Leonhard Euler (1707-1783) - famous Swiss and Russian mathematician, member of the St. Petersburg Academy of Sciences, lived most of his life in Russia. The most famous in mathematical analysis, statistics, computer science and logic is the Euler circle (Euler-Venn diagram), used to indicate the scope of concepts and sets of elements.

John Venn (1834-1923) - English philosopher and logician, co-author of the Euler-Venn diagram.

Compatible and incompatible concepts

A concept in logic means a form of thinking that reflects the essential features of a class of homogeneous objects. They are denoted by one or a group of words: “world map”, “dominant fifth chord”, “Monday”, etc.

In the case when the elements of the scope of one concept fully or partially belong to the scope of another, we speak of compatible concepts. If not a single element of the scope of a certain concept belongs to the scope of another, we have a situation with incompatible concepts.

In turn, each type of concept has its own set of possible relationships. For compatible concepts these are the following:

• identity (equivalence) of volumes;
• intersection (partial coincidence) of volumes;
• subordination (subordination).

For incompatibles:

• subordination (coordination);
• opposite (contrary);
• contradiction (contradiction).

Schematically, relationships between concepts in logic are usually denoted using Euler-Venn circles.

Relationships of equivalence

In this case, the concepts imply the same subject. Accordingly, the scope of these concepts completely coincides. For example:

A - Sigmund Freud;

B is the founder of psychoanalysis.

A - square;

B - equilateral rectangle;

C is an equiangular rhombus.

Fully coinciding Euler circles are used for notation.

Intersection (partial match)

A - teacher;

B is a music lover.

As can be seen from this example, the scope of the concepts partially coincide: a certain group of teachers may turn out to be music lovers, and vice versa - among the music lovers there may be representatives of the teaching profession. A similar relationship will be in the case when the concept A is, for example, “city dweller”, and the concept B is “driver”.

Subordination (subordination)

Schematically designated as Euler circles of different scales. The relationship between concepts in this case is characterized by the fact that the subordinate concept (smaller in scope) is completely included in the subordinate (larger in scope). At the same time, the subordinate concept does not completely exhaust the subordinating one.

For example:

A - tree;

B - pine.

Concept B will be subordinate to concept A. Since pine belongs to trees, concept A becomes subordinate in this example, “absorbing” the volume of concept B.

Subordination (coordination)

A relationship characterizes two or more concepts that exclude each other, but at the same time belong to a certain general generic circle. For example:

A - clarinet;

B - guitar;

C - violin;

D - musical instrument.

Concepts A, B, C do not overlap with each other, however, they all belong to the category of musical instruments (concept D).

Opposite (contrary)

Opposite relations between concepts imply that these concepts belong to the same genus. Moreover, one of the concepts has certain properties (signs), while the other denies them, replacing them with opposite ones in nature. Thus, we are dealing with antonyms. For example:

A - dwarf;

B is a giant.

With opposite relations between concepts, the Euler circle is divided into three segments, the first of which corresponds to concept A, the second to concept B, and the third to all other possible concepts.

Contradiction (contradiction)

In this case, both concepts represent species of the same genus. As in the previous example, one of the concepts indicates certain qualities (signs), while the other denies them. However, unlike the relation of opposition, the second, opposite concept does not replace the denied properties with other, alternative ones. For example:

A - difficult task;

B is an easy task (not-A).

Expressing the scope of concepts of this kind, Euler's circle is divided into two parts - there is no third, intermediate link in this case. Thus, the concepts are also antonyms. In this case, one of them (A) becomes positive (affirming some attribute), and the second (B or non-A) becomes negative (denying the corresponding attribute): “white paper” - “not white paper”, “domestic history” - “foreign history”, etc.

Thus, the ratio of the volumes of concepts in relation to each other is the key characteristic that defines Euler circles.

Relationships between sets

You should also distinguish between the concepts of elements and sets, the volume of which is reflected by Euler circles. The concept of set is borrowed from mathematical science and has a fairly broad meaning. Examples in logic and mathematics display it as a certain collection of objects. The objects themselves are elements of this set. “A set is many things conceived as one” (Georg Cantor, founder of set theory).

Sets are designated in capital letters: A, B, C, D... etc., elements of sets are designated in lowercase letters: a, b, c, d... etc. Examples of a set can be students in the same classroom, books standing on a certain shelf (or, for example, all the books in a certain library), pages in a diary, berries in a forest clearing, etc.

In turn, if a certain set does not contain a single element, then it is called empty and denoted by the sign Ø. For example, the set of intersection points of parallel lines, the set of solutions to the equation x 2 = -5.

