Venna charts - a graphical way to task and analyze logical and mathematical theories and their formulas. It is constructed by splitting a part of the plane on cells (subsets) with closed contours (Jordan curves). In the cells, information characterizing the theory or formula is presented. The purpose of building diagrams is not only illustrative, but also operator - algorithmic processing of information. The venna chart apparatus is usually used with analytical.

The method of partitioning, the number of cells, as well as the problems of recordings in them depend on the theory under consideration, which can also be entered (described) graphically - some of the venna charts, asked initially, in particular, together with the algorithms of their transformations, when some diagrams can act as operators acting on other diagrams. For example, in the case of classical logic statements For formulas composed of the n wide propositional variables, part of the plane (university) is divided into 2 "cells corresponding to constituents (in conjunctive or in a disjunctive form). The venna diagram of each formula is considered to be such a plane in the cells of which is set (or not) asterisk *. So, formula

(¬ A & ¬ B & C) V (A & ¬ B & C) V (¬ A & B & ¬ C)

with three propositional variables A, B and C, the diagram shown in the figure, where the stars in the cells correspond to the conjunctive components of this perfect normal disjunctive formula. If there are no cells marked with asterisks, then the venna diagram is compared, for example, the definitely false formula, say (A & ¬ A).

The inductive way to split the plane on 2 "cells dates back to the writings of the English logic of J. Venn, is called the way of Venna and is as follows:

1. With n \u003d 1, 2, the circles are apparently used obviously. (On the shown figure n \u003d 3.)

2. Suppose that at n \u003d k (k ≥ 3), such a position is indicated to the figures that the plane is divided into 2K cells.

Then, for the location of K + 1, the shapes on this plane are enough, firstly, choose an impaired curve (Wed without self-intersection points, i.e. an unlocked curve of Jordan, belonging to the boundaries of all 2K cells and having only one common piece with each of these borders. Secondly, obli φ Closed curve Jordan Ψ k + 1 so that the curve Ψ K + 1 passed through all 2K cells and crossed the border of each cell only twice. Thus, the location is n \u003d k + 1 figures such that the plane is divided into 2K + 1 cells.

For the presentation of other logic and mathematical theories, the method of Vennovsky diagrams expands. The theory itself is written so as to highlight its language elements in a graphic image suitable. For example, atomic formulas of the classical predicate logic are written as the words of the type P (y1..yr), where P is predicate, and Y1, ..., Yr - subject variables, not necessarily different; The word y1, ..., yr is an object infix. The obvious theoretical-multiple nature of Venna diagrams allows to represent and explore with their help, in particular, theoretical and multiple calculus, for example, the calculus of ZF the theory of cummer-frenkel sets. Graphic methods in logic and mathematics have developed long. Such, in particular, the logical square, the circles of Euler and the original diagrams of L. Carroll. However, the method of Venna diagrams is significantly different from the well-known method of Euler's circles used in traditional syllogistics. At the heart of Vennovsky diagrams lies the idea of \u200b\u200bthe decomposition of the Boolean function to Constituent - the central in the algebra of logic, which causes their operational character. The Venne diagrams applied primarily to solve the tasks of the logic of classes. Its diagrams can be effectively used to solve problems of logic of statements and predicates, review of the consequences from the parcels, solving logical equations, as well as other issues, up to the problem of solvability. The venna diagram apparatus is applied in the applements of mathematical logic and the theory of automata, in particular when solving problems associated with neural circuits and the problem of synthesizing reliable schemes from relatively little reliable elements.

A. S. Kuzichev

New philosophical encyclopedia. In four volumes. / In-t philosophy RAS. Scientifies. Tip: VS Stepin, A.A. Huseynov, G.Yu. Semigin. M., Thought, 2010, t. I, a - d, p. 645.

Literature:

Venn J. Symbolic Logic. L., 1881. ED. 2, Rev. L., 1894;

Kuzichev A. S. Venna Charts. History and applications. M., 1968;

He is Solving some mathematical logic tasks using Venna charts. - In the book: Study of logical systems. M., 1970.

Euler-Venna chart - Visual means to work with sets. On these diagrams, all possible variants of the intersection of sets are depicted. The number of intersections (regions) n is determined by the formula:

n \u003d 2 n,

where n is the number of sets.

