Application of Euler-Venn diagrams in solving logical problems. Euler circles on the example of solving a problem
Leonard Euler (1707-1783) - a famous Swiss and Russian mathematician, a member of the St. Petersburg Academy of Sciences, spent most of his life in Russia. The most famous in statistics, computer science and logic is the Euler circle (Euler-Venn diagram), used to denote 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 designated by one or a group of words: "world map", "dominant fifth-seventh chord", "Monday", etc.
In the case when the volume elements of one concept fully or partially belong to the volume of another, one speaks of compatible concepts. If not one element of the volume of a certain concept belongs to the volume of another, then we have a place with incompatible concepts.
In turn, each of the concept types has its own set of possible relationships. For compatible concepts, these are the following:
- identity (equivalence) of volumes;
- intersection (overlap) of volumes;
- subordination (subordination).
For incompatible:
- subordination (coordination);
- opposite (contrariness);
- contradiction (contradictory).
Schematically, the relationship between concepts in logic is usually denoted using Euler-Venn circles.
Equivalence relations
In this case, the concepts mean the same subject. Accordingly, the volumes of these concepts completely coincide. For example:
A - Sigmund Freud;
B - the founder of psychoanalysis.
A - square;
B - equilateral rectangle;
C - conformal rhombus.
Fully matching Euler circles are used for designation.
Intersection (partial match)
A - teacher;
B is a music lover.
As can be seen from this example, the scope of concepts overlaps: a certain group of teachers may turn out to be music lovers, and vice versa, there may be representatives of the teaching profession among music lovers. A similar attitude will be in the case when in A acts, for example, "city dweller", and in the capacity of B - "driver".
Subordination (subordination)
They are 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 volume) is fully included in the subordinate (larger in volume). At the same time, the subordinate concept does not completely exhaust the subordinate one.
For example:
A - tree;
B - pine.
Concept B will be subordinate to concept A. Since pine refers to trees, concept A becomes in this example subordinating, "absorbing" the scope of the concept of V.
Subordination (coordination)
A relationship characterizes two or more concepts that exclude each other, but at the same time belong to a certain common family circle. For example:
A - clarinet;
B - guitar;
С - violin;
D is a musical instrument.
The concepts A, B, C are not overlapping in relation to each other, however, they all belong to the category musical instruments(concept D).
Opposite (contrariness)
Opposite relationships between concepts imply the attribution of these concepts to the same genus. In this case, one of the concepts has certain properties (features), while the other denies them, replacing them with the opposite in character. Thus, we are dealing with antonyms. For example:
A - dwarf;
B is a giant.
With the opposite relationship between concepts, Euler's 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.
Controversy (contradictory)
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 is 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 - the third, intermediate link in this case does not exist. Thus, concepts are also antonyms. In this case, one of them (A) becomes positive (affirming any sign), and the second (B or not-A) - negative (denying the corresponding sign): "white paper" - "not white paper", " National history"-" foreign history ", etc.
Thus, the ratio of the volumes of concepts in relation to each other is the key characteristic that determines the Euler circles.
Relationships between sets
You should also distinguish between the concepts of elements and sets, the volume of which is displayed by the Euler circles. The concept of a set is borrowed from mathematical science and has a fairly broad meaning. Examples in logic and mathematics display it as a kind of collection of objects. The objects themselves are elements of this set. “Many are many, conceivable as one” (Georg Cantor, founder of the theory of sets).
The designation of the sets is carried out by A, B, C, D ... and so on, the elements of the sets - in lowercase: a, b, c, d ... and others. Examples of the set can be students in the same classroom, books standing on a certain shelf (or, for example, all the books in a particular library), pages in a diary, berries in a forest glade, etc.
In turn, if a certain set does not contain any element, then it is called empty and denoted by the sign Ø. For example, the set of intersection points is the set of solutions to the equation x 2 = -5.
Solving problems
Euler circles are actively used to solve a large number of problems. Examples in logic clearly demonstrate the connection with set theory. In this case, the truth tables of concepts are used. For example, the circle designated by the name A represents the region of truth. Thus, 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 that define the Euler circle in which its values for elements A and B will be true.
The use of Euler circles has found wide practical use in different industries. For example, in a situation with a professional choice. If the subject is preoccupied with choice future profession, he can be guided by the following criteria:
W - what do I like to do?
D - what do I do?
P - how can I earn good money?