Problem solving

Euler circles are actively used to solve a large number of problems. Examples in logic clearly demonstrate the connection between logical operations and set theory. In this case, concept truth tables are used. For example, the circle designated by the name A represents the region of truth. So the area outside the circle will represent a lie. To determine the area of ​​the diagram for a logical operation, you should shade the areas defining the Euler circle in which its values ​​for elements A and B will be true.

The use of Euler circles has found wide practical application in various industries. For example, in a situation with professional choice. If a subject is concerned about choosing a future profession, he can be guided by the following criteria:

W - what do I like to do?

D - what am I doing?

P - how can I make good money?

Let's depict this in the form of a diagram: Euler circles (examples in logic - intersection relation):

The result will be those professions that will be at the intersection of all three circles.

Euler-Venn circles occupy a special place in mathematics (set theory) when calculating combinations and properties. Euler circles of the set of elements are enclosed in the image of a rectangle denoting the universal set (U). Instead of circles, other closed figures can also be used, but the essence does not change. The figures intersect with each other, according to the conditions of the problem (in the most general case). Also, these figures must be marked accordingly. The elements of the sets under consideration can be points located inside various segments of the diagram. Based on it, specific areas can be shaded, thereby designating newly formed sets.

With these sets it is possible to perform basic mathematical operations: addition (sum of sets of elements), subtraction (difference), multiplication (product). In addition, thanks to Euler-Venn diagrams, it is possible to compare sets by the number of elements included in them, without counting them.

Leonhard Euler - greatest of mathematicians wrote more than 850 scientific papers.In one of them these circles appeared.

The scientist wrote that“they are very suitable for facilitating our reflections.”

Euler circles is a geometric diagram that helps to find and/or make logical connections between phenomena and concepts more clear. It also helps to depict the relationship between a set and its part.

Of the 90 tourists going on a trip, 30 people speak German, 28 people speak English, 42 people speak French.8 people speak English and German at the same time, 10 people speak English and French, 5 people speak German and French, 3 people speak all three languages. How many tourists don’t speak any language?

Solution:

Let's show the condition of the problem graphically - using three circles

Answer: 10 people.

Problem 2

Many children in our class love football, basketball and volleyball. And some even have two or three of these sports. It is known that 6 people from the class play only volleyball, 2 - only football, 5 - only basketball. Only 3 people can play volleyball and football, 4 can play football and basketball, 2 can play volleyball and basketball. One person from the class can play all the games, 7 can’t play any game. Need to find:

How many people are in the class?

How many people can play football?

How many people can play volleyball?

Problem 3

There were 70 children at the children's camp. Of these, 20 are involved in the drama club, 32 sing in the choir, 22 are fond of sports. There are 10 choir kids in the drama club, 6 athletes in the choir, 8 athletes in the drama club, and 3 athletes attend both the drama club and the choir. How many kids don’t sing in the choir, aren’t interested in sports, and aren’t involved in the drama club? How many guys are only involved in sports?

Problem 4

Of the company’s employees, 16 visited France, 10 – Italy, 6 – England. In England and Italy - five, in England and France - 6, in all three countries - 5 employees. How many people have visited both Italy and France, if the company employs 19 people in total, and each of them has visited at least one of these countries?

Problem 5

Sixth graders filled out a questionnaire asking about their favorite cartoons. It turned out that most of them liked “Snow White and the Seven Dwarfs,” “SpongeBob SquarePants,” and “The Wolf and the Calf.” There are 38 students in the class. 21 students like Snow White and the Seven Dwarfs. Moreover, three of them also like “The Wolf and the Calf,” six like “SpongeBob SquarePants,” and one child equally likes all three cartoons. “The Wolf and the Calf” has 13 fans, five of whom named two cartoons in the questionnaire. We need to determine how many sixth graders like SpongeBob SquarePants.

Problems for students to solve

1. There are 35 students in the class. All of them are readers of school and district libraries. Of these, 25 borrow books from the school library, 20 from the district library. How many of them:

a) are not readers of the school library;

b) are not readers of the district library;

c) are readers only of the school library;

d) are readers only of the regional library;

e) are readers of both libraries?

2.Each student in the class studies English or German, or both. English is studied by 25 people, German by 27 people, and both by 18 people. How many students are there in the class?

3. On a sheet of paper, draw a circle with an area of ​​78 cm2 and a square with an area of ​​55 cm2. The area of ​​intersection of a circle and a square is 30 cm2. The part of the sheet not occupied by the circle and square has an area of ​​150 cm2. Find the area of ​​the sheet.

