzzy rules of inference. Here are some examples of this: the length of the article is approximately 12 pages, most of territory, overwhelming superiority in the game, a group of several people.

Let's dwell on last example. It is clear that a group of people of 3, 5, or 9 people belongs to the concept: “a group of people consisting of several people.” However, they will not have the same degree of confidence in belonging to this concept, which depends on various, including subjective, circumstances. These circumstances can be formalized if we assume that the membership function can take any values ​​on the interval. Moreover extreme values are prescribed if the element definitely does not belong or definitely belongs this concept. In particular, a set of people A consisting of several people can be described by an expression of the form:

A = ((1½0), 2½0.1), 3½0.4), (4½1), (5½1), (6½1), (7½0.8), (8½0.3), (9½0.1), (a½0)

Let us give the definition of a fuzzy set given by the founder of the theory of fuzzy sets L.A. Zade. Let x be an element of a specific universal (so-called basic) set E. Then fuzzy(fuzzy) multitude A defined on the base set E is the set of ordered pairs

A= (xúm A((x)), "x О E,

where m A(X) - membership function, mapping the set E into a unit interval, i.e. m A (x): E ® .

Obviously, if the range of values ​​m A (x) is limited to two numbers 0 and 1, then this definition will coincide with the concept of an ordinary (crisp) set.

The membership function of a fuzzy set can be specified not only by listing all its values ​​for each element of the base set, but also in the form analytical expression. For example, many real numbers Z very close to the number 2, can be given like this:

Z= (xúm Z(x)), "x О R,

where m Z(x) = .

The set of real numbers Y that are sufficiently close to the number 2 is

Y= (xúm Y(x)), "x О R,

M Y Z(x) = .

Graphic image these two membership functions are given in Fig. 3.9.

Definition. Fuzzy set A called fuzzy subset B, if A And B are defined on the same base set E and "x О E: m A(x) £ m B(x), which is denoted as AÌ B.

Conditions for the equality of two fuzzy sets A And B, defined on the same base set E, has the following form

A = B or "x О E: m A(x) = m B(x).

Comment. There is some similarity between the inherently different concepts of “fuzzy” and “probability”. Firstly, these concepts are used in tasks where there is uncertainty or inaccuracy of our knowledge or fundamental impossibility accurate predictions results of decisions. Secondly, the intervals of change and probability and membership functions coincide:

and P О and m A(x) О .

At the same time, probability is an objective characteristic and conclusions obtained based on the application of probability theory can, in principle, be tested experimentally.

The membership function is determined subjectively, although it usually reflects the real relationships between the objects under consideration. The effectiveness of using methods based on fuzzy set theory is usually judged after obtaining specific results.

If probability theory assumes that the probability reliable event equal to one, i.e.

then the corresponding sum of all values ​​of the membership function can take any value from 0 to ¥.

So, to define a fuzzy set A needs to be determined base set elements of E, and form a membership function m A(x), which is a subjective measure of confidence with which each element x of E belongs to a given fuzzy set A.

Fuzzy set- key concept fuzzy logic. Let E- universal set, X- element E, a R is some property. Regular (crisp) subset A universal set E, whose elements satisfy the property R is defined as the set of ordered pairs

A = ( μA(x) / x},

Where μ A (x) —characteristic function, taking the value 1 if X satisfies property R, and 0 otherwise.

Fuzzy subset is different from regular topics, which is for elements X from E there is no clear yes-no answer regarding property R. In this regard, the fuzzy subset A universal set E is defined as the set of ordered pairs

A = ( μA(x) / x},

Where μ A (x) — characteristic membership function(or simply membership function), taking values ​​in some completely ordered set M(For example, M = ).

The membership function indicates the degree (or level) of membership of an element X subset A. A bunch of M called a set of accessories. If M= (0, 1), then the fuzzy subset A can be considered as an ordinary or crisp set.

Examples of writing a fuzzy set

Let E = {x 1 , x 2 , x z,x 4 , x 5 ), M = ; A is a fuzzy set for which μ A ( x 1 )= 0.3; μ A ( x 2)= 0; μ A ( X 3) = 1; μ A (x 4) = 0.5; μ A ( x 5)= 0,9.

Then A can be represented in the form

A ={0,3/x 1 ; 0/X 2 ; 1/X 3 ; 0,5/X 4 ; 0,9/X 5 } ,

or

A={0,3/x 1 +0/X 2 +1/X 3 +0,5/X 4 +0,9/X 5 },

or

Comment. Here the “+” sign does not denote the operation of addition, but has the meaning of union.

Basic characteristics of fuzzy sets

Let M= and A— fuzzy set with elements from the universal set E and many accessories M.

The quantity is called height fuzzy set A. Fuzzy set It's okay if its height is 1, i.e. upper limit its membership function is 1 (= 1). At< 1нечеткое множество называется subnormal.

Fuzzy set empty if ∀ xϵ E μ A ( x) = 0. A non-empty subnormal set can be normalized using the formula

Fuzzy set unimodal, If μ A ( x) = 1 on only one X from E.

. Carrier fuzzy set A is an ordinary subset with the property μ A ( x)>0, i.e. carrier A = {x/x ϵ E, μ A ( x)>0}.

Elements xϵ E, for which μ A ( x) = 0,5 , are called transition points sets A.

