SOLVING PROBLEMS USING “EULER CIRCLES”
Rybina Angelina
Class 5 “D”, Municipal Educational Institution “Secondary School No. 59 with UIP”, Russian Federation,Saratov
Bagaeva Irina Viktorovna
scientific adviser,teacher of the highest category, mathematics teacher,Municipal educational institution "Secondary school No. 59 with UIP", Russian Federation,Saratov
“... circles are very suitable for facilitating our thinking”
Leonard Euler
There is no scientist whose name is mentioned in educational mathematical literature as often as the name of Euler. Even in high school, logarithms and trigonometry are still taught largely “according to Euler.”
In 1741, Euler wrote “Letters on various physical and philosophical matters, written to a certain German princess...”, where “Euler’s circles” appeared for the first time. Euler wrote then that “circles are very suitable for facilitating our thinking.”
When solving a number of problems, Leonhard Euler used the idea of representing sets using circles, and they were called “Eulerian circles.”
Using these circles, Euler also depicted the set of all real numbers:
N - set of natural numbers,
Z - set of integers,
Q - set of rational numbers,
· R is the set of all real numbers.
Figure 1. Illustration of the set of real numbers
What is a set?
There is no precise definition of this concept in mathematics. The concept of “set” is not defined, it is explained with examples: many apples in a basket; set of points on a line segment. A set consists of elements. In the examples given, these are apples, letters, dots.
Sets are designated by capital letters of the Latin alphabet: A, B, C, ... K, M, N ... X, ...; elements of the set - in lowercase letters of the alphabet: a, b, c, ... k, m, n ... x, y, .... A = (a; b; c; d) - set A consists of elements a, b , c, d, or, they say that the element a belongs to the set A, is written: aA (the sign reads: “belongs”). Element 5 is not included in set A, they say that “5 does not belong to A”: 5 A, or . If the set B does not contain a single element, then it is said to be empty, denoted: B =.
A set can be understood as a collection of any objects called set elements. Examples of sets can be houses on our street, and the alphabet is a collection of letters, and our 5th “D” class is a set of students.
Sets can be:
· Finite (elements of which can be counted; for example, a set of numbers)
· Empty (not containing a single element; for example, a lot of hares that study in our class).
A set K is called a subset of a set N if every element of the set K is an element of the set N. Denoted by: KÍN. The set K is said to be included in the set N.
Subsets can be illustrated by Euler circles.
Figure 2. Subset image
Actions with sets
In mathematics, there are several operations on sets. We will look at two of them: intersection and union.
1. Intersection of sets
Intersection of sets M And N is a set consisting of elements that simultaneously belong to M And N. Intersection of many M And N denoted by .
Example. Set N = ( A N D R E Y );
set K = ( A L E K S E Y ); set M = ( D M I T R I Y )
Figure 3. Example of intersection of sets
2. Union of sets
A union of sets is a set that contains all the elements of the original sets. Union of sets M And N denoted by .
Example ; 2) the union of the set of all dog breeds and the set of pugs is the set of all dogs.
The operations of union and intersection of sets are very conveniently shown using Euler circles.
By definition, the intersection of two sets M and N includes elements that belong to the sets M and N simultaneously
Example. Let D be the set of the 12 nicest girls, M be the set of the 12 smartest boys. We got our class.
Figure 4. Example of merging sets
3. Nested sets.
Example. There are three sets: “children”, “schoolchildren”, “primary school students”. We see that these 3 sets are located one inside the other . A set located inside another set is said to be nested.
Figure 5. Example of nested sets
Problems that can be solved using Euler diagrams
Task No. 1
Two 10 cm x 10 cm napkins were thrown onto the table. They covered an area of the table equal to 168. What is the overlap area?
1)168 – 10 x 10 = 68;
2)10 x 10 – 68 = 32.
Answer: 32 cm
Figure 6. Drawing for task No. 1
Problem No. 2
80% of the class went on a hike, and 60% went on excursions, and everyone was on a hike or on an excursion. What percentage of the class were both there and there?
A - many students who went on a hike
B - many students who were on the excursion
100 % – 80 % = 20 %
60 % – 20 % = 40 %
Answer: 40%
Figure 7. Drawing for task No. 2
Task No. 3
There are 24 students in our class. They all had a good winter holiday. 10 people went skiing, 16 went to the skating rink, and 12 made snowmen. How many students were able to ski, skate, and build a snowman?