4. There are 25 people in the group of tourists. Among them, 20 people are under 30 years old and 15 people are over 20 years old. Could this be true? If so, in what case?

5. There are 52 children in the kindergarten. Each of them loves cake or ice cream, or both. Half of the children like cake, and 20 people like cake and ice cream. How many children love ice cream?

6. There are 36 people in the class. Pupils of this class attend mathematical, physical and chemical clubs, with 18 people attending the mathematical club, 14 - physical, 10 - chemical. In addition, it is known that 2 people attend all three clubs, 8 people attend both mathematical and physical, 5 - both mathematical and chemical, 3 - both physical and chemical circles. How many students in the class do not attend any clubs?

7. After the holidays, the class teacher asked which of the children went to the theater, cinema or circus. It turned out that out of 36 students, two had never been to the cinema, theater, or circus. 25 people attended the cinema; in the theater - 11; in the circus - 17; both in cinema and theater - 6; both in the cinema and in the circus - 10; both in the theater and in the circus - 4. How many people visited the theater, cinema and circus at the same time?

Solving Unified State Exam problems using Euler circles

Problem 1

In the search engine query language, the symbol "|" is used to denote the logical "OR" operation, and the symbol "&" is used for the logical "AND" operation.

Cruiser & Battleship? It is assumed that all questions are executed almost simultaneously, so that the set of pages containing all the searched words does not change during the execution of queries.

Solution:

Using Euler circles we depict the conditions of the problem. In this case, we use the numbers 1, 2 and 3 to designate the resulting areas.

Based on the conditions of the problem, we create the equations:

1. Cruiser | Battleship: 1 + 2 + 3 = 7000
2. Cruiser: 1 + 2 = 4800
3. Battleship: 2 + 3 = 4500

To find Cruiser & Battleship(indicated in the drawing as area 2), substitute equation (2) into equation (1) and find out that:

4800 + 3 = 7000, from which we get 3 = 2200.

Now we can substitute this result into equation (3) and find out that:

2 + 2200 = 4500, from which 2 = 2300.

Answer: 2300 - number of pages found by requestCruiser & Battleship.

In search engine query language to denote

Solution

To solve the problem, let's display the sets of Cakes and Pies in the form of Euler circles.

A B C ).

From the problem statement it follows:

Cakes │Pies = A + B + C = 12000

Cakes & Pies = B = 6500

Pies = B + C = 7700

To find the number of Cakes (Cakes = A + B ), we need to find the sector A Cakes│Pies ) subtract the set of Pies.

Cakes│Pies – Pies = A + B + C -(B + C) = A = 1200 – 7700 = 4300

Sector A equals 4300, therefore

Cakes = A + B = 4300+6500 = 10800

Problem 3

|", and for the logical operation "AND" - the symbol "&".

How many pages (in thousands) will be found for the query? Bakery?

It is believed that all queries were executed almost simultaneously, so that the set of pages containing all the searched words did not change during the execution of the queries.

Solution

To solve the problem, we display the sets Cakes and Baking in the form of Euler circles.

Let us denote each sector with a separate letter ( A B C ).

From the problem statement it follows:

Cakes & Pastries = B = 5100

Cake = A + B = 9700

Cake │ Pastries = A + B + C = 14200

To find the quantity of Baking (Baking = B + C ), we need to find the sector IN , for this from the general set ( Cake │ Baking) subtract the set Cake.

Sector B is equal to 4500, therefore Baking = B + C = 4500+5100 = 9600

Problem 4
descending
To indicateThe logical operation "OR" uses the symbol "|", and for the logical operation "AND" - the symbol "&".
Solution

Let's imagine sets of shepherd dogs, terriers and spaniels in the form of Euler circles, denoting the sectors with letters ( A B C D ).

With paniels │(terriers & shepherds) = G + B

With paniel│shepherd dogs= G + B + C

spaniels│terriers│shepherds= A + B + C + D

terriers & shepherds = B

Let's arrange the request numbers in descending order of the number of pages:3 2 1 4

Problem 5

The table shows queries to the search server. Place the request numbers in order increasing the number of pages that the search engine will find for each request.
To indicateThe logical operation "OR" uses the symbol "|", and for the logical operation "AND" - the symbol "&".

1	baroque | classicism | empire style
2	baroque | (classicism & empire style)
3	classicism & empire style
4	baroque | classicism

Solution

Let us imagine the sets classicism, empire style and classicism in the form of Euler circles, denoting the sectors with letters ( A B C D ).

Let us transform the problem condition in the form of a sum of sectors:

baroque│ classicism│empire = A + B + C + D
Baroque │(classicism & empire) = G + B

From the sector sums we see which request produced more pages.

