If you think you don't know anything about Euler circles, you're wrong. In fact, you've probably encountered them more than once, you just didn't know what it was called. Where exactly? Schemes in the form of Euler circles formed the basis of many popular Internet memes (images circulated online on a specific topic).
Let's figure out together what kind of circles these are, why they are called that and why they are so convenient to use to solve many problems.
Origin of the term
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.
It’s not very clear yet, right? Look at this picture:
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.
Well, has it become clearer? That is why Euler circles are a method that clearly demonstrates: it is better to see once than to hear a hundred times. Its merit is that clarity simplifies reasoning and helps to get an answer faster and easier.
The author of the method is the scientist Leonhard Euler (1707-1783). He said this about the diagrams named after him: “circles are suitable to facilitate our thinking.” Euler is considered a German, Swiss and even Russian mathematician, mechanic and physicist. The fact is that he worked for many years at the St. Petersburg Academy of Sciences and made a significant contribution to the development of Russian science.
Before him, the German mathematician and philosopher Gottfried Leibniz was guided by a similar principle when constructing his conclusions.
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. For example, Czech mathematician Bernard Bolzano used the same method, but with rectangular circuits.
The German mathematician Ernest Schroeder also made his contribution. But the main merits belong to the Englishman John Venn. He was a specialist in logic and published the book “Symbolic Logic”, in which he outlined in detail his version of the method (he used mainly images of intersections of sets).
Thanks to Venn's contribution, the method is even called Venn diagrams or Euler-Venn diagrams.
Why are Euler circles needed?
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.
If we talk about the types of Euler circles, we can divide them into those that describe the unification of some concepts (for example, the relationship between genus and species) - we looked at them using an example at the beginning of the article.
And also those that describe the intersection of sets according to some characteristic. John Venn was guided by this principle in his schemes. And it is this that underlies many popular memes on the Internet. Here is one example of such Euler circles:
It's funny, isn't it? And most importantly, everything immediately becomes clear. You can spend a lot of words explaining your point of view, or you can just draw a simple diagram that will immediately put everything in its place.
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.
Solving problems using Euler circles
Let's look at some examples of problems that can be solved using Euler circles.
Here on this site - http://logika.vobrazovanie.ru/index.php?link=kr_e.html Elena Sergeevna Sazhenina offers interesting and simple problems, the solution of which will require the Euler method. Using logic and mathematics, we will analyze one of them.
Problem about favorite cartoons
Sixth graders filled out a questionnaire asking about their favorite cartoons. It turned out that most of them like “Snow White and the Seven Dwarfs”, “SpongeBob” Square Pants" and "Wolf and 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.
Solution:
Since according to the conditions of the problem we are given three sets, we draw three circles. And since the guys’ answers show that the sets intersect with each other, the drawing will look like this:
We remember that according to the terms of the task, among fans of the cartoon “The Wolf and the Calf”, five guys chose two cartoons at once:
It turns out that:
21 – 3 – 6 – 1 = 11 – the guys chose only “Snow White and the Seven Dwarfs”.
13 – 3 – 1 – 2 = 7 – the guys only watch “The Wolf and the Calf.”
It remains only to figure out how many sixth-graders prefer the cartoon “SpongeBob SquarePants” to the other two options. From the total number of students we subtract all those who love the other two cartoons or chose several options:
38 – (11 + 3 + 1 + 6 + 2 + 7) = 8 – people only watch “SpongeBob SquarePants.”
Now we can safely add up all the resulting numbers and find out that:
cartoon “SpongeBob SquarePants” was chosen by 8 + 2 + 1 + 6 = 17 people. This is the answer to the question posed in the problem.
Let's also look at task, which was put on display in 2011 Unified State Exam test in computer science and ICT (source - http://eileracrugi.narod.ru/index/0-6).
Conditions of the problem:
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) |
| Cruiser | Battleship | 7000 |
| Cruiser | 4800 |
| Battleship | 4500 |
How many pages (in thousands) will be found for the query? 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:
- Cruiser | Battleship: 1 + 2 + 3 = 7000
- Cruiser: 1 + 2 = 4800
- 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 - the number of pages found by request Cruiser & Battleship.
As you can see, Euler circles help to quickly and easily solve even quite complex or simply confusing problems at first glance.