Let's depict this in the form of a diagram: in logic - the 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 when calculating combinations and properties. Euler's circles of the set of elements are enclosed in a rectangle representing the universal set (U). Instead of circles, other closed shapes can also be used, but the essence does not change. The figures intersect each other according to the conditions of the problem (in the most general case). Also, these figures should be marked accordingly. Points located inside different segments of the diagram can act as elements of the sets under consideration. On its basis, it is possible to shade specific areas, thereby denoting the newly formed sets.
With these sets, it is permissible to perform basic mathematical operations: addition (sum of sets of elements), subtraction (difference), multiplication (product). In addition, thanks to the Euler-Venn diagrams, it is possible to perform comparison operations of sets by the number of elements included in them, not counting them.
1. Introduction
The Computer Science and ICT course in basic and high school examines such important topics like “Basics of Logic” and “Searching for Information on the Internet”. When solving a certain type of problems, 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, for 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 are considered, similar in meaning to the unions “and”, “or” of the Russian language. The meaning of logical connectives becomes clearer if you illustrate them using a graphical diagram - Euler circles (Euler-Venn diagrams).
| Logical connective | Request example | Explanation | Euler circles |
| & - "AND" | Paris & university | All pages where both words are mentioned: Paris and University will be selected | Fig. 1 |
| | - "OR" | Paris | university | All pages that mention the words Paris and / or University will be selected | Fig. 2
|
3. Connection of logical operations with set theory
With the help of Euler-Venn diagrams, one can visualize the connection between logical operations and set theory. For demonstration, you can use the slides in Annex 1.
Logical operations are specified by their own truth tables. V Annex 2 graphic illustrations of logical operations are discussed in detail along with their truth tables. Let us explain the principle of constructing a diagram in the general case. On the diagram, the area of the circle with the name A reflects the truth of the statement A (in set theory, the circle A is the designation of all the elements included in this 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). Shading sequentially: 1) the area outside the two intersecting circles, which corresponds to the values A = 0, B = 0; 2) the area related only to the 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 union of these three areas will be a graphical representation of the logical operation of implication.
4. Using Euler circles in proving logical equalities (laws)
In order to prove logical equalities, you can apply the method of Euler-Venn diagrams. Let us prove the following equality ¬ (AvB) = ¬A & ¬B (de Morgan's law).
For a visual representation of the left side of the equality, we will execute sequentially: shade both circles (apply disjunction) in gray, then to display the inversion, shade the area outside the circles in black:
Fig. 3 
Fig. 4 
For visual representation of the right-hand side of the equality, we will execute sequentially: shade the area for displaying the inversion (¬A) with gray color and, similarly, the area ¬B with gray color; then, to display the conjunction, you need to take the intersection of these gray areas (the result of the overlay is shown 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. Tasks in the GIA and USE format on the topic: "Information search on the Internet"
Problem number 18 from the demo version of the GIA 2013.
The table lists the requests 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 diminishing the number of pages that the search engine will find for each request.
| Code | Inquiry |
| A | (Fly & Money) | Samovar |
| B | Fly & Money & Bazaar & Samovar |
| V | Fly | Money | Samovar |
| G | Fly & Money & Samovar |
For each request, let's build an Euler-Venn diagram:
| Request A | Request B
|
Request B
|
Request D
|
Answer: VGB.
Problem B12 from the demo version of the Unified State Exam-2013.
The table shows the requests and the number of pages found on them for a certain segment of the Internet.
| Inquiry | Pages found (in thousands) |
| Frigate | Destroyer | 3400 |
| Frigate & Destroyer | 900 |
| Frigate | 2100 |
How many pages (in thousands) will be found by request Destroyer?
It is assumed that all queries were executed almost simultaneously, so that the set of pages containing all the search words did not change during the execution of the queries.
Ф - number of pages (in thousands) upon request Frigate;
E - the number of pages (in thousands) on request Destroyer;
X is the number of pages (in thousands) for the request that mentions Frigate and not mentioned Destroyer;
Y - the number of pages (in thousands) for the request that mentions Destroyer and not mentioned Frigate.
Let's build Euler-Venn diagrams for each request:
| Inquiry | 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. Hence we find Y = 3400-2100 = 1300.
- E = 900 + Y = 900 + 1300 = 2200.
Answer: 2200.