A - a lot of guys skiing
B - a lot of guys skating
C - a lot of guys making snowmen
Let x be the number of guys,
who managed to do everything during these holidays!
(12 - x) + (16 - x) + (10 - x) + x = 24
Answer: 7 guys
Figure 8. Drawing for task No. 3
Problem No. 4
9 of my friends like bananas, 8 like oranges, 7 like plums, 5 like bananas and oranges, 3 like bananas and plums, 4 like oranges and plums, 2 like bananas, oranges and plums. How many friends do I have?
5 – 2 = 3 3 – 2 = 1 4 – 2 = 2
9 – 6 = 3 8 – 7 = 1 7 – 5 = 2
3 + 1 + 2 + 3 + 2 + 1 + 2 = 14
Answer: 14 friends
Figure 9. Drawing for problem No. 4
Problem No. 5
In the Dubki pioneer camp, 30 excellent students, 28 Olympiad winners and 42 athletes rested during the shift. 10 people were both excellent students and winners of Olympiads, 5 were excellent students and athletes, 8 were athletes and winners of Olympiads, 3 were both excellent students and athletes and winners of Olympiads.
How many guys were at the camp?
A - many excellent students
B - many Olympiad winners
C - many athletes
10 – 3 = 7 5 – 3 = 2 8 – 3 = 5
30 – 12 = 18 28 – 15 = 13 42 – 10 = 32
18 + 13 + 32 + 7 + 2 + 5 + 3 = 80
Answer: 80 guys
Figure 10. Drawing for problem No. 5
3. Conclusion
Euler diagrams are the general name for a number of methods of graphic illustration, widely used in various fields of mathematics: set theory, probability theory, logic, statistics, computer science, etc. The use of Euler circles allows even a fifth grader to easily solve problems that can only be solved in the usual way in high school.
Bibliography:
1.Alexandrova R.A., Potapov A.M. Elements of set theory and mathematical logic. Workshop / Kaliningrad. 1997. - 66 p.
2.Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. A manual for students of grades 5-6. M.: Education, 1999. p. 189-191, 231.
3.Tasks for extracurricular work in mathematics in grades V-VI: A manual for teachers / Comp. V.Yu. Safonova. Ed. D.B. Fuksa, A.L. Gavronsky. M.: MIROS, 1993. - p. 42.
4. Entertaining mathematics. 5-11 grades. How to make lessons not boring / Author. comp. T.D. Gavrilova. Volgograd: Teacher, 2005. - p. 32-38.
5. Smykalova E.V. Additional chapters on mathematics for 5th grade students. St. Petersburg: SMIO Press, 2009. - p. 14-20.
6.Encyclopedia for children. T. 11. Mathematics Chief editor. M.D. Aksenov. M.: Avanta +, 2001. - p. 537-542.
Each object or phenomenon has certain properties (signs).
It turns out that forming a concept about an object means, first of all, the ability to distinguish it from other objects similar to it.
We can say that a concept is the mental content of a word.
Concept - it is a form of thought that displays objects in their most general and essential characteristics.
A concept is a form of thought, and not a form of a word, since a word is only a label with which we mark this or that thought.
Words can be different, but still mean the same concept. In Russian - “pencil”, in English - “pencil”, in German - bleistift. The same thought has different verbal expressions in different languages.
RELATIONS BETWEEN CONCEPTS. EULER CIRCLES.
Concepts that have common features in their content are called COMPARABLE(“lawyer” and “deputy”; “student” and “athlete”).
Otherwise, the concepts are considered INCOMPARABLE(“crocodile” and “notebook”; “man” and “steamboat”).
If, in addition to general characteristics, concepts also have common elements of volume, then they are called COMPATIBLE.
