Euler circles are used to display what ratio. Euler circles are figures that conventionally represent sets

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 1

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 a choir, aren’t interested in sports, and don’t participate in a 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 school libraries e, 20 - in the district. 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.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 2

In search engine query language to denote

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

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 "&".

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

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 = 4500

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
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.

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 depicting 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 depicting 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 a 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 the 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. - For a large ship, a long voyage. - 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 no less than their geometric mean, i.e. p 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 the polar expedition, from eight applicants A, B, C, D J5, F, G and I, six specialists must be selected: a biologist, a hydrologist, a weather forecaster, a radio operator, 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 he is on an expedition, will be able to 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?

Logics. Tutorial Gusev Dmitry Alekseevich

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 relationship options considered apply 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 celestial 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 celestial 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

Mutual position circles shows that 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.

From the book Epiphany author Efimov Viktor Alekseevich

From the book Philosophy of Science and Technology author Stepin Vyacheslav Semenovich

Theoretical schemes and abstract objects of technical theory Theoretical schemes are a set of abstract objects oriented, on the one hand, to the use of the corresponding mathematical apparatus, and on the other, to thought experiment,

From the book Dialectics of Myth author Losev Alexey Fedorovich

2. Dialectics of scheme, allegory and symbol What types of this relationship are generally possible? There are a lot of them. But, following Schelling, three main types can be identified. At the same time, we will keep in mind that our terms “internal” and “external” are very general terms and can be

From the book Course of the Age of Aquarius. Apocalypse or rebirth author Efimov Viktor Alekseevich

From the book Selected works author Shchedrovitsky Georgy Petrovich

From the book Man Among Teachings author Krotov Viktor Gavrilovich

Comments and diagrams The teaching, which is based on the inner work of the individual, could not survive this personality itself without the tides of new inner work of new personalities. Those who saw a special meaning for themselves in this teaching. The conditions of existence change, it comes

From the book The Art of Thinking Correctly author Ivin Alexander Arkhipovich

SCHEMES OF CORRECT REASONING Here are two examples of deductive conclusions from the story of the Russian humorist of the beginning of the century V. Bilibin. “If the sun did not exist in the world, we would have to constantly burn candles and kerosene. If we had to constantly burn candles and kerosene, then officials

From the book Ethics of Love and Metaphysics of Self-will: Problems of Moral Philosophy. author Davydov Yuri Nikolaevich

The moral philosophy of Tolstoy and Dostoevsky within the framework of the Nietzschean scheme of nihilism Since the last quarter of the last century, the problem of nihilism has come to one of the first places among the most important problems of Western European philosophy. With her “status” she is primarily

From the book Norms in the space of language author Fedyaeva Natalya Dmitrievna

2.1.1. Norms and schemes of speech communication: speech etiquette The choice of the first problem area - speech etiquette - is due to the following. When determining the essential characteristics of the norm, we started moving from social norms, while noticing that their existence is fully

From the book Spiral Dynamics [Managing Values, Leadership and Change in the 21st Century] by Beck Don

2.1.2. Semiotically fixed norms-schemes: genres The basis for the opposition of socially and semiotically fixed norms, as was said in Chapter I, is the way they are consolidated in sociocultural practice. The first - unwritten laws - become programs, schemes

From the book Logic and Argumentation: Textbook. manual for universities. author Ruzavin Georgy Ivanovich

From the book Architecture and Iconography. “The body of the symbol” in the mirror of classical methodology author Vaneyan Stepan S.

9.1. Graphic diagrams of the structure of argumentation Any argumentation begins with the establishment and discussion of certain facts, which will be further called data, and with the help of which a certain conclusion is put forward and justified. In addition, to move from

From the author's book

Iconography as a system of methods: schemes and threats The very practice of iconographic analysis has formed a “tested scheme” of sequential research actions. The scheme implies: – clarification historical significance motive - from the point of view of time (moment

Sections: Informatics

1. Introduction

In the course of Computer Science and ICT of the basic and high school, the following are discussed: important topics like “Fundamentals of Logic” and “Searching for Information on the Internet”. When solving a certain type of problem, it is convenient to use Euler circles (Euler-Venn diagrams).