6. Solution of logical meaningful problems by the method of Euler-Venn diagrams
There are 36 people in the class. Pupils of this class attend the math, physics and chemistry circles, and the math circle is attended by 18 people, the physical - 14 people, the chemistry - 10. In addition, it is known that 2 people attend all three circles, 8 people - 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, the use of Euler circles is very convenient and illustrative.
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 ( NS) circles.
Let be MFH- a lot of children, each of whom attends all three circles. МФ¬Х- a lot of children, each of whom attends math and physics circles and not attends chemical. ¬M¬FH- a lot of children, each of whom attends the chemistry circle and does not attend the physics and mathematics circles.
Similarly, we introduce the sets: ¬MFH, M¬FH, M¬F¬H, ¬MF¬H, ¬M¬F¬H.
It is known that all three circles are attended by 2 people, therefore, in the region MFH write the number 2. Since 8 people attend both mathematical and physics circles, and among them there are already 2 people who attend all three circles, then to the region МФ¬Х we will enter 6 people (8-2). Similarly, we will 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 the circles.
This means that 36-28 = 8 students do not attend circles.
After winter break classroom 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 not been to the cinema. neither in the theater nor in the circus. The cinema was visited by 25 people, in the theater - 11, in the circus - 17 people; both in the cinema and in the theater - 6; both in the cinema and in the circus - 10; and in the theater and in the circus - 4.
How many people have visited the cinema, the theater, and the circus?
Let x be the number of children who have been to the cinema, to the theater, and to the circus.
Then you can build the following diagram and count the number of children in each area:
|
|
6 people visited the cinema and theater, which means that only cinema and theater (6) people. Similarly, only in the cinema and circus (10) people. Only in the theater and circus (4) people. 25 people visited the cinema, which means that only 25 of them were in the cinema - (10th) - (6th) - x = (9 + x). Similarly, only in the theater there were (1 + x) people. There were (3 + x) people only in the circus. Have not been to the theater, cinema and circus - 2 people. This means that 36-2 = 34 people. visited the events. On the other hand, we can sum up the number of people who have been to the theater, cinema and circus: (9 + x) + (1 + x) + (3 + x) + (10-x) + (6-x) + (4-x) + 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 USE and GIA format and in solving meaningful logical tasks.
Literature
- V.Yu. Lyskova, E.A. Rakitin. Logic in computer science. M .: Informatics and Education, 2006.155 p.
- L.L. Bosov. Arithmetic and logical foundations of computers. Moscow: Informatics and Education, 2000.207 p.
- L.L. Bosova, A. Yu. Bosov. Textbook. Informatics and ICT for Grade 8: BINOM. Knowledge Laboratory, 2012.220 p.
- L.L. Bosova, A. Yu. Bosov. Textbook. Informatics and ICT for Grade 9: BINOM. Knowledge Laboratory, 2012.244 p.
- FIPI website: http://www.fipi.ru/
Euler-Venn diagrams are geometric representations of sets. The construction of a diagram consists in the image of a large rectangle representing the universal set U, and inside it - circles (or some other closed figures) representing the sets.
The shapes should intersect in the most general way required by the problem and should be marked accordingly. Points lying within different areas of the diagram can be considered as elements of the corresponding sets. Having constructed a diagram, it is possible to shade certain areas to denote the newly formed sets.
Operations on sets are considered to obtain new sets from existing ones.
Definition. The union of sets A and B is a set consisting of all those elements that belong to at least one of the sets A, B (Fig. 1):
Definition. The intersection of sets A and B is a set consisting of all those and only those elements that belong simultaneously to both the set A and the set B (Fig. 2):
Definition.
The difference of sets A and B is the set of all those and only those elements of A that are not contained in B (Fig. 3):
Definition. The symmetric difference of the sets A and B is the set of elements of these sets that belong either only to the set A, or only to the set B (Fig. 4):
Definition. The absolute complement of the set A is the set of all those elements that do not belong to the set A (Fig. 5):
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Leonard Euler - the greatest of mathematicians, wrote over 850 scientific papers.In one of them, these circles appeared.
The scientist wrote that"They are very suitable for making our thinking easier."
Euler circles Is a geometric scheme that helps to find and / or make logical connections between phenomena and concepts more visible. It also helps to depict the relationship between any set and its part.
Problem 1Of the 90 tourists going on a trip, 30 people speak German, 28 people speak English, and 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 do not speak any language?
Solution:
Let us show the condition of the problem graphically - using three circles
Answer: 10 people.