Thus, if two sets are used in the problem, then n \u003d 2 2 \u003d 4, if three sets, then n \u003d 2 3 \u003d 8, if four sets, then n \u003d 2 4 \u003d 16. Therefore, the Euler-Venn diagrams are used mainly for two or three sets.

Sets are depicted in the form of circles (if 2-3 sets are used) and ellipses (if 4 sets are used) placed in a rectangle (universo).

Universal set (Universum) u (In the context of the problem) - a set containing all the elements of the problem under consideration: the elements of all sets of tasks and elements that are not included in them.

Empty set Ø (In the context of the task) - a set that does not contain a single element of the problem under consideration.

In the diagram, intersecting sets are built, they enclose them in the union. Select areas whose number is equal to the number of intersections.

Euler-Venna charts are also used for visual representation of logical operations.

We will analyze examples of constructing Euler-Venna charts for two and three sets.

Example 1.

Universum U \u003d (0,1,2,3,4,5,6)

Euler-Venna charts for two sets A and B:

Example 2.

Let there be the following multiple numbers:

University service u \u003d (0,1,2,3,4,5,6,7)

Euler-Venna charts for three sets A, B, C:

We define the areas and the numbers that they belong:

Example 3.

Let there be the following multiple numbers:

A \u003d (0,1,2,3,4,5,6,7)

B \u003d (3,4,5,7,8,9,10,13)

C \u003d (0,2,3,7,8,10,11,12)

D \u003d (0,3,4,6,9,10,11,14)

Universum u \u003d (0,1,2,3,4,5,6,7,8,1,11,12,13,11,15)

Euler-Venna charts for four sets A, B, C, D:

We define the areas and the numbers that they belong:

If you want to determine typical tasks on the set, then go to the article.

If you think that you don't know anything about Euler's circles, you are mistaken. In fact, you probably have come across with them, just did not know how it is called. Where exactly? The circles in the form of Euler's circles have formed the basis of many popular Internet memes (submitted in the network of images on a specific topic).

Let's see together, what are these circles, why they are so called and why they are so convenient to use to solve many tasks.

The origin of the term

- This is a geometric scheme that helps find and / or make more visual logical links between phenomena and concepts. And also helps to portray relations between any set and its part.

So far not very clear, right? Look at this picture:

The picture shows the set - all possible toys. Some of the toys are designers - they are highlighted in a separate oval. This is part of a large set of "toys" and at the same time a separate set (because the designer can be "Lego", and primitive constructors from cubes for kids). Some part of a large set of "toys" can be plated toys. They are not designers, so we draw a separate oval for them. Yellow oval "Clockwork car" refers simultaneously to a set of "toys" and is part of a smaller set of "clockwork toy". Therefore, it is depicted inside both oval at once.

Well, so it became clearer? That is why the circles of Euler is the method that clearly demonstrates: it is better to see once than hearing a hundred times. His merit is that clarity simplifies reasoning and helps to get an answer faster and easier.

The author of the method is a scientist Leonard Euler (1707-1783). He also spoke about the schemes named after him: "Circles are suitable in order to alleviate our reflections." Euler is considered German, Swiss and even a Russian mathematician, mechanic and physicist. The fact is that he worked for many years at the St. Petersburg Academy of Sciences and made a significant contribution to the development of Russian science.

Before it, a German mathematician and philosopher Gottfried Labitz were guided by such a principle when building his conclusions.

The Euler method received well-deserved recognition and popularity. And after him, many scientists used him in their work, and also modified on their own way. For example, Czech Mathematics Bernard Bolzano used the same method, but with rectangular schemes.

The German mathematics Ernest Schroeder also made his contribution. But the main merit belongs to the Englishman John Venna. He was a specialist in logic and published a book "Symbolic Logic", in which it outlined in detail its version of the method (used mainly images of the intersections of sets).

Thanks to the contribution of the Venna, the method is even called the charts of Venna or another Euler-Venna.

Why do you need Euler's circles?

Euler's circles have an applied destination, that is, with their help, in practice, tasks are solved on the combination or intersection of sets in mathematics, logic, management and not only.

If we talk about the types of Euler's circles, then they can be divided into those that describe the union of some concepts (for example, the ratio of the genus and species) - we considered them on the example at the beginning of the article.