Mathematical reference. Euler-Venn diagrams are used primarily in set theory as a schematic representation of all possible intersections of several sets. In general, they represent all 2 n combinations of n properties. For example, with n=3, the Euler-Venn diagram is usually depicted as three circles with centers at the vertices of an equilateral triangle and the same radius, approximately equal to the length of the side of the triangle.

2. Representation of logical connectives in search queries

When studying the topic “Searching for information on the Internet”, examples of search queries using logical connectives, similar in meaning to the conjunctions “and”, “or” of the Russian language, are considered. The meaning of logical connectives becomes clearer if you illustrate them using a graphical diagram - Euler circles (Euler-Venn diagrams).

Logical connective	Example request	Explanation	Euler circles
& - "AND"	Paris & university	All pages that mention both words: Paris and university will be selected	Fig.1
\| - "OR"	Paris \| university	All pages where the words Paris and/or university are mentioned will be selected	Fig.2

3. Connection of logical operations with set theory

Euler-Venn diagrams can be used to visualize the connection between logical operations and set theory. For demonstration, you can use the slides in Appendix 1.

Logical operations are specified by their truth tables. IN Appendix 2 Graphic illustrations of logical operations along with their truth tables are discussed in detail. Let us explain the principle of constructing a diagram in the general case. In the diagram, the area of the circle with the name A displays the truth of statement A (in set theory, circle A is the designation of all elements included in a given set). Accordingly, the area outside the circle displays the “false” value of the corresponding statement. To understand which area of the diagram will display a logical operation, you need to shade only those areas in which the values of the logical operation on sets A and B are equal to “true”.

For example, the implication value is true in three cases (00, 01, and 11). Let's shade sequentially: 1) the area outside the two intersecting circles, which corresponds to the values A=0, B=0; 2) an area related only to circle B (crescent), which corresponds to the values A=0, B=1; 3) the area related to both circle A and circle B (intersection) - corresponds to the values A=1, B=1. The combination of these three areas will be a graphical representation of the logical operation of implication.

4. Use of Euler circles in proving logical equalities (laws)

In order to prove logical equalities, you can use the Euler-Venn diagram method. Let us prove the following equality ¬(АvВ) = ¬А&¬В (de Morgan's law).

To visually represent the left side of the equality, let’s do it sequentially: shade both circles (apply disjunction) with gray color, then to display the inversion, shade the area outside the circles with black color:

Fig.3 Fig.4

To visually represent the right side of the equality, let’s do it sequentially: shade the area for displaying the inversion (¬A) in gray and, similarly, the area ¬B also in gray; then to display the conjunction you need to take the intersection of these gray areas (the result of the overlay is represented in black):

Fig.5 Fig.6 Fig.7

We see that the areas for displaying the left and right parts are equal. Q.E.D.

5. Problems in the State Examination and Unified State Exam format on the topic: “Searching for information on the Internet”

Problem No. 18 from the demo version of GIA 2013.

The table shows queries to the search server. For each request, its code is indicated - the corresponding letter from A to G. Arrange the request codes from left to right in order descending the number of pages that the search engine will find for each request.

Code	Request
A	(Fly & Money) \| Samovar
B	Fly & Money & Bazaar & Samovar
IN	Fly \| Money \| Samovar
G	Fly & Money & Samovar

For each query, we will build an Euler-Venn diagram:

Request A

Request B

Request G

Answer: VAGB.

Problem B12 from the demo version of the Unified State Exam 2013.

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.

Ф – number of pages (in thousands) on request Frigate;

E – number of pages (in thousands) on request Destroyer;

X – number of pages (in thousands) for a query that mentions Frigate And Not mentioned Destroyer;

Y – number of pages (in thousands) for a query that mentions Destroyer And Not mentioned Frigate.