Task 2
Many guys in our class love football, basketball and volleyball. And some even 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 volleyball and football can be played by 3 people, football and basketball - 4, volleyball and basketball - 2. One person from the class can play all the games, 7 do not know how to play any game. It is required to find:
How many people are there in the class?
How many people know how to play football?
How many people can play volleyball?
Problem 3
70 children rested in the children's camp. Of these, 20 are engaged in the drama club, 32 sing in the choir, 22 are fond of sports. There are 10 children from the choir 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 guys do not sing in the choir, do not play sports and do not go in for a drama club? How many guys are engaged only in sports?
Problem 4
Of the company's employees, 16 have visited France, 10 - in Italy, 6 - in 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, and each of them has visited at least one of the named countries?
Problem 5
Sixth graders filled out a questionnaire with questions 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. And three of them also like "The Wolf and the Calf", six - "SpongeBob SquarePants", and one child equally loves all three cartoons. The Wolf and the Calf has 13 fans, five of whom named two cartoons in their profile. We need to determine how many sixth graders like SpongeBob SquarePants.
Tasks for students to solve
1. There are 35 students in the class. All of them are readers of the school and district libraries. Of these, 25 borrow books from the school library, 20 - from the district library. How many of them are:
a) are not readers of the school library;
b) are not readers of the district library;
c) are readers of the school library only;
d) are readers of the district library only;
e) are readers of both libraries?
2.Each student in the class learns English or German, or both. English 25 people study, German - 27 people, and he and the other - 18 people. How many students are there in the class?
3. A circle with an area of 78 cm2 and a square with an area of 55 cm2 were drawn on a sheet of paper. The intersection area of a circle and a square is 30 cm2. The part of the sheet that is not occupied by a circle and a square has an area of 150 cm2. Find the area of the leaf.
4. There are 25 people in a group of tourists. Among them, 20 people are under 30 years old and 15 people are over 20 years old. Could it be so? If so, in what case?
5.In kindergarten 52 children. Each of them loves cake or ice cream, or both. Half of the children love cake and 20 people love cake and ice cream. How many kids love ice cream?
6. There are 36 people in the class. Pupils of this class attend mathematics, physics and chemistry circles, and the mathematical circle is attended by 18 people, the physical - 14, the chemistry - 10. In addition, it is known that 2 people attend all three circles, 8 people - both mathematical and physical, 5 - both mathematical and chemical, 3 - both physics and chemistry 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 were neither in the cinema, nor in the theater, nor in the circus. The cinema was attended by 25 people; in the theater - 11; in the circus - 17; both in the cinema and in the theater - 6; both in the cinema and in the circus - 10; and in the theater, and in the circus - 4. How many people have been to the theater, cinema and circus at the same time?
Solution tasks of the exam using Euler circles
Problem 1
In the search engine query language, the symbol "|" is used to denote the logical operation "OR", and the symbol "&" for the logical operation "AND".
Cruiser & Battleship? It is assumed that all questions are executed almost simultaneously, so that the set of pages containing all the search words did not change during the execution of the queries.
| Inquiry | Pages found (in thousands) |
| Cruiser | Battleship | 7000 |
| Cruiser | 4800 |
| Battleship | 4500 |
Solution:
With the help of Euler's circles, we represent the conditions of the problem. In this case, the numbers 1, 2 and 3 are used to indicate the resulting areas.
Based on the conditions of the problem, we compose 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) in equation (1) and find out that:
4800 + 3 = 7000, whence we get 3 = 2200.
Now we can substitute this result into equation (3) and find out that:
2 + 2200 = 4500, whence 2 = 2300.
Answer: 2300 - the number of pages found by requestCruiser & Battleship.
Task 2In the search engine query language, to denote
| Inquiry | Pages found (in thousands) |
| Cakes | Pies | 12000 |
| Cakes & Pies | 6500 |
| Pies | 7700 |
How many pages (in thousands) will be found by request Cakes?
Solution 
To solve the problem, let's display the sets of Cakes and Pies in the form of Euler circles.
A B C ).
It follows from the condition of the problem:
Cakes │Pies = A + B + B = 12000
Cakes & Pies = B = 6500
Pies = B + B = 7700
To find the number of Cakes (Cakes = A + B ), you need to find a sector A Cakes│Pies ) we subtract the set Pies.