There are six types of relationships between comparable concepts. It is convenient to denote relationships between the volumes of concepts using Euler circles (circular diagrams, where each circle denotes the volume of a concept).
| KIND OF RELATIONSHIP BETWEEN CONCEPTS | IMAGE USING EULER CIRCLES |
| EQUIVALITY (IDENTITY) The scopes of the concepts completely coincide. Those. These are concepts that differ in content, but the same elements of volume are thought of in them. | 1) A - Aristotle B - founder of logic 2) A - square B - equilateral rectangle |
| SUBORDINATION (SUBORDINATION) The scope of one concept is completely included in the scope of another, but does not exhaust it. | 1) A - person B - student 2) A - animal B - elephant |
| INTERSECTION (CROSSING) The volumes of two concepts partially coincide. That is, concepts contain common elements, but also include elements that belong to only one of them. | 1) A - lawyer B - deputy 2) A - student B - athlete |
| COORDINATION (COORDINATION) Concepts that do not have common elements are completely included in the scope of the third, broader concept. | 1) A - animal B - cat; C - dog; D - mouse 2) A - precious metal B - gold; C - silver; D - platinum |
| OPPOSITE (CONTRAPARITY) Concepts A and B are not simply included in the scope of the third concept, but seem to be at its opposite poles. That is, concept A has in its content such a feature, which in concept B is replaced by the opposite one. | 1) A - white cat; B - red cat (cats are both black and gray) 2) A - hot tea; iced tea (tea can also be warm) I.e. concepts A and B do not exhaust the entire scope of the concept they are included in. |
| CONTRADITION (CONTRADITIONALITY) The relationship between concepts, one of which expresses the presence of some characteristics, and the other - their absence, that is, it simply denies these characteristics, without replacing them with any others. | 1) A - tall house B - low house 2) A - winning ticket B - non-winning ticket I.e. the concepts A and not-A exhaust the entire scope of the concept into which they are included, since no additional concept can be placed between them. |
Exercise : Determine the type of relationship based on the scope of the concepts below. Draw them using Euler circles.
1) A - hot tea; B - iced tea; C - tea with lemon
Hot tea (B) and iced tea (C) are in an opposite relationship.
Tea with lemon (C) can be either hot,
so cold, but it can also be, for example, warm.
2)A- wood; IN- stone; WITH- structure; D- house.
Is every building (C) a house (D)? - No.
Is every house (D) a building (C)? - Yes.
Something wooden (A) is it necessarily a house (D) or a building (C) - No.
But you can find a wooden structure (for example, a booth),
You can also find a wooden house.
Something made of stone (B) is not necessarily a house (D) or building (C).
But there may be a stone building or a stone house.
3)A- Russian city; IN- capital of Russia;
WITH- Moscow; D- city on the Volga; E- Uglich.
The capital of Russia (B) and Moscow (C) are the same city.
Uglich (E) is a city on the Volga (D).
At the same time, Moscow, Uglich, like any city on the Volga,
are Russian cities (A)
P O N I T I E
Each object or phenomenon has certain properties (signs).
It turns out that forming a concept about an object means, first of all, the ability to distinguish it from other objects similar to it.
We can say that a concept is the mental content of a word.
A concept is a form of thought that reflects objects in their most general and essential characteristics*.
A concept is a form of thought, and not a form of a word, since a word is only a label with which we mark this or that thought.
Words can be different, but still mean the same concept. In Russian – “pencil”, in English – “pencil”, in German – bleistift. The same thought has different verbal expressions in different languages.
RELATIONS BETWEEN CONCEPTS. EULER CIRCLES.
Concepts that have common features in their content are called COMPARABLE(“lawyer” and “deputy”; “student” and “athlete”).
Otherwise, the concepts are considered INCOMPARABLE(“crocodile” and “notebook”; “man” and “steamboat”).
If, in addition to common features, concepts also have common elements of volume, then they are called COMPATIBLE.
There are six types of relationships between comparable concepts. It is convenient to denote relationships between the volumes of concepts using Euler circles (circular diagrams, where each circle denotes the volume of a concept).