And also on those that describe the intersection of sets according to some sign. This principle was guided by John Venn in his schemes. And it is he who is the basis of many people popular on the Internet. Here is one example of such circles of Euler:

Funny, really? And most importantly, everything immediately becomes clear. You can spend a lot of words, explaining your point of view, but you can simply draw a simple scheme that immediately separates everything in places.

By the way, if you can not decide what profession to choose, try drawing a scheme in the form of circles of Euler. Perhaps the drawing like this will help you decide on the choice:

Those options that will be at the intersection of all three circles, and there is a profession that will not only be able to feed you, but I will like you.

Solving tasks using Euler's circles

Let's look at a few examples of tasks that can be solved using Euler's circles.

Here on this site - http://logika.vobrazovanie.ru/index.php?link\u003dkr_e.html Elena Sergeyevna Sazhenina offers interesting and simple tasks to solve the Euler method. Using logic and mathematics, we will analyze one of them.

Task about favorite cartoons

Six graders filled out the questionnaire with questions about their favorite cartoons. It turned out that most of them like "Snow White and Seven Dwarfs", "Sponge Bob Square Pants" and "Wolf and Calf". In class 38 students. Snow White and Seven Dwarfs Like 21 Pupils. Moreover, three among them also like "Wolf and Calf", six - "Sponge Bob Square Pants", and one child loves all three cartoon equally. The "wolf and calf" 13 fans, five of which were called two cartoons in the questionnaire. It is necessary to determine how many six-graders like "Sponge Bob Square Pants."

Decision:

Since, according to the terms of the task, we have three sets, blacks are trained. And since the answers of the guys come out that the sets intersect with each other, the drawing will look like this:

We remember that, according to the terms of the task among the fans of the cartoon "Wolf and Calf", five guys chose two cartoons at once:

It turns out that:

21 - 3 - 6 - 1 \u003d 11 - The guys chose only "Snow White and Seven Dwarfs".

13 - 3 - 1 - 2 \u003d 7 - the guys watch only the "wolf and calf".

It remains only to figure out how many six-graders two other options prefers the cartoon "Sponge Bob Square Pants". From the total number of students, we take all those who love two other cartoon or chose several options:

38 - (11 + 3 + 1 + 6 + 2 + 7) \u003d 8 - people watch only "Sponge Bob Square Pants".

Now we can safely fold all the numbers received and find out that:

the cartoon "Sponge Bob Square Pants" chose 8 + 2 + 1 + 6 \u003d 17 people. This is the answer to the question that delivered in the task.

And let's consider taskwhich in 2011 was put on a demonstration test exam on computer science and ICT (source - http://eileracrugi.narod.ru/index /0-6).

Conditions of the problem:

In the search server requests, the "|" symbol is used to indicate the logical operation "or", and the "&" symbol is used.

The table shows the requests and the number of pages found on them some segment of the Internet.

How many pages (in thousands) will be found on request Cruiser & Linor.?

It is believed that all questions are performed almost simultaneously, so that the set of pages containing all the sked words has not changed during the execution time of the requests.

Decision:

With the help of Euler circles, you will show the conditions of the task. At the same time, the numbers 1, 2 and 3 use to designate the areas obtained as a result.

Based on the conditions of the problem, to make an equation:

1. Cruiser | Linkor: 1 + 2 + 3 \u003d 7000
2. Cruiser: 1 + 2 \u003d 4800
3. Linkor: 2 + 3 \u003d 4500

To find Cruiser & Linor. (indicated in the drawing as region 2), we substitute equation (2) to equation (1) and find out that:

4800 + 3 \u003d 7000, from where we get 3 \u003d 2200.

Now we can substitute this result in equation (3) and find out that:

2 + 2200 \u003d 4500, from where 2 \u003d 2300.

Answer: 2300 - the number of pages found on request Cruiser & battle.

As you can see, Euler's circles help quickly and simply solve even enough complex or simply tangled for the task.

Conclusion

I suppose we managed to convince you that Euler's circles are not just an entertaining and interesting thing, but also a very useful method of solving problems. And not only abstract tasks for school lessons, but also quite life problems. Choosing a future profession, for example.

You will also probably be curious to know that in modern mass culture, Euler circles were reflected not only in the form of memes, but also in popular TV shows. Such as the "Theory of the Big Explosion" and "4).

Use this useful and visual method to solve problems. And be sure to tell about him friends and classmates. To do this, under the article there are special buttons.

the site, with full or partial copying of the material reference to the original source is required.