Cakes│Pies - Pies = A + B + C - (B + C) = A = 1200 - 7700 = 4300
Sector A is equal to 4300, therefore
Cakes = A + B = 4300 + 6500 = 10800
Problem 3
| ", and for the logical" AND "operation - the symbol" & ".
| Inquiry | Pages found (in thousands) |
| Cake & Bakery | 5100 |
| Cake | 9700 |
| Cake | Bakery | 14200 |
How many pages (in thousands) will be found by request Bakery?
It is assumed that all queries were executed almost simultaneously, so that the set of pages containing all the search words did not change during the execution of the queries.
Solution
To solve the problem, we display the sets Cake and Pastries in the form of Euler circles.
Let's designate each sector with a separate letter ( A B C ).
It follows from the condition of the problem:
Cake & Pastry = B = 5100
Cake = A + B = 9700
Cake │ Baking = A + B + B = 14200
To find the amount of Baking (Baking = B + C ), you need to find a sector V , for this, from the general set ( Cake │ Baking) subtract the set Cake.
Pastry │ Baking - Pastry = A + B + C - (A + B) = B = 14200-9700 = 4500Sector B is equal to 4500, therefore Baking = B + B = 4500 + 5100 = 9600
Problem 4diminishing
To denotelogical operation "OR" uses the symbol "| ", and for the logical" AND "operation - the symbol" & ".
Solution
We represent the sets of shepherd dogs, terriers and spaniels in the form of Euler circles, denote the sectors by letters (
A B C D ).with paniels │ (terriers & shepherds) = D + B
with paniels= 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 lists the requests to the search server. Arrange the request numbers in order increases the number of pages that the search engine will find for each request.
To denotelogical operation "OR" uses the symbol "| ", and for the logical" AND "operation - the symbol" & ".
| 1 | baroque | classicism | empire |
| 2 | baroque | (classicism & empire) |
| 3 | classicism & empire |
| 4 | baroque | classicism |
Solution
We represent the sets of classicism, empire and classicism in the form of Euler circles, denote the sectors by letters ( A B C D ).
Let's transform the condition of the problem as a sum of sectors:
baroque classicism ampir = A + B + C + Dbaroque │ (classicism & empire) = D + B
classicism & empire = B
baroqueclassicism = G + B + A
From the sums of sectors, we see which request issued more quantity pages.
Let's arrange the request numbers in ascending order of the number of pages:3 2 4 1
Problem 6
The table lists the requests to the search server. Arrange the request numbers in order increases the number of pages that the search engine will find for each request.
To denotelogical operation "OR" uses the symbol "| ", and for the logical" AND "operation - the symbol" & ".
| 1 | canaries | goldfinches | content
|
| 2 | canaries & content |
| 3 | canaries & goldfinches & content |
| 4 | breeding & keeping & canaries & goldfinches |
Solution
To solve the problem, we represent the queries in the form of Euler circles.
K - canaries,
Щ - goldfinches,
R - breeding.
| canaries | terriers | content | canaries & content | canaries & goldfinches & content | breeding & keeping & canaries & goldfinches |
 |  |  |  |
The first query has the largest area of filled sectors, then the second, then the third, and the fourth query has the smallest.
In ascending order by the number of pages, requests will be presented in the following order: 4 3 2 1
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 (USE 2013)
In the search engine query language, the symbol "|" is used to denote the logical operation "OR", and the symbol "&" for the logical operation "AND".
The table shows the requests and the number of pages found on them for a certain segment of the Internet.
| Inquiry | Pages found (in thousands) |
|---|---|
| Frigate | Destroyer | 3400 |
| Frigate & Destroyer | 900 |
| Frigate | 2100 |
How many pages (in thousands) will be found by request Destroyer?
It is assumed that all queries were executed almost simultaneously, so that the set of pages containing all the search words did not change during the execution of the queries.
Euler's circles, in fact, are quite common in our life. Even in elementary school, students begin to work with schematic figures that clearly explain the relationship between objects and concepts.
Description of the Euler circle scheme
Euler's circles are geometric constructions used to simplify the perception of logical connections between objects, concepts and phenomena.
They are divided into groups, depending on the type of relationship between sets:
- equivalent (Fig. 1);
- intersecting (Fig. 2);
- subordinates (Fig. 3);
- subordinate (Fig. 4);
- contradicting (fig. 5);
- opposite (fig. 6).
A typical example of such a diagram:
The largest set, marked in green, represents all variations of toys.