Let's arrange the request numbers in ascending order of the number of pages:3 2 4 1

Solution

To solve the problem, let's imagine queries in the form of Euler circles.

K - canaries,

Ш – goldfinches,

R – breeding.

canaries | terriers | content	canaries & content	canaries & goldfinches & contents	breeding & keeping & canaries & goldfinches

The first request has the largest area of ​​shaded sectors, then the second, then the third, and the fourth request has the smallest.

In ascending order by number of pages, requests will be presented in the following order: 4 3 2 1

Please note that in the first request, the filled sectors of the Euler circles contain the filled sectors of the second request, and the filled sectors of the second request contain the filled sectors of the third request, and the filled sectors of the third request contain the filled sector of the fourth request.

Only under such conditions can we be sure that we have solved the problem correctly.

Problem 7 (Unified State Exam 2013)

In the search engine query language, the symbol "|" is used to denote the logical "OR" operation, and the symbol "&" is used for the logical "AND" operation.

The table shows the queries and the number of pages found for a certain segment of the Internet.

Request	Pages found (in thousands)
Frigate | Destroyer	3400
Frigate & Destroyer	900
Frigate	2100

How many pages (in thousands) will be found for the query? Destroyer?
It is believed that all queries were executed almost simultaneously, so that the set of pages containing all the searched words did not change during the execution of the queries.

Logics. Textbook Gusev Dmitry Alekseevich

1.6. Euler circle diagrams

1.6. Euler circle diagrams

As we already know, in logic there are six options for relationships between concepts. Any two comparable concepts are necessarily in one of these relations. For example, concepts writer And Russian are in relation to intersection, writer And Human– submission, Moscow And capital of Russia– equivalence, Moscow And Petersburg– subordination, wet road And dry road– opposites, Antarctica And mainland– submission, Antarctica And Africa– subordination, etc., etc.

We must pay attention to the fact that if two concepts denote a part and a whole, for example month And year, then they are in a relationship of subordination, although it may seem that there is a relationship of subordination between them, since the month is included in the year. However, if the concepts month And year were subordinates, then it would be necessary to assert that a month is necessarily a year, and a year is not necessarily a month (remember the relationship of subordination using the example of the concepts crucian carp And fish: crucian carp is necessarily a fish, but fish is not necessarily crucian carp). A month is not a year, and a year is not a month, but both are a period of time, therefore, the concepts of month and year, as well as the concepts book And book page, car And car wheel, molecule And atom etc., are in a relationship of subordination, since part and whole are not the same as species and genus.

At the beginning it was said that concepts can be comparable and incomparable. It is believed that the six options of relations considered are applicable only to comparable concepts. However, it is possible to assert that all incomparable concepts are related to each other in a relationship of subordination. For example, such incomparable concepts as penguin And heavenly body can be considered as subordinate, because a penguin is not a celestial body and vice versa, but at the same time the scope of concepts penguin And heavenly body are included in the broader scope of a third concept, generic in relation to them: this may be the concept object of the surrounding world or form of matter(after all, both the penguin and the celestial body are different objects of the surrounding world or different forms of matter). If one concept denotes something material, and the other – immaterial (for example, tree And thought), then the generic concept for these (as it can be argued) subordinate concepts is form of being, because a tree, a thought, and anything else are different forms of being.

As we already know, the relationships between concepts are depicted by Euler's circular diagrams. Moreover, until now we have schematically depicted the relationship between two concepts, and this can be done with a large number of concepts. For example, relationships between concepts boxer, black And Human

The relative position of the circles shows that the concepts boxer And black person are in relation to intersection (a boxer may be a black man and may not be, and a black man may be a boxer and may not be one), and the concepts boxer And Human, just like concepts black person And Human are in a relationship of subordination (after all, any boxer and any Negro is necessarily a person, but a person may not be either a boxer or a Negro).

Let's consider the relationships between concepts grandfather, father, man, person using a circular diagram:

As we see, these four concepts are in a relationship of sequential subordination: a grandfather is necessarily a father, and a father is not necessarily a grandfather; any father is necessarily a man, but not every man is a father; and, finally, a man is necessarily a person, but not only a man can be a person. Relationships between concepts predator, fish, shark, piranha, pike, living creature are depicted by the following diagram:

Try to comment on this diagram yourself, establishing all the types of relationships between concepts present on it.