Let's build Euler-Venn diagrams for each query:

Request	Euler-Venn diagram	Number of pages
Frigate \| Destroyer	Fig.12	3400
Frigate & Destroyer	Fig.13	900
Frigate	Fig.14	2100
Destroyer	Fig.15	?

According to the diagrams we have:

X + 900 + Y = F + Y = 2100 + Y = 3400. From here we find Y = 3400-2100 = 1300.
E = 900+U = 900+1300= 2200.

Answer: 2200.

6. Solving logical meaningful problems using the Euler-Venn diagram method

There are 36 people in the class. Pupils of this class attend mathematical, physics and chemical circles, with 18 people attending the mathematical circle, 14 people attending the physical circle, 10 people attending the chemical circle. In addition, it is known that 2 people attend all three circles, 8 people attend both mathematical and physical, 5 and mathematical and chemical, 3 - both physical and chemical.

How many students in the class do not attend any clubs?

To solve this problem, it is very convenient and intuitive to use Euler circles.

The largest circle is the set of all students in the class. Inside the circle there are three intersecting sets: members of the mathematical ( M), physical ( F), chemical ( X) circles.

Let MFC- a lot of guys, each of whom attends all three clubs. MF¬X- a lot of kids, each of whom attends math and physics clubs and Not visits chemical. ¬M¬FH- a lot of guys, each of whom attends the chemistry club and does not attend the physics and mathematics clubs.

Similarly, we introduce sets: ¬МФХ, М¬ФХ, М¬Ф¬Х, ¬МФ¬Х, ¬М¬Ф¬Х.

It is known that all three circles are attended by 2 people, therefore, in the region MFC Let's enter the number 2. Because 8 people attend both mathematical and physical circles, and among them there are already 2 people attending all three circles, then in the region MF¬X let's enter 6 people (8-2). Let us similarly determine the number of students in the remaining sets:

Let's sum up the number of people in all regions: 7+6+3+2+4+1+5=28. Consequently, 28 people from the class attend clubs.

This means 36-28 = 8 students do not attend clubs.

After winter break class teacher asked which of the guys went to the theater, cinema or circus. It turned out that out of 36 students in the class, two had never been to the cinema. neither in the theater nor in the circus. 25 people went to the cinema, 11 to the theater, 17 to the circus; both in cinema and theater - 6; both in the cinema and in the circus - 10; and in the theater and circus - 4.

How many people have been to the cinema, the theater, and the circus?

Let x be the number of children who have been to the cinema, the theater, and the circus.

Then you can build the following diagram and count the number of guys in each area:

6 people visited the cinema and theater, which means only 6 people went to the cinema and theater.

Similarly, only in cinema and circus (10th) people.

Only in theater and circus (4) people.

25 people went to the cinema, which means that 25 of them only went to the cinema - (10's) - (6's) - x = (9+x).

Similarly, only in the theater there were (1+x) people.

Only there were (3+x) people in the circus.

Haven’t been to the theatre, cinema or circus – 2 people.

So, 36-2=34 people. attended events.

On the other hand, we can sum up the number of people who were in the theater, cinema and circus:

(9+x)+(1+x)+(3+x)+(10's)+(6's)+(4's)+x = 34

It follows that only one person attended all three events.

Thus, Euler circles (Euler-Venn diagrams) find practical application in solving problems in the Unified State Examination and State Examination format and in solving meaningful logical problems.

Literature

V.Yu. Lyskova, E.A. Rakitina. Logic in computer science. M.: Informatics and Education, 2006. 155 p.
L.L. Bosova. Arithmetic and logical foundations of computers. M.: Informatics and Education, 2000. 207 p.
L.L. Bosova, A.Yu. Bosova. Textbook. Computer science and ICT for grade 8: BINOM. Knowledge Laboratory, 2012. 220 p.
L.L. Bosova, A.Yu. Bosova. Textbook. Computer science and ICT for grade 9: BINOM. Knowledge Laboratory, 2012. 244 p.
FIPI website: http://www.fipi.ru/

Objective of the lesson: Introduce students to solving simple logical problems using the circle method

Lesson Objectives

Educational: give students an idea of the Euler circle method;
Developmental: development of logical and analytical thinking;
Educational: developing the ability to listen to the opinions of other students and defend their point of view.