Constructors are one of the options for toys. They are highlighted with a blue oval. Constructors are a separate set, and, at the same time, part of the set of "Toys".
Clockwork toys are also part of the Toys set, but not part of the Constructors set. Therefore, they stand out with a purple oval. But the set of "Clockwork Cars" is independent, but at the same time, it is a subset of "Clockwork Toys".
The method was developed by the famous Swiss and Russian mathematician Leonard Euler.
Using this method, the scientist solved the most difficult math problems... The use of simple shapes made it possible to reduce the solution to any, even the most difficult task, to symbolic logic - the maximum simplification of reasoning.
Later, this way was modified by the Englishman John Wenn, who introduced the concept of the intersection of several sets.
The technique is very easy to use - Euler circles for preschoolers from 4-5 years old begin to teach already in kindergarten. At the same time, it is so convenient that it is used even in the highest academic environment.
Applying Euler circles
The main purpose of using diagrams is the practical solution of problems of union or intersection of sets.
Fields of application: mathematics, logic, management, statistics, computer science, etc. In fact, there are much more of them, but it is simply impossible to list them all.
There are two types of charts.
The first describes the unification of concepts, the nesting of one into another. An example is given in the article above.
The second describes the intersection of two different sets by some common features... One example
Examples of tasks and solutions
Consider the problems in which Euler's circles help to understand, examples of solving problems in logic and mathematics.
Tasks for preschoolers
First in line: Euler's circles for preschoolers, tasks with answers to which will help you understand how kids first get acquainted with the method of simplifying complex mathematical and logical problems.
Task number 1 - initial level.
Purpose: to teach a child to identify an object that is most consistent with two properties at the same time.
Correct answer: Rubik's cube.
Task number 2
Correct answer: frog.
Task number 3
Correct answer: pear.
Task number 4 - intermediate level.
The tasks are complicated by the fact that more sets are used.
Correct answer: the sun.
Task number 5
Correct answer: dress.
Task number 6
Correct answer: helpful.
Assignments for schoolchildren
The following logical problems with answers, in which Euler's circles are the basis for the solution, concern junior schoolchildren... Such tasks teach children to disassemble logical intersections according to certain criteria.
Task number 1
35 students are registered in school or city libraries. Of these, 25 regularly visit school library, and 20 - urban.
How many students:
- Visiting both libraries?
- Don't go to the city library?
- Don't go to the school library?
- Do they only go to the city library?
- Do they only go to the school library?
Answer:
- Let's determine the number of visitors to two libraries - the common part on the diagram:
(25 + 20) – 35 = 10.
- Students not attending the city library:
35 - 20 = 15 - the left sector of the blue zone.
- Students not attending the school library:
35 - 25 = 10 - the right sector is purple.
- City library visitors only:
35 - 25 = 10 - also, the right sector is purple.
- School Library Visitors Only:
35 - 20 = 15 - also, the left sector is blue.
Task number 2 - is also intended for elementary grades, but is more difficult.
There are 38 students in 7-A. Students are attracted to different sports games: 16 - basketball, 17 - hockey, 18 - football. At the same time, 4 people love basketball and hockey, 3 people love basketball and football, 5 people love hockey and football, and 3 students are not interested in sports.
- Are there any students who are fond of all sports games?
- How many students are interested in just one of the sports games?
Answer:
All students in the class are the largest circle.
Circle "B" - basketball players, "X" - hockey players, "F" - football players, "Z" - universal athletes. Three unsportsmanlike students are simply in a common circle.
Basketball players included in the set "B", but not included in the intersection zones with the sets "X" and "F".
16 - (4 + Z + 3) = 9 - Z.
By analogy, we find the number of hockey players.
17 - (4 + Z + 5) = 8 - Z.
Football players.
18 - (3 + Z + 5) = 10 - Z.
To limit the value of Z, you need to sum the sets of students.
3 + (9 - Z) + (8 - Z) + (10 - Z) + 3 + 4 + 5 + Z = 38;
42 - 2 * Z = 38;
Accordingly, B = 7, F = 8, X = 6.
The use of pie charts allows you to clearly demonstrate all relationships different groups students.
Method schematic image the relationship of sets is not just a fascinating thing. Euler's circles, examples of solving problems whose logic is not obvious, show that the method can be used not only to unleash mathematical tasks, but also to find a way out of everyday situations.