Federal Agency for Education

State Educational Institution of Higher Professional Education

National Research

Tomsk Polytechnic University

Institute of Natural Resources

Department of VM

ESSAY

Subject : « Euler-Venna chart»

Executor:

Student group 2,00

Leader:

Introduction ......................................................................... ......... ..3

1. From the story ...........................................................................................4

2. Chart of Euler-Venna .........................................................................4

3. Operations on sets of Euler-Venna chart ......................5

a) Association ...................................................................... 7

b) intersection, addition ................................................................... ..7

c) Pier Arrow, Shaffer Barcode and Difference ... ..............................8

d) difference .............................................................................. 8

e) symmetric difference and equivalence ......................... ...... .9

Conclusion ................................................................................. 10.

List of references ............................................................. .......... ..11

Introduction

Euler circles - a geometric scheme, with which you can depict the relationship between subsets for a visual representation. Circles were invented by Leonard Euler. Used in mathematics, logic, management and other applied directions.

An important private case of Euler circles - Euler chart - Venna, depicting all 2N combinations of n properties, that is, the final Boolean algebra. At n \u003d 3, the Euler diagram - Venna is usually depicted as three laps with centers in the tops of the equilateral triangle and the same radius, approximately equal length of the side of the triangle.

When solving a number of tasks, Leonard Euler used the idea of \u200b\u200ba set of sets using circles. However, this method also to Euler used an outstanding German philosopher and mathematician (1646-1716). Leibniz used them for the geometric interpretation of logical connections between concepts, but still preferred to use linear schemes.

But quite thoroughly developed this method L. Euler himself. The German mathematician Ernst Schröder (1841-1902) was used by Euler (1841-1902) in the Logic Algebra book. Special flourishing Graphic methods reached in the writings of the English logic of John Venna (1843-1923), in detail them in the book "Symbolic Logic", published in London in 1881. Therefore, such schemes are sometimes called the Euler charts - Venna.

1. Stories

Leonard Euler (1707 - 1783, St. Petersburg, Russian Empire) - Mathematics, mechanic, physicist. Adjunct on physiology, professor of physics, professor of higher mathematics, made a significant contribution to the development of mathematics, as well as mechanics, physics, astronomy and a number of applied sciences.

Euler is the author of more than 800 works on mathematical analysis, differential geometry, theory of numbers, approximate calculations, celestial mechanics, mathematical physics, optics, ballistics, shipbuilding, music theory, etc.

Almost half of him spent in Russia, where there was a significant contribution to the formation of Russian science. In 1726, he was invited to work in Vnct-Petersburg, where he was moving a year later. From 1711 to 1741, and from 1766 he was an academician of the St. Petersburg Academy of Sciences (in 1741-1766 he worked in Berlin, while remaining an honorary member of the St. Petersburg Academy at the same time). He knew Russian well and part of his writings (especially textbooks) published in Russian. The first Russian academicians-mathematics (S. K. Kotelnikov) and astronomers (S. Ya. Rumovsky) were Euler students. Some of his descendants still live in Russia.

John Venn. (1, English Logic. He worked in the field of classes logic, where he created a special graphics apparatus (the so-called Venna diagrams), which was found in the logical-mathematical theory of "formal neural networks". The venna belongs to the rationale for reverse operations in logical calculus J. Bul. John's interest was logic, and he published three works on this topic. These were the "logic of the case", in which the interpretation of frequency or frequency theory of probabilities is introduced in 1866; "Symbolic logic" with which Venna charts in 1881; "Principles Empirical logic "in 1889, in which the rationale for reverse operations in Boolean logic is given.

In mathematics, drawings in the form of circles depicting sets are used for a very long time. One of the first who used this method was an outstanding German mathematician and a philosopher (1 in its draft sketches were discovered drawings with such circles. Then this method was quite thorough and Leonard Euler. He worked for many years in the St. Petersburg Academy of Sciences. By this time belongs His famous "letters to the German princess", written in the period from 1761 to 1768. In some of these "letters ..." Euler just talks about his method. After Euler, the same method was developed by Czech Mathematics Bernard Bolzano (1T. Difference from Euler he painted not circular, but rectangular schemes. The German mathematician Ernest Schroeder also used the method of Euler's circles (1Tot. The method is widely used in the book "Logic algebra". But the greatest flourishing graphics methods reached in the writings of the English logic of John Venna (1C of the greatest completeness of this The method is set forth by him in the book "Symbolic Logic", published in London in 1881. In the honor of Venna, instead of the circles of Euler, the corresponding Figures are sometimes called Venna diagrams; In some books, they are also called diagrams (or circles) Euler-Venna.

