Each object or phenomenon has certain properties (signs).

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

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

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

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

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

RELATIONS BETWEEN CONCEPTS. EULER CIRCLES.

Concepts that have in their content general signs, 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)

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Leonard Euler Leonard Euler, the greatest mathematician of the 18th century, was born in Switzerland. In 1727 by invitation St. Petersburg Academy sciences, he came to Russia. Euler found himself in the circle of outstanding mathematicians and received great opportunities to create and publish his works. He worked with passion and soon became, according to the unanimous recognition of his contemporaries, the first mathematician in the world. One of the first to use circles to solve problems was the outstanding German mathematician and philosopher Gottfried Wilhelm Leibniz (1646 - 1716). In his rough sketches, drawings with circles were found. This method was then thoroughly developed by the Swiss mathematician Leonhard Euler (1707 – 1783). (1707-1783)

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From 1761 to 1768, he wrote the famous “Letters to a German Princess,” where Euler talked about his method, about depicting sets in the form of circles. That is why drawings in the form of circles are usually called “Eulerian circles”. Euler noted that the representation of sets as circles “is very suitable to facilitate our reasoning.” It is clear that the word “circle” here is very conditional; sets can be depicted on a plane in the form of arbitrary figures.

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After Euler, the same method was developed by the Czech mathematician Bernard Bolzano (1781 - 1848). Only, unlike Euler, he drew not circular, but rectangular diagrams. Euler's circle method was also used by the German mathematician Ernst Schroeder (1841 – 1902). This method is widely used in his book Algebra Logic. But graphical methods reached their greatest flowering in the works of the English logician John Venn (1843 - 1923). He outlined this method most fully in his book “Symbolic Logic,” published in London in 1881. In honor of Venn, instead of Euler circles, the corresponding drawings are sometimes called Venn diagrams; in some books they are also called Euler–Venn diagrams (or circles).

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Euler depicted the set of all real numbers using these circles: N-set natural numbers, Z – set of integers, Q – set rational numbers, R is the set of all real numbers. Well, how do Euler circles help in solving problems? R Q Z N

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Euler circles This is a new type of problem in which you need to find some intersection of sets or their union, observing the conditions of the problem.

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EULER circles are a geometric diagram with which you can depict the relationships between subsets for a visual representation.

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Solving the problems "Inhabited Island" and "Hipsters" Some guys from our class like to go to the movies. It is known that 15 children watched the film “Inhabited Island”, 11 people watched the film “Hipsters”, of which 6 watched both “Inhabited Island” and “Hipsters”. How many people have only watched the movie “Hipsters”?

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Solution We draw two sets in this way: we place 6 people who watched the films “Inhabited Island” and “Hipsters” at the intersection of the sets. 15 – 6 = 9 – people who watched only “Inhabited Island”. 11 – 6 = 5 – people who watched only “Hipsters”. We get: Answer. 5 people watched only “Hipsters”. 6 “inhabited island” “Hipsters” “inhabited island” “Hipsters” 9 6 5

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“World of Music” 35 customers came to the “World of Music” store. Of these, 20 people bought the singer Maxim’s new disc, 11 bought Zemfira’s disc, 10 people did not buy a single disc. How many people bought CDs of both Maxim and Zemfira? Solution Let us represent these sets on Euler circles.

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Now let’s count: In total, there are 35 buyers inside the large circle, and 35–10 = 25 buyers inside the two smaller ones. According to the conditions of the problem, 20 buyers bought the new CD of the singer Maxim, therefore, 25 – 20 = 5 buyers bought only Zemfira’s CD. And the problem says that 11 buyers bought Zemfira’s disc, which means 11 – 5 = 6 buyers bought both Maxim’s and Zemfira’s discs: Answer: 6 buyers bought both Maxim’s and Zemfira’s discs.

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Consideration of the simplest cases of Euler–Venn circles a) Let a certain set be given and property A indicated. Obviously, the elements of this set may or may not have this property. Therefore, this set splits into two parts, which can be denoted by A and A*. This can be depicted in two ways in the figure. The large circle represents the given set, the small circle A represents that part of the elements of the given set that has property A, and the ring-shaped part A* represents that part of the elements that do not have property A.

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b) Let a certain set be given and two properties indicated: A, B. Since the elements of a given set may or may not have each of these properties, then four cases are possible: AB, AB*, A*B, A*B*. Consequently, this set splits into 4 subsets. This can also be depicted in two ways: in the form of circles or diagrams. In the first figure, circle A is a subset of those elements of a given set that have property A, and the area outside the circle, i.e. area A* is a subset of those elements that do not possess property A. Similarly, circle B and the area outside it. In the second figure, subsets A, A*, B*, B are depicted differently: subset A is the area to the left of the vertical line, and subset A* is the area to the right of this line. B and B* are depicted similarly: area B is the upper semicircle, and area B* is the lower semicircle.