Conclusion
I think we have managed to convince you that Euler circles are not just a fun and interesting thing, but also a very useful method for solving problems. And not only abstract problems on school lessons, but also quite a lot of everyday problems. Choice future profession, For example.
You will probably also be curious to know that in modern popular culture Euler’s circles are reflected not only in the form of memes, but also in popular TV series. Such as “The Big Bang Theory” and “4Isla”.
Use this useful and visual method to solve problems. And be sure to tell your friends and classmates about it. There are special buttons under the article for this.
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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 in different languages has a different verbal expression.
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 scopes of concepts using Euler circles (circular diagrams where each circle denotes the scope 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)
Problem 1.
Each of the 35 sixth-graders is a reader of at least one of two libraries: school and district. Of these, 25 people borrow books from the school library, 20 from the district library.
How many sixth graders:
1. Are readers of both libraries;
2. Are not readers of the district library;
3. Are not readers of the school library;
4. Are readers only of the regional library;
5. Are readers only at the school library?
Note that the first question is key to understanding and solving this problem. After all, you won’t immediately understand how it turns out 20 + 25 = 45 out of 35. The first question provides a hint for understanding the condition: there are students who visit both libraries. And if the condition of the problem is depicted on a diagram, then the answer to the first question becomes obvious.
Solution.
1. 20 + 25 – 35 = 10 (people) – are readers of both libraries. In the diagram this is the common part of the circles. We have determined the only quantity unknown to us. Now, looking at the diagram, we can easily answer the questions posed.
2. 35 – 20 = 15 (people) – are not readers of the district library. (In the diagram, the left part of the left circle)
3. 35 – 25 = 10 (people) – are not readers of the school library. (In the diagram, the right part of the right circle)
4. 35 – 25 = 10 (people) – are readers only of the regional library. (In the diagram, the right part of the right circle)
5. 35 – 20 = 15 (people) – are readers only of the school library. (In the diagram, the left part of the left circle).
It's obvious that 2 and 5 , and 3 and 4 – are equivalent and the answers to them are the same .
When solving this problem, we used a method of its graphical representation using the so-called Euler circles. This method was proposed by Leonhard Euler and is widely used in solving logical problems.
Leonard Euler(4(15) April 1707, Basel, Switzerland - 7(18) September 1783, St. Petersburg, Russian empire) – Swiss, German and 
Russian mathematician who made a significant contribution to the development of mathematics, as well as mechanics, physics, astronomy and a number of applied sciences. Some of his descendants still live in Russia.
Let's look at another example.
Task 2.
Some of the residents of our building subscribe only to the Komsomolskaya Pravda newspaper, some - only the Izvestia newspaper, and some - both newspapers. What percentage of residents of the house subscribe to both newspapers, if 85% of them subscribe to the Komsomolskaya Pravda newspaper, and 75% to Izvestia?
Solution.
There is no fundamental difference here from the previous solution. In the finished drawing, replace the data: 25 with 85% and 20 with 75%. Considering that all residents of the house make up 100%, we replace 35 with 100% and get a ready-made solution: 85% + 75% – 100% = 60%.
Answer: 60% of residents subscribe to both newspapers.
The more complex and intricate the logical problem associated with sets, the more obvious the effect of using Euler circles. Only after drawing up the drawing does their solution become quite obvious.
Task 3.
There are 70 children in three seventh grades. Of these, 27 are involved in the drama club, 32 sing in the choir, 22 are fond of sports. There are 10 guys from the choir in the drama club, 6 athletes in the choir, 8 athletes in the drama club; 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?
Solution.
Let
D - drama club,
X – choir,
S – sport.
Then
in circle D - 27 guys,
in circle X – 32 people,
in circle C – 22 students.
Those 10 guys from the drama club who sing in the choir will be in the common part of circles D and X. Three of them are also athletes, they will be in the common part of all three circles. The remaining seven are not interested in sports. Similarly, 8 – 3 = 5 athletes who do not sing in the choir and 6 – 3 = 3 who do not attend the drama club.
It is easy to see that 5 + 3 + 3 = 11 athletes attend a choir or drama club,
22 – (5 + 3 + 3) = 11 do only sports;
70 – (11 + 12 + 19 + 7 + 3 + 3 + 5) = 10 – do not sing in a choir, do not participate in a drama club, do not enjoy sports.