2.Diagram Euler-Venna

The concepts of set and subsets are used in determining many concepts of mathematics and, in particular, when determining the geometric shape. We define as a universal set plane. Then you can give the following definition of a geometric shape in planimetry:

Geometric figureit is called any set of plane points. To clearly display the set and relationship between them, draw geometric shapes that are among themselves in this relationship. Such images of sets and are called Euler-Venna diagrams. Charts Euler-Venna make visual various statements regarding sets. The universal set is depicted in the form of a rectangle, and its subsets - circles. Used in mathematics, logic, management and other applied directions.

Euler-Venna charts in the image of a large rectangle representing a universal set U., and inside it - circles (or any other closed figures) representing the set. Figures should intersect in the most general case required in the task, and must be appropriately designated. Points underlying inside various areas of diagrams can be considered as elements of the corresponding sets. Having a constructed diagram, you can shave certain areas to designate newly formed sets.

Basic operations on sets:

Crossing Union difference

3. Operations over the sets of Euler-Venna chart

Operations on sets are considered to obtain new sets from existing ones.

Now in more detail on the examples.

Let a certain set of objects, which, after recalculation, it would be necessary to designate as

A \u003d (1, 2, 4, 6) and b \u003d (2, 3, 4, 8, 9)

round and white items. Can be called source set fundamental, and subsets a and b - just sets.

As a result, we obtain four classes of elements:

C.0 \u003d (5, 7, 10, 11) - elements do not possess any of the named properties,

C.1 \u003d (1, 6) - elements have only property A (round),

C.2 \u003d (3, 8, 9) - elements only have the property B (white),

C.3 \u003d (2, 4) - elements have simultaneously two properties A and B.

In fig. 1.1. These classes are depicted using chart Euler - Venna.

Fig. 1.1

Often the charts do not have the entire completeness of the generality, for example, that is shown in Fig. 1.2. It is already a set A fully included in B. For such a case, a special inclusion symbol is used (ì): a ì B \u003d (1, 2, 4) ì (1, 2, 3, 4, 6).

If two conditions are performed at the same time: a ì B and B ì A, then a \u003d b, in this case it is said that the sets a and b fully equivalent.

Fig. 1.2.

After four classes of elements are defined and the necessary information about Euler charts - Venna, we introduce operations on sets. As the first, consider the operation association.

a) Association

Associationsets a \u003d (1, 2, 4, 6) and b \u003d (2, 3, 4, 8, 9)

let's call many

A è b \u003d (1, 2, 3, 4, 6, 8, 9),

where è is the symbol of unification of sets. Thus, the association covers three classes of elements - C.1, C.2 I. C.3, which on the diagram (Fig. 1.3) are shaded.

Logically, the operation of the combination of two sets can be characterized by words: element x. It belongs to the set A or a set of B. At the same time, the bundle "or" simultaneously means the bundle "and". Fact belonging element x. A set A is indicated as x. Î A. So what x. Belongs A. or / I. B, expressed by the formula:

x. Î A è b \u003d ( x. Î a) ú ( x. Î b),

where ú is a symbol of a logical ligament or called disjunction.

b) intersection, addition

Intersection Sets a and b are called the set A ç B, containing those elements from A and B, which are included simultaneously in both sets. For our numerical example, we will have:

A ç B \u003d (1, 2, 4, 6) ç (2, 3, 4, 8, 9) \u003d (2, 4) \u003d C.3.

The Euler chart - the venna for the intersection is shown in Fig. 1.4.

What x. It belongs simultaneously with two sets A and B can be represented by the expression:

x. Î A ç B \u003d ( x. Î a) ù ( x. Î b),

where ù is a symbol of logical bundle "and" called conjunction.

Imagine the operation as a result of which will be shaded areas C.1 I. C.3, forming the set A (Fig. 1.5). Then another operation that will be covered by two other areas - C.0 I. C.2 not included in A, which is indicated as A. (Fig.1.6).

If you combine the shaded areas on both diagrams, we obtain all the shaded set 1; Crossing A and A. It will give an empty set 0, in which no element is contained:

A è. A. \u003d 1, a ç A. = 0.