|
KIND OF RELATIONSHIP BETWEEN CONCEPTS |
IMAGE USING EULER CIRCLES |
|
EQUIVALUE(IDENTITY) The scope of the concepts completely coincides. Those. These are concepts that differ in content, but the same elements of volume are thought of in them. |
1) A – Aristotle B – founder of logic 2) A – square B – equilateral rectangle |
|
SUBORDINATION(SUBORDINATION) The scope of one concept is completely included in the scope of the other, but does not exhaust it. |
1) A – person B – student 2) A – animal |
|
CROSSING(CROSSTALK) The scope of the two concepts partially coincides. That is, concepts contain common elements, but also include elements that belong to only one of them. |
1) A – lawyer B – deputy 2) A – student B – athlete |
|
SUBORDINATION(COORDINATION) Concepts that do not have common elements are completely included in the scope of the third, broader concept. |
1) A – animal B – cat; C – dog; D – mouse 2) A – precious metal B – gold; C – silver; D - platinum |
|
OPPOSITE(CONTRAPARITY) Concepts A and B are not simply included in the scope of the third concept, but seem to be at its opposite poles. That is, concept A has in its content such a feature, which in concept B is replaced by the opposite one. |
1) A – white cat; B – red cat (cats come in both black and gray) 2) A – hot tea; cold tea (tea can be warm) |
|
Those. concepts A and B do not exhaust the entire scope of the concept they are included in. CONTRADICTION (CONTRADITIONALITY) |
The relationship between concepts, one of which expresses the presence of any characteristics, and the other - their absence, that is, simply denies these characteristics, without replacing them with any others. 1) A – high house B – low house 2) A – winning ticket B – non-winning ticket |
Those. the concepts A and not-A exhaust the entire scope of the concept into which they are included, since no additional concept can be placed between them. Exercise:
1) Determine the type of relationship based on the scope of the concepts below. Draw them using Euler circles.
A – hot tea; B – iced tea; C – tea with lemon
Hot tea (B) and iced tea (C) are located
Tea with lemon (C) can be either hot,
so cold, but it can also be, for example, warm.
2
)
A- wood; IN– stone; WITH– structure; D- house.
Is every building (C) a house (D)? - No.
Is every house (D) a building (C)? - Yes.
Something wooden (A) is it necessarily a house (D) or a building (C) – No.
But you can find a wooden structure (for example, a booth),
You can also find a wooden house.
Something made of stone (B) is not necessarily a house (D) or building (C).
But there may be a stone building or a stone house.
3
)
A– Russian city; IN- capital of Russia;
WITH- Moscow; D- a city on the Volga; E- Uglich.
The capital of Russia (B) and Moscow (C) are the same city.
Uglich (E) is a city on the Volga (D).
At the same time, Moscow, Uglich, like any city on the Volga,
are Russian cities (A)
Euler circles- one of the simplest themes that you need for admission to the 5th grade of physics and mathematics lyceums. In fact, Euler circles is nothing more than a graphical representation of sets. Objects with a certain property are located inside Euler-Venn circle those who do not possess are outside. Of course, usually the diagram contains not one circle, but several, each of which combines objects with some kind of property. Any task from this block boils down to the fact that it is necessary to count the number of elements in any area. Let's look at examples of what needs to be done:
Tasks for many people
There are students in the class. study English, German and French. People don't know any language. It is also known that of all the children, only one boy studies languages: English and French. How many people study a language?
To solve the problem, let's denote the number of required students as (those who study the language). The number of students studying a different number of languages can be expressed through and the conditions in the problem. Euler-Venn diagram in this case it will look like this: For example, guys who know only English are indicated in red and their number. 
Note that we have not used the total number of students in any way - this condition will generate the very equation with which the problem will be solved: 
It turns out that all languages are studied by humans (Now, knowing , you can independently reconstruct how many students were in the class and check the answer)
Divisibility problems (complex divisibility)
These are tasks of increased complexity. We recommend that you study the topic first. A must-read only for those who are planning to win prizes.
For how many numbers between and is the following statement true: the number is divisible by or not divisible by?
Such a terrible and incomprehensible condition becomes simple if you use Euler circles. It is clear that in this problem we consider numbers that - we are interested in those inside the corresponding circle. There are also numbers that vdots 12 - we are interested in the numbers that are outside. But what about the numbers that belong to both sets? Firstly, what common property do they have, and secondly, are they of interest to us?
Let's answer the first question first. It turns out that if a number is simultaneously divisible by two other numbers, then it is divisible by Least Common Multiple these two numbers, that is, by the minimum number that is divisible without a remainder by both of them. For numbers and LCM there is nothing more than the number , since and , and there is no smaller number with such properties. In total, at the intersection of our sets there are numbers that .
Next, it should be noted that the word is used in the condition "OR". This means that for the required numbers, AT LEAST ONE of the proposed statements must be true (possibly both). That is, we are suitable for numbers that are inside the circle of numbers, which are, as well as all the numbers that are outside the circle.