Answer: 10 people and 11 people.
Task 4.
There are 30 people in the class. 20 of them use the metro every day, 15 use the bus, 23 use the trolleybus, 10 use both the metro and trolleybus, 12 use both the metro and bus, 9 use both trolleybus and bus. How many people use all three modes of transport every day?
Solution.
1 way.
To solve, we again use Euler circles. Let x person 
uses all three modes of transport. Then enjoy
only by metro and trolleybus – (10 – x) people,
only by bus and trolleybus – (9 – x) people,
only by metro and bus – (12 – x) people.
Let's find how many people use the metro alone:
20 – (12 – x) – (10 – x) – x = x – 2.
Similarly, we get: x – 6 – only by bus and x + 4 – only by trolleybus, since there are only 30 people, we create the equation:
x + (12 – x) + (9 – x) + (10 – x) + (x + 4) + (x – 2) + (x – 6) = 30,
from here x = 3.
Method 2. But you can solve this problem in another way: 20 + 15 + 23 – 10 – 12 – 9 + x = 30, 27 + x = 30, x = 3. Here we added up the number of students who use at least one type of transport and subtracted from the resulting amount the number of those who use two or three types and, therefore, entered the total 2-3 times. Thus, we got the number of all students in the class.
Answer. 3 people use all three modes of transport every day.
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Euler circles are figures that conventionally represent sets and visually illustrate some properties of operations on sets. In the literature, Euler circles are sometimes called Venn diagrams (or Euler-Venn diagrams). Euler circles, illustrating the basic operations on sets, are presented in Fig. 1.2 (the sets obtained as a result of these operations are marked with shading). AR 00 ABV Fig. 1.2 Example 1.8. Using Euler circles, we first establish the validity of the first relation, which expresses the distributive property of the operations of union and intersection of sets. In Fig. 1.3, and the circle representing the set A is vertically shaded, and the area corresponding to the intersection of sets B and C is horizontally shaded. As a result, the area representing the set A U (BPS) is shaded in one way or another. In Fig. 1.3.5, the area corresponding to the union of the sets A and B is vertically shaded, and horizontally - the union of the sets A and C, so that in both ways the area representing the set (A U B) P (A U C) and coinciding with the area shaded by any method in Fig. 1.3,a. Thus, Euler circles make it possible to establish the validity of (1.10). Now consider De Morgan's second law (1.7) Shaded in Fig. 1.4, and the area depicts the set of LIVs, and the unshaded part of the rectangle Q (external to the shaded part) corresponds to the set of LIVs. In Fig. 1.4,5 parts of rectangle 12, shaded vertically and horizontally, correspond to A and B, respectively. Then the Lie set B corresponds to the area shaded in at least one of the indicated ways. It coincides with the area not shaded in Fig. 1.4,a and corresponding to the set of LPBs, which establishes the validity of (1.11). Questions and tasks 1.1. The notation m|n, where m,n € Z, means that the number m completely divides the number n (then it is a divisor of n). Describe the given sets provided that x € N: 1.2. Prove the following relations and illustrate them with Euler circles: . 1.3. Establish in what relation (X C Y, X E Y or X = Y) the sets X and Y are located if: a Use Euler circles for illustration. 1.4. Let Aj be the set of points forming the sides of some triangle inscribed in a given circle. Describe the union and intersection of all such sets if the triangles are: a) arbitrary; b) correct; c) rectangular. Find IK and flAi ieN i en for given families of sets: 1.6. Indicate which of the following relationships are incorrect and explain why: 1. 7. Indicate which of the sets are equal to each other: . 1.8. Find the Lie sets B, AG\B, A\B, BA\A and depict them on the number line if A = (1.0. Considering the segment to be a universal set, find and depict on the number line the complements of the sets: . 1.10. According to the descriptions below sets of people, select statements in the language of sets for each entry suitable proverb or a saying. We hope that this will allow us to once again analyze the meaning of folk sayings. For example, if Z is a set of people who themselves do not properly know what they are talking about, then the entry x £ Z can be attributed to the proverb “He heard a ringing, but does not know where it is, since this is exactly what they say about a person endowed with specified property(in this case, a characteristic property of the set Z, see 1.1). Sets of people ft - the universal set of all people, L - kind, 5e B - extraordinary, with great abilities, S - stupid, D - smart, E - acting in their own way, not listening to advice, F - connected by selfish relationships, G - promising a lot , I - those who do not keep their promises, J - those who abuse