Lots of A. complements set A to the fundamental set V (or 1); Hence the name: additional Set A, or additionas an operation. Supplement to the logical variable x., i.e. x. (not- x.), is called most often denial X..

After the introduction of intersection and addition operations, all four areas CI On the Euler diagram - Venna can be expressed as follows:

C.0 = A. Ç B., C.1 \u003d A ç B., C.2 = A. Ç b C.3 \u003d A ç B.

By combining the relevant regions CI You can submit any multiple operation, including the merger itself:

A è b \u003d (a ç B.) È ( A. Ç b) è (a ç b).

On the Euler diagram - Venna for implication (Fig. 1.10) is shown partialthe inclusion of the set A in the set B to be distinguished from full inclusion (Fig. 1.2).

If it is argued that "the elements of the set A are included in the set B", then the area C.3 must be shaded, and the area C.1 With the same need to be left white. Regarding areas C.0 I. C.1 located in A., we note that we do not have the right to leave them white, but we are obliged to still fall in A.Sharp.

E) symmetric difference and equivalence

Two other mutually complementary operations remain a symmetric difference and equivalence. The symmetric difference of two sets A and B is the combination of two differences:

A + b \u003d (a - b) è (b - a) \u003d C.1 è. C.2 = {1, 3, 6, 8, 9}.

Equivalence is determined by those elements of sets a and b, which are common to them. However, elements that are not incorporated in neither in B are also considered equivalent:

A ~ B \u003d ( A. Ç b) è (a ç B.) = C.0 È C.3 = {2, 4, 5, 7, 10, 11}.

In fig. 1.11 and 1.12 shows the hatching of the Euler charts - Venna.

In conclusion, we note that the symmetric difference has several names: strict disjunction, excluding alternative, the sum of the module is two. This operation can be conveyed by words - "either, or in", i.e. it is a logical bunch of "or", but without the ligament included in it "and".

Conclusion

Euler-Venna charts - geometric representations of sets. Simple diagram building provides a visual image representing a universal set. U., and inside it - circles (or any other closed figures) representing the set. The figures intersect in the most general case required in the task, and correspond to the shape. Points underlying inside various areas of diagrams can be considered as elements of the corresponding sets. Having a constructed diagram, you can shave certain areas to designate newly formed sets. This allows us to have the most complete picture of the task and solutions. The simplicity of Euler-Venna charts allows you to use this technique in such directions such as mathematics, logic, management and other applied directions.

Bibliography

1. Dictionary on logic. - M.: Tumanit, ed. Center Vlados. . 1997.

2. Weistein, Eric W. Venna Chart (eng.) On the Wolfram Mathworld website.

Vennia diagram is a circuit with intersecting circles, which shows how many common have various sets. To construct the venna diagram, several groups of objects are chosen and placed them in separate circles, and objects that combine the properties of these sets fall into the area of \u200b\u200bcrossing circles.

We give the simplest example. Suppose we have two groups of objects - lighting devices (we denote them in the first round) and energy-saving technologies (we denote them in the second round). In this case, the area of \u200b\u200bcrossing the circles will cover objects that can be attributed to the first, and to the second group, that is, energy-saving lighting devices.

Venna diagrams are successfully used in mathematics, logic, management and other applied areas to compare any sets and establish relationships between them.

The only minus of such diagrams - they can only be used to determine the general qualities of the objects under consideration and do not provide information on the number of objects.

Venna charts: what they need

The Venn diagrams resort to the comparison of the source data in two cases:

• the data is too complex for understanding;
• there are problems to identify relationships between these data.

Due to the visual form of information feeding and simplicity, the transcript of the Venna chart greatly facilitate the process of comprehension and analyzing compared objects. That is why they were widely used when conducting presentations.

Drawing Venna Chart is a not a difficult process that includes only four stages:

1. Consider groups of objects that you need to compare - their number should be equal to the number of circles in your diagram.
2. A bit retreating from the center, draw the first circle. Given that each circle will contain information about the characteristics of the object under consideration, personality, place, etc., it should be quite large.
3. Draw the second round, so that it partially overlap the first circle. In this case, both circles must be of one size. Make sure that inside the intersection area there is also enough space - here you will mention the objects that reveal the similarity between groups.
4. Assign the name to each group of items and sign the circles.