So, Euler-Venn diagram looks like this: The shading indicates the numbers that need to be found. Now, I hope, it is obvious that we need to find how many numbers there are in the problem under consideration, from this quantity subtract the number of numbers that and add the number of numbers that . 
So let's get started:
It turns out that the required numbers
So, let's summarize. If you are going enter the 5th grade of the physics and mathematics lyceum, then general knowledge of Euler-Venn circles You need it. The main area of application is problems where there are sets of objects that have certain properties, and it is necessary to find the number of objects that have (or do not have) a set of specified properties.
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Nowadays, a huge amount of information has been collected around us, and it can be difficult to understand it. Therefore, many do not know that behind the name “Euler Circles” lies a practical and convenient method for solving various problems. Everyone has heard about them, but few can explain what they are. However, I believe that Euler Circles are useful both in everyday life and in science, so everyone should know how to use them. In this work, I collected all the necessary information to understand what Euler Circles are and where they are convenient to use.
Euler circles are a geometric diagram that can be used to visualize the relationships between different sets and subsets. This scheme helps to find logical connections between phenomena and concepts; it was invented by Leonhard Euler and is used in mathematics and other scientific disciplines. Using Euler Circles simplifies reasoning and helps you get an answer faster and easier. (1),(2)
Euler circles are inextricably linked with the concept of set. Therefore, in order to better understand what is depicted on Euler circles, you need to know what a set is and what kinds of sets there are.
A set can be understood as a collection of any objects called elements of the set. Sets can combine any objects with a common characteristic. For example, the set of students in gymnasium 11 and students in grade 7 “B” constitute a separate set. There can be sets of inanimate objects. For example, many books written by some author. With the help of Euler circles, a set is denoted as an empty circle, and its elements are designated as dots. (5)
Let's draw a lot of numbers. In the figure, the outline indicates a set, and the elements of this set are indicated by dots.
There are three types of sets:
· Finite (for example - a lot of numbers)
· Infinite (for example - a set of numbers)
· Empty (set of natural numbers
less than zero). (5)
A group of objects that forms a set within a larger set is depicted as a smaller circle drawn inside a larger circle and is called a subset. This relationship is formed between a large set of animals and its subset of flatworms. (5)
In cases where two concepts coincide only partially, the relationship between such sets is depicted using two intersecting circles. This relationship is formed between many students in grade 7 “B” and many C students. Some elements of the set of students in grade 7 “B” also belong to the set of C students. (5)
When no object from one set can simultaneously belong to the second set, then the relationship between them is depicted by means of two circles drawn one outside the other. Such sets are the set of negative and the set of positive numbers. (5)
Euler circles were invented and named after Leonard Euler (portrait on the left). He was a Swiss mathematician who made a significant contribution to the development of mathematics, as well as mechanics, physics, astronomy and a number of applied sciences. Euler was born in Switzerland, studied in Germany, but worked and died in Russia. This scientist is the author of 800 works. Leonhard Euler was born in 1707 into a pastor's family. His father was a friend of the Bernoulli family. Euler showed early mathematical abilities. While studying at the gymnasium, the boy enthusiastically studied mathematics, and later began attending university lectures by Johann Bernoulli. On October 20, 1720, Leonhard Euler became a student at the Faculty of Arts at the University of Basel. The gifted young man attracted the attention of Professor Johann Bernoulli. He gave the student mathematical articles to study, and also invited him to come to his home to jointly analyze the incomprehensible. At his teacher’s house, Euler met and began to communicate with Bernoulli’s sons, Daniel (portrait on the left) and Nikolai (portrait on the right), who were also engaged in mathematics. (6)
Young Euler wrote several scientific papers. “Physics Dissertation on Sound” received a favorable review. At that time, the number of scientific vacancies in Switzerland was small. Therefore, the brothers Daniil and Nikolai Bernoulli left for Russia, where the Russian Academy of Sciences began to be created; they promised to work there for a position for Euler. At the beginning of the winter of 1726, Euler received a letter from St. Petersburg: on the recommendation of the Bernoulli brothers, he was invited to the position of adjunct in physiology with a salary of 200 rubles. Euler spent a lot of time in Russia, where he made significant contributions to Russian science. In 1731 he was elected academician of the St. Petersburg Academy. He knew the Russian language well, and published essays and textbooks in Russian. (6)
Then Euler describes in detail his method of solving certain problems using Euler circles. In 1741, Euler writes “Letters on various physical and philosophical matters, to a certain German princess...”, which mentions “Euler circles”. Euler wrote that “circles are very suitable for facilitating our thinking.” (3)
Euler's method has received well-deserved recognition and popularity. And after him, many scientists used it in their work, and also modified it in their own way. Bernard Bolzano used the same method, but with rectangular patterns. Thanks to Venn's contribution, the method is even called Venn diagrams or Euler-Venn diagrams. Euler circles have an applied purpose, that is, with their help, problems involving the union or intersection of sets in mathematics, logic, management and more are solved in practice. (1)
Here are a few problems to solve that are convenient to use Euler circles:
Task 1.