Lesson material: task cards, portrait of L. Euler, blackboard.

Lesson progress

Organizational moment (3 min)
Warm-up (5 min)
Learning new material (5 min)
Initial testing of the solution method (30 min)
Summing up the lesson (2 min)
Organizational moment.

Teacher: Hello guys! Today in class we will introduce you to a new method for solving logical problems - Euler circles. We will learn to solve some of those problems that are included in the group of competitive and olympiad problems. The purpose of our lesson: is to get acquainted with solving the simplest logical problems using the circle method.

Warm-up

Students are offered several humorous logical tasks aimed at activating students' thinking.

A goose costs 20 rubles and half of what it actually costs. How much did the goose cost?
Two athletes ran 8 laps around the stadium at the competition. How many laps did each person run?
Name two numbers whose difference is equal to their sum.
What is two plus two times two?

Learning new material

Teacher: In mathematics, drawings in the form of circles representing sets have been used for a very long time. One of the first to use this method was the outstanding German mathematician and philosopher Gottfried Wilhelm Leibniz (1646-1716). Drawings with such circles were found in his rough sketches. Then this method was developed quite thoroughly by Leonhard Euler. He worked for many years at the St. Petersburg Academy of Sciences.

For a visual geometric illustration of concepts and relationships between them, Euler-Venn diagrams (Euler circles) are used. If there are any concepts A, B, C, etc., then the volume of each concept (set) can be represented as a circle, and the relationships between these objects (sets) can be represented as intersecting circles.

Before solving the problem, answer the following questions:

How many sets are we talking about in this problem?
Which of the data listed in the problem belong to different sets at the same time?

Initial development of the solution method. Students are offered the following tasks. The first task is discussed in detail. Subsequent problems are solved by students at the board.

Task 1. Pets. All my friends have pets. Six of them love and keep cats, and five love dogs. And only two have both. Guess how many girlfriends I have?

Solution: Let's draw two circles, since we have two types of pets. In one we will record the owners of cats, in the other - owners of dogs. Since some friends have both animals, we will draw circles so that they have a common part. In this general part we put the number 2 since both of them have cats and dogs. In the remaining part of the “cat” circle we put the number 4 (6 - 2 = 4). In the free part of the “dog” circle we put the number 3 (5 - 2 = 3). And now the picture itself suggests that in total I have 4 + 2 + 3 = 9 girlfriends.

Answer. 9 girlfriends.

Task 2. Libraries. There are 30 students in the class. All of them are readers of school and district libraries. Of these, 20 children borrow books from the school library, 15 from the district library. How many students are not readers of the school library?

Solution: Let circle W represent readers only of the school library, circle P - only the district one. Then ShR is an image of readers of both district and school libraries at the same time. It follows from the figure that the number of students who are not readers of the school library is equal to:

(not Ш) = R - ШР. There are 30 students in total, W = 20 people, P = 15 people. Then the value of ШР can be found as follows (see figure): ШР = (Ш + Р) - 30 = (20 + 15) - 30 = = 5, i.e. 5 students are readers of the school and district libraries at the same time. Then (not Ш) = = Р - ШР= 15 - 5= 10.

Answer: 10 students are not readers of the school library.

Task 3. Favorite cartoons. A survey was conducted among fifth grade schoolchildren on their favorite cartoons. The most popular were three cartoons: “Snow White and the Seven Dwarfs”, “Winnie the Pooh”, “Mickey Mouse”. There are 28 people in total in the class. "Snow White and the Seven Dwarfs" was chosen by 16 students, among whom three named "Mickey Mouse", six - "Winnie the Pooh", and one wrote all three cartoons. The cartoon “Mickey Mouse” was named by 9 children, among whom five chose two cartoons each. How many people chose the cartoon "Winnie the Pooh"?