their official position, K - those who are too self-important, too self-important, L - those who interfere in something other than their own business, M - those who are enterprising, dexterous, who know how to get organized, P - those who take on several things at once, Q - fruitfully working, S - making mistakes, T - feeling guilty and the possibility of retribution, U - not achieving results, V - betraying themselves with their behavior, W - short-sighted, X - acting together, not betraying each other, U - experienced, experienced people. Recording statements in the language of sets heK; xeGnH; xCBCiQ; x£jr\U; xeJ; heM; heSPE; xCTnV; xEPDU; xGE; x € FnX; xeYnS; xeDOW. Proverbs and sayings - God does not give a horn to a lively cow. - big ship- great swimming. - Free will. - A raven will not peck out a crow's eye. - There is no law for fools. - If you chase two hares, you won’t catch either. - The cat knows whose meat it ate. - Cricket know your nest. - And the old woman can get into trouble. - The chicken is not the aunt, the pig is not the sister. - He who dared ate it. - Simplicity is enough for every wise man. - The titmouse made a name for itself, but didn’t set the sea on fire. - The world is not without good people. 1.11. Prove the validity of relations (1.2). 1.12. Prove the validity of the second of the relations of the distributivity property of the operations of union and intersection directly and by contradiction. 1.13. Using the method of mathematical induction, we can prove that for any natural number n the inequalities n^2n~1 and (l + :r)n ^ 1 + ns, Vs>-1 (Bernoulli’s inequality) are valid. 1.14. Prove that the arithmetic mean of n positive real numbers is not less than their geometric mean, i.e. clause 1.15. Brown, Jones and Smith are charged with complicity in bank robbery. The thieves fled in a car that was waiting for them. During the investigation, Brown testified that it was a blue Buick, Jones a blue Chrysler, and Smith a Ford Mustang, but not blue. What color was the car and what make, if it is known that, wanting to confuse the investigation, each of them indicated correctly either only the make of the car, or only its color? 1.1c. For polar expedition from eight applicants A, B, C, D J5, F, G and Z, six specialists must be selected: biologist, hydrologist, weather forecaster, radio operator, mechanic and doctor. The duties of a biologist can be performed by E and G, a hydrologist - B and F, a weather forecaster - F and G, a radio operator - C and D, a mechanic - C and Z, a doctor - A and D, but each of them, if on an expedition, can perform only one duty. Who and by whom should be taken on the expedition if F cannot go without D - without I and without C, C cannot go with G, and D cannot go with B?
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.
Problem 1Of 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 district library;
e) are readers of both libraries?
2.Every student in the class learns English or German, or both of these languages. English language 25 people study German, 27 people study German, and 18 people study both. 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. B kindergarten 52 children. 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, and 18 people attend the mathematical club, 14 - physical, 10 - chemical. In addition, it is known that 2 people attend all three clubs, 8 people - 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?
Solution 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.
| Request | Pages found (in thousands) |
| Cruiser | Battleship | 7000 |
| Cruiser | 4800 |
| Battleship | 4500 |
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:
- Cruiser | Battleship: 1 + 2 + 3 = 7000
- Cruiser: 1 + 2 = 4800
- 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.
Problem 2In search engine query language to denote
| Request | Pages found (in thousands) |
| Cakes | Pies | 12000 |
| Cakes & Pies | 6500 |
| Pies | 7700 |
How many pages (in thousands) will be found for the query? Cakes?
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 "&".
| Request | Pages found (in thousands) |
| Cakes & Baking | 5100 |
| Cake | 9700 |
| Cake | Bakery | 14200 |
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.
Cake │ Baking – Cake = A + B + C -(A + B) = C = 14200–9700 = 4500Sector B is equal to 4500, therefore Baking = B + C = 4500+5100 = 9600
Problem 4descending
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 5The 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 + DBaroque │(classicism & empire) = G + B
classicism & empire style = B
baroque│classicism = G + B + A
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
Problem 6
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 | canaries | goldfinches | content
|
| 2 | canaries & content |
| 3 | canaries & goldfinches & contents |
| 4 | breeding & keeping & canaries & goldfinches |
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.