Children from one school were asked about their pets. 100 of them answered that they have a dog and/or cat at home. 87 guys had one dog, and 63 guys had one cat. How many guys have both a dog and a cat?
Solution:
To solve this problem without using Euler circles, you need to count how many dogs and cats the students had. To do this you need to add 87 and 63. 87+63=150 pets. There were only 100 students, and a fractional number of pets cannot be obtained. This means that if each student has 1 pet, there are still 50 extra. Therefore, 50 students have 2 pets. And since the problem states that none of the students have 2 cats or 2 dogs, this means that 50 students have both a cat and a dog.
But this method is long and suitable only for simple tasks. It is much more convenient to solve such a problem using Euler circles.
We will depict the set of dog owners with a red circle, and the set of cat owners with a blue circle. There were 100 students in total. Those who have both a cat and a dog X. To find the number of students who only have a dog, you need to subtract X from 87. Since there are 100 students in total, we get:
X=50 students
Answer: 50 students have both a cat and a dog
Task 2.
One day the students were asked which of them liked mathematics, which liked the Russian language, and which liked physics. It turned out that out of 36 students, 2 did not like mathematics, Russian, or physics. 25 students like mathematics, 11 students like Russian, 17 students like physics; both mathematics and Russian - 6; both mathematics and physics - 10; Russian language and physics - 4.
How many people love all three subjects?
Solution:
Let's depict 3 sets. The red set is those who love mathematics, the blue ones are those who love the Russian language, and the green set is physics.
Now let’s enter the number of elements into the sets. 6 people love both Russian and mathematics. Of these, X people also love physics. This means that only 6 people like mathematics and Russian. Only mathematics and physics 10-X people, only Russian and physics 4-X people. 25 people love mathematics. But X, 6-X, 10-X people also love other objects. This means that only mathematics is loved by 25-(6-X)-(10-X)-X= 25-6+X-10+X -X=5+X people. Only Russian is loved by 11-(6-Х)-(4-Х)-Х= 11-10+2Х-Х=1+Х students, only physics by 17-(10-Х)-(4-Х)-Х= 17-14+2X-X= 3+X.
Since 2 people do not like either of these items, then:
3+X+9+X+1+X+6-X+10-X+4-X+X=36-2
Answer: 1 person likes all three items
Task 3.
The table shows the queries and the number of pages found for a certain segment of the Internet.
How many pages (in thousands) will be found for the query nature? (4)
Solution :
At the request of people, 2,100 thousand pages were found. 900 of them are also about nature. This means there are 2100-900=200 thousand pages only about man, and X-900 thousand only about nature. We get that:
2100-900+X-900+900=3400
2100-900+X=3400
X=2200 thousand pages
Answer: the query nature will find 2,200 thousand pages.
As you can see, Euler Circles are a useful and important discovery for mathematics in general and for each of us in particular. Euler circles are found not only in exams, but we also need them in everyday life. This is an interesting and necessary thing that should not be forgotten.
Literature:
https://www.tutoronline.ru/blog/krugi-jejlera
https://ru.wikipedia.org/wiki/%D0%9A%D1%80%D1%83%D0%B3%D0%B8_%D0%AD%D0%B9%D0%BB%D0%B5%D1 %80%D0%B0
http://sibac.info/shcoolconf/science/xvii/42485
http://www.jwy.narod.ru/logic/_04_eiler.html
https://ru.wikipedia.org/wiki/%D0%AD%D0%B9%D0%BB%D0%B5%D1%80,_%D0%9B%D0%B5%D0%BE%D0%BD %D0%B0%D1%80%D0%B4