Solution: There are 3 sets in this problem; from the conditions of the problem it is clear that they all intersect with each other. Only Snow White was chosen by 16-6-3-1=6 people. Only "Mickey Mouse" was chosen by 9-3-2-1=3 people.

Only “Winnie the Pooh” was chosen by 28-(6+3+3+2+6+1)=7 people. Then, taking into account that some people chose several cartoons, we get that “Winnie the Pooh” was chosen by 7+6+1+2=16 people.

Task 4. Hobby. Of the 24 students in grade 5, 10 people attend music school, 8 people attend art school, 12 people attend sports school, 3 people attend music and art school, 2 people attend art and sports school, 2 people attend music and sports school, 1 person attends all three schools . How many students attend only one school? How many students do not develop themselves in anything?

Solution: There are 3 sets in this problem, from the conditions of the problem it is clear that they all intersect with each other. Only the music school is attended by 10-3-2-1=4 students. Only the art school is attended by 8-3-2-1=2 students. Only the sports school is attended by 12-2-2-1=7 students.

Only one school is attended by 4+2+7=13 students.

24-(4+2+7+3+2+2+1)=3 students do not develop themselves in anything.

Answer. 13 students attend only one school, 3 students do not develop themselves.

Task 5. About puzzles. There were 26 different math games- puzzles. Both Grisha and Sasha played in 4 of them. Igor tried to play 7 games that neither Grisha nor Sasha touched, and two puzzles that Grisha played. In total, Grisha played 11 mathematical games - puzzles. How many puzzles did Sasha play?

Solution: Since Grisha lost a total of 11 games, of which 4 puzzles were solved by Sasha and 2 puzzles by Igor, then 11 - 4 - 2 = 5 - games were lost only by Grisha. Therefore, 26 - 7 - 2 - 5 - 4 = 8 - puzzles were solved by Sasha alone. But all Sasha played was games.

Answer. 12 games were solved by Sasha.

Objective 7. Sports for everyone. There are 38 people in the class. Of these, 16 play basketball, 17 play hockey, 18 play football. Four are fond of two sports - basketball and hockey, three - basketball and football, five - football and hockey. Three are not interested in basketball, hockey, or football. How many kids are interested in three sports at the same time? How many kids are interested in only one of these sports?

Solution. Let's use Euler circles.

Let the large circle represent all the students in the class, and the three smaller circles B, X and F represent basketball, hockey and football players, respectively. Then the figure Z, the common part of circles B, X and F, depicts children who are fond of three sports. From the examination of Euler circles it is clear that only one sport - basketball - is played by 16 - (4 + z + 3) = 9 - z; hockey alone 17 - (4 + z + 5) = 8 - z; just football

18 - (3 + z + 5) = 10 - z. We make up an equation, taking advantage of the fact that the class is divided into separate groups of children; the number of guys in each group is circled in the figure: 3 + (9 - z) + (8 - z) + (10 - z) + 4 + 3 + 5 + z = 38,z = 2. Thus, two guys get carried away all three sports. Adding the numbers 9 - z, 8 - z and 10 - z, where z = 2, we find the number of children who are interested in only one sport: 21 people.

Answer: Two guys are fond of all three human sports. Those who are interested in only one sport: 21 people.

Homework. Task 6. Sports class. There are 35 students in the class. 24 of them play football, 18 play volleyball, 12 play basketball. 10 students play football and volleyball at the same time, 8 play football and basketball, and 5 play volleyball and basketball. How many students play football, volleyball, and basketball at the same time?

Summing up the lesson

Students summarize the lesson independently or by answering guiding questions:

What did we learn in class?
What is this method? What is it?
What did we learn in class today?
Where do you need to start solving the problem?
What tasks gave you the most difficulty? Why?

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