To summarize, we note that the relations between concepts are the relations between their volumes. This means that in order to be able to establish relationships between concepts, their volume must be sharp and the content, accordingly, clear, i.e. these concepts must be definite. As for the indefinite concepts discussed above, it is quite difficult, in fact impossible, to establish exact relationships between them, because due to the vagueness of their content and blurred volume, any two indefinite concepts can be characterized as equivalent or intersecting, or as subordinate, etc. For example, is it possible to establish relationships between vague concepts sloppiness And negligence? Whether it will be equivalence or subordination is impossible to say for sure. Thus, the relations between indefinite concepts are also indefinite. It is clear, therefore, that in those situations of intellectual and speech practice where accuracy and unambiguity in determining the relationships between concepts is required, the use of vague concepts is undesirable.

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If you think that you know nothing about such a concept as Euler circles, then you are deeply mistaken. Even from elementary school, schematic images, or circles, are known that allow one to visually comprehend the relationships between concepts and elements of the system.

The method, invented by Leonhard Euler, was used by the scientist to solve complex mathematical problems. He depicted sets in circles and made this diagram the basis of such a concept as symbolic. The method is designed to simplify as much as possible the reasoning aimed at solving a particular problem, which is why the technique is actively used both in primary school and in the academic environment. Interestingly, a similar approach was previously used by the German philosopher Leibniz, and was later taken up and applied in various modifications by famous minds in the field of mathematics. For example, rectangular diagrams of the Czech Bolzano, Schroeder, Venn, famous for creating a popular diagram based on this simple but surprisingly effective method.

Circles are the basis of the so-called “visual Internet memes,” which are based on the similarity of characteristics of individual sets. It’s funny, visual, and most importantly, understandable.

Circles of thought

Circles allow you to clearly describe the conditions of a problem and instantly make the right decision, or identify the direction of movement towards the correct answer. Typically, Euler circles are used to solve logical-mathematical problems involving sets, their unions, or partial superpositions. The intersection of circles includes objects that have the properties of each of the sets depicted in a circle. Objects not included in the set are located outside of one or another circle. If the concepts are absolutely equivalent, they are denoted by one circle, which is the union of two sets that have equal properties and volumes.

Logic of relationships

Using Euler circles, you can solve a number of everyday problems and even decide on the choice of a future profession, you just need to analyze your capabilities and desires and choose their maximum intersection.

Now it becomes clear that Euler’s circles are not at all an abstract mathematical and philosophical concept from the category of theoretical knowledge, they have a very applied and practical significance, allowing you to deal not only with the simplest mathematical problems, but also to solve important life dilemmas in a visual and understandable way for everyone.

Euler circles are a geometric diagram. With its help, you can depict the relationships between subsets (concepts) for a visual representation.

The way of depicting concepts in the form of circles allows you to develop imagination and logical thinking not only for children, but also for adults. Starting from 4-5 years old, children can solve simple problems with Euler circles, first with explanations from adults, and then independently. Mastering the method of solving problems using Euler circles develops the child’s ability to analyze, compare, generalize and group their knowledge for wider application.

Example

The picture shows a variety of all possible toys. Some of the toys are construction sets - they are highlighted in a separate oval. This is part of a large set of “toys” and at the same time a separate set (after all, a construction set can be “Lego” or primitive construction sets made from blocks for kids). Some part of the large variety of “toys” may be wind-up toys. They are not constructors, so we draw a separate oval for them. The yellow oval “wind-up car” refers both to the set “toy” and is part of the smaller set “wind-up toy”. Therefore, it is depicted inside both ovals at once.

Here are some logical thinking tasks for young children:

• Identify the circles that fit the description of the object. In this case, it is advisable to pay attention to those qualities that the object possesses permanently and that it has temporarily. For example, a glass glass with juice always remains glass, but there is not always juice in it. Or there is some kind of broad definition that includes different concepts; such a classification can also be depicted using Euler circles. For example, a cello is a musical instrument, but not every musical instrument is a cello.

For older children, you can offer options for problems with calculations - from fairly simple to very complex. Moreover, independently coming up with these tasks for children will provide parents with a very good workout for the mind.

• 1. Of the 27 fifth-graders, all are studying foreign languages ​​- English and German. 12 are studying German and 19 are learning English. It is necessary to determine how many fifth-graders are studying two foreign languages; how many people don’t study German; how many people don’t study English; How many study only German and only English?

At the same time, the first question of the problem hints in general at the path to solving this problem, informing that some students study both languages, in which case the use of the diagram also makes it easier for children to understand the problem.

By the way, if you can’t decide which profession to choose, try drawing a diagram in the form of Euler circles. Perhaps a drawing like this will help you make your choice:

Those options that will be at the intersection of all three circles are the profession that will not only be able to feed you, but will also please you.

And one more sign...

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