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c) Let a certain set be given and three properties indicated: A, B, C. In this case, this set is divided into eight parts. This can be depicted in two ways.

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Problems solved using Euler circles Problem No. 1. How many natural numbers from the first ten are not divisible by either 2 or 3? Solution. To solve the problem, it is convenient to use Euler circles. In our case, there are three circles: the large circle is a set of numbers from 1 to 10, inside the large circle there are two smaller circles intersecting with each other. Let the set of numbers that are multiples of 2 be set A, and the set of numbers that are multiples of 3 be set B. Let’s reason. Every second number is divisible by 2. This means that there will be 10:2=5 such numbers. 3 is divisible by 3 numbers (10:3). Those numbers that are divisible by 6 are divisible by 2 and 3. There is only one such number. Therefore, set A consists of 5-1=4 numbers, set B – 3-1=2 numbers. It follows that the first ten contains 10-(4+1+2)=3 numbers.

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Problem No. 2. Problem solved using the Euler–Venn diagram. The guys were tasked with making cubes. Several cubes were made from cardboard, and the rest from wood. The cubes came in two sizes: large and small. Some of them were painted green, others red. This made 16 green cubes. There were 6 large green cubes. There were 4 large green cardboard cubes. There were 8 red cardboard cubes. There were 9 red wooden cubes. There were 7 large wooden cubes, and 11 small wooden cubes. How many cubes were there in total? Solution. Let's do the drawing.

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Preparation of problems of practical importance. Problem 1. There are 35 students in the class. 12 of them are in the math club, 9 are in the biology club, and 16 kids do not attend these clubs. How many biologists are interested in mathematics? Solution: We see that 19 children attend clubs, since 35 - 16 = 19, of which 10 people attend only a math club (19-9 = 10) and 2 biologists (12-10 = 2) are interested in mathematics. Answer: 2 biologists. With the help of Euler circles it is easy to see another way to solve the problem. Let's depict the number of students using a large circle, and place smaller circles inside. Obviously, in the general part of the circles there will be the very biologists-mathematicians about whom the problem asks. Now let's count: Inside the large circle there are 35 students, inside circles M and B: 35-16 = 19 students, inside circle M - 12 guys, which means that in that part of circle B, which has nothing to do with circle M, there are 19-12 =7 students, therefore, there are 2 students in the MB (9-7=2). Thus, 2 biologists are interested in mathematics. 1)35-16=19(persons); 2) 12+9=21 (persons); 3)21-19=2(persons). Answer: 2 biologists.

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Fill out the diagram. 1) We must start with the subset for which three properties are indicated. These are large green cubes made of cardboard - there are 4 such cubes. 2) Next, we look for a subset for which two of the listed three properties are indicated. These are large green cubes - 6. But this subset consists of cardboard and wood. There were 4 cardboard ones. So, 6-4 = 2 wooden ones. 3) There are 7 large wooden cubes. Of these, 2 are green. This means that there will be 7-2=5 red ones. 4) 9 red wooden cubes, 5 of which are large. This means that there will be 9-5=4 small red wooden cubes. 5) There are 11 small wooden cubes. Of these, 4 are red. So, there are 11-4 = 7 small green wooden cubes. 6) Total green cubes are 16. Green cubes are placed in a ring-shaped part consisting of four parts. This means there are 16 small green cardboard cubes - (4+2+7) = 3. 7) The last condition remains: there were 8 red cardboard cubes. We don’t need to know how many of them are small and how many are large. 8) We count: 2+5+8+4+4+7+3=33. Answer: A total of 33 cubes were made.

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"Mathematical Encyclopedia". To prepare this work, materials were used from the site http://minisoft.net.ru/ http://logika.vobrazovanie.ru/index.php?link=kr_e.html http://reshizadachu.ucoz.ru/index/ krugi_ehjlera/0-18

Logics. Tutorial Gusev Dmitry Alekseevich

1.6. Euler circle diagrams

1.6. Euler circle diagrams

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

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

At the beginning it was said that concepts can be comparable and incomparable. It is believed that the six options of relations considered are applicable only to comparable concepts. However, it is possible to assert that all incomparable concepts are related to each other in a relationship of subordination. For example, such incomparable concepts as penguin And heavenly body can be considered as subordinate, because a penguin is not a celestial body and vice versa, but at the same time the scope of concepts penguin And heavenly body are included in the broader scope of a third concept, generic in relation to them: this may be the concept object of the surrounding world or form of matter(after all, both the penguin and the celestial body are different objects of the surrounding world or various shapes 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 big amount concepts. For example, relationships between concepts boxer, black And Human

Mutual arrangement 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 relation consistent submission: 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 being 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.