Graphic representation of concepts using Euler circles. Logical problems and Euler circles. Compatible and incompatible concepts

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

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 "&".

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:

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.

Euler circles- one of the simplest themes that you need for admission to the 5th grade of physics and mathematics lyceums. In fact, Euler circles is nothing more than a graphical representation of sets. Objects with a certain property are located inside Euler-Venn circle those who do not possess are outside. Of course, usually the diagram contains not one circle, but several, each of which combines objects with some kind of property. Any task from this block boils down to the fact that it is necessary to count the number of elements in any area. Let's look at examples of what needs to be done:

Tasks for many people

There are students in the class. study English, German and French. People don't know any language. It is also known that of all the children, only one boy studies languages: English and French. How many people study a language?

To solve the problem, let's denote the number of required students as (those who study the language). The number of students studying a different number of languages ​​can be expressed through and the conditions in the problem. Euler-Venn diagram in this case it will look like this: For example, guys who know only English are indicated in red and their number.

Note that we have not used the total number of students in any way - this condition will generate the very equation with which the problem will be solved:





It turns out that all languages ​​are studied by humans (Now, knowing , you can independently reconstruct how many students were in the class and check the answer)

Divisibility problems (complex divisibility)

These are tasks of increased complexity. We recommend that you study the topic first. A must-read only for those who are planning to win prizes.

For how many numbers between and is the following statement true: the number is divisible by or not divisible by?

Such a terrible and incomprehensible condition becomes simple if you use Euler circles. It is clear that in this problem we consider numbers that - we are interested in those inside the corresponding circle. There are also numbers that vdots 12 - we are interested in the numbers that are outside. But what about the numbers that belong to both sets? Firstly, what common property do they have, and secondly, are they of interest to us?

Let's answer the first question first. It turns out that if a number is simultaneously divisible by two other numbers, then it is divisible by Least Common Multiple these two numbers, that is, by the minimum number that is divisible without a remainder by both of them. For numbers and LCM there is nothing more than the number , since and , and there is no smaller number with such properties. In total, at the intersection of our sets there are numbers that .

Next, it should be noted that the word is used in the condition "OR". This means that for the required numbers, AT LEAST ONE of the proposed statements must be true (possibly both). That is, we are suitable for numbers that are inside the circle of numbers, which are, as well as all the numbers that are outside the circle.

So, Euler-Venn diagram looks like this: The shading indicates the numbers that need to be found. Now, I hope, it is obvious that we need to find how many numbers there are in the problem under consideration, from this quantity subtract the number of numbers that and add the number of numbers that .

So let's get started:

So, let's summarize. If you are going enter the 5th grade of the physics and mathematics lyceum, then general knowledge of Euler-Venn circles You need it. The main area of ​​application is problems where there are sets of objects that have certain properties, and it is necessary to find the number of objects that have (or do not have) a set of specified properties.

Sections: Computer science

1. Introduction

In the course of Computer Science and ICT of the basic and senior school, such important topics as “Fundamentals of Logic” and “Searching for Information on the Internet” are discussed. 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).

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 Annex 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.

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

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.

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:

According to the diagrams we have:

1. X + 900 + Y = F + Y = 2100 + Y = 3400. From here we find Y = 3400-2100 = 1300.
2. 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 the winter holidays, the class teacher asked which of the children 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:

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

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

Each object or phenomenon has certain properties (signs).

It turns out that forming a concept about an object means, first of all, the ability to distinguish it from other objects similar to it.

We can say that a concept is the mental content of a word.

Concept - it is a form of thought that displays objects in their most general and essential characteristics.

A concept is a form of thought, and not a form of a word, since a word is only a label with which we mark this or that thought.

Words can be different, but still mean the same concept. In Russian - “pencil”, in English - “pencil”, in German - bleistift. The same thought has different verbal expressions in different languages.

RELATIONS BETWEEN CONCEPTS. EULER CIRCLES.

Concepts that have common features in their content are called COMPARABLE(“lawyer” and “deputy”; “student” and “athlete”).

Otherwise, the concepts are considered INCOMPARABLE(“crocodile” and “notebook”; “man” and “steamboat”).

If, in addition to 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).

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)