Examples of fuzzy sets

1. Let E = {0, 1, 2, . . ., 10}, M =. Fuzzy set"Several" can be defined as follows:

“Several” = 0.5/3 + 0.8/4 + 1/5 + 1/6 + 0.8/7 + 0.5/8; its characteristics:height = 1, carrier = {3, 4, 5, 6, 7, 8}, transition points — {3, 8}.

2. Let E = {0, 1, 2, 3,…, n,… ). The fuzzy set “Small” can be defined:

3. Let E= (1, 2, 3,..., 100) and corresponds to the concept “Age”, then the fuzzy set “Young” can be defined using

Fuzzy set “Young” on the universal set E"= (IVANOV, PETROV, SIDOROV,...) is specified using the membership function μ Young ( x) on E =(1, 2, 3, ..., 100) (age), called in relation to E" compatibility function, while:

Where X— SIDOROV’s age.

4. Let E= (ZAPOROZHETS, ZHIGULI, MERCEDES,...) - a set of car brands, and E"= is the universal set “Cost”, then on E" we can define fuzzy sets of the type:

Rice. 1.1. Examples of membership functions

“For the poor”, “For the middle class”, “Prestigious”, with affiliation functions like Fig. 1.1.

Having these functions and knowing the cost of cars from E V this moment time, we will thereby determine E" fuzzy sets with the same names.

So, for example, the fuzzy set “For the poor”, defined on the universal set E =(ZAPOROZHETZ, ZHIGULI, MERCEDES,...), looks as shown in Fig. 1.2.

Rice. 1.2. An example of specifying a fuzzy set

Similarly, you can define the fuzzy set “High-speed”, “Medium”, “Slow-speed”, etc.

5. Let E- set of integers:

E= {-8, -5, -3, 0, 1, 2, 4, 6, 9}.

Then the fuzzy subset of numbers, according to absolute value close to zero, can be defined, for example, like this:

A ={0/-8 + 0,5/-5 + 0,6/-3 +1/0 + 0,9/1 + 0,8/2 + 0,6/4 + 0,3/6 + 0/9}.

On methods for constructing membership functions of fuzzy sets

The above examples used straight methods when an expert either simply sets for each X ϵ E meaning μ A (x), or defines a compatibility function. As a rule, direct methods for specifying the membership function are used for measurable concepts such as speed, time, distance, pressure, temperature, etc., or when polar values ​​are distinguished.

In many problems, when characterizing an object, it is possible to select a set of features and for each of them determine polar values ​​corresponding to the values ​​of the membership function, 0 or 1.

For example, in the task of face recognition, we can distinguish the scales given in table. 1.1.

Table 1.1. Scales in the face recognition task

For a specific personAthe expert, based on the given scale, setsμ A(x)ϵ, forming the vector membership function (μ A(x 1) , μ A(x 2),…, μ A(x 9)}.

With direct methods, group direct methods are also used, when, for example, a group of experts is presented with a specific person and everyone must give one of two answers: “this person is bald” or “this person is not bald”, then the number of affirmative answers divided on total number experts, gives meaning μ bald ( of this person). (In this example, you can act through the compatibility function, but then you will have to count the number of hairs on the head of each person presented to the expert.)

Indirect methods for determining the values ​​of the membership function are used in cases where there are no elementary measurable properties through which the fuzzy set of interest to us is determined. As a rule, these are pairwise comparison methods. If the values ​​of the membership functions were known to us, for example, μ A(X-i) = ω i , i= 1, 2, ..., n, then pairwise comparisons can be represented by a matrix of relations A= ( a ij ), where a ij= ωi/ ω j(division operation).

In practice, the expert himself forms the matrix A, in this case it is assumed that the diagonal elements are equal to 1, and for elements that are symmetrical with respect to the diagonal a ij = 1/a ij , i.e. if one element evaluates to α times stronger than the other, then this latter must be 1/α times stronger than the first. IN general case the problem reduces to finding a vector ω that satisfies an equation of the form Aw= λmax w, where λ max is the largest eigenvalue of the matrix A. Since the matrix A is positive by construction, a solution to this problem exists and is positive.

Two more approaches can be noted:

• use of standard forms curves for specifying membership functions (in the form of (L-R)-Type - see below) with clarification of their parameters in accordance with experimental data;
• use of relative frequenciesaccording to the experiment as membership values.

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 a 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 behavior 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, the concept of a fuzzy set was introduced.

In the theory of crisp sets, the characteristic function of a crisp set in the universal space was considered , equal to 1 if the 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.” A fuzzy set (the same notation is left as for the crisp set) in the universal space through the membership function (the same notation as for the characteristic function) is defined as follows

The membership function is most often interpreted as follows: the value means subjective assessment the degree of membership of an element in a fuzzy set, for example, means that it belongs 80%. 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, elements of the universal set with membership function values ​​equal to zero are omitted. 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, a fuzzy subset of a universal set is defined as a set of ordered pairs

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 for surgery fuzzy sets the long-developed apparatus of random sets is suitable...

Concept of fuzzy set - 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 membership function of a fuzzy 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 height fuzzy set A. Fuzzy set A Fine , 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 subnormal.

A fuzzy set is called empty, 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. .

Carrier 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.