The Euler circle. Circles Euler - examples in logic

Leonard Euler (1707-1783) - a famous Swiss and Russian mathematician, a member of the Petersburg Academy of Sciences, spent most of his life in Russia. The most famous in mathematical analysis, statistics, informatics and logic is the Euler circle (Euler-Venn diagram), used to denote the scope of concepts and sets of elements.

John Venn (1834-1923) is an English philosopher and logician, co-author of the Euler-Venn diagram.

Compatible and incompatible concepts

By concept in logic is meant a form of thinking reflecting the essential attributes of a class of homogeneous objects. They are denoted by one or a group of words: "world map", "dominant quintuptakkord", "Monday", etc.

In the case when the elements of the volume of one concept fully or partially belong to the volume of the other, they speak of compatible concepts. If no element of the volume of a certain concept belongs to the volume of the other, we have a place with incompatible concepts.

In turn, each of the types of concepts has its own set of possible relationships. For compatible concepts this is the following:

• Identity (equivalence) of volumes;
• Intersection (partial coincidence) of volumes;
• Subordination.

For incompatible:

• Subordination (coordination);
• Contrast (contrast);
• Contradiction.

Schematically, the relationship between concepts in logic is usually denoted by Euler-Venn circles.

Equivalence relations

In this case, the concepts mean the same thing. Accordingly, the volumes of these concepts completely coincide. For example:

A - Sigmund Freud;

B - the founder of psychoanalysis.

Or:

A is a square;

B is an equilateral rectangle;

C is a conformal rhombus.

For the designation, the completely coinciding Euler circles are used.

Intersection (partial coincidence)

This category includes concepts that have common elements that are related to crossing. That is, the volume of one of the concepts is partly included in the scope of another:

A - the 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 - among the music lovers there can be representatives of the pedagogical profession. A similar relation will be in the case when, for example, the "citizen" appears as the concept A, and the "autoconductor" as B.

Subordination

Schematically denoted as different in scale Euler circles. Relationships between concepts in this case are characterized by the fact that the subordinate concept (less in scope) is fully part of the subordinate (larger in volume). In this case, the subordinate concept does not fully exhaust the subordinate.

For example:

A is a tree;

B - pine.

The concept of B will be subordinate to the concept of A. Since pine refers to trees, the concept A becomes in this example subordinate, "absorbing" the scope of the concept of B.

Subordination (coordination)

The relation characterizes two or more concepts that exclude each other, but belong to a certain common generic circle. For example:

A - clarinet;

B - guitar;

C - violin;

D is a musical instrument.

The concepts A, B, C are not intersecting with each other, nevertheless, they all belong to the category of musical instruments (concept D).

Contrast (Contrast)

Opposite relations between concepts imply the attribution of these concepts to the same genus. In this case, one of the concepts has certain properties (attributes), while the other denies them, replacing the opposite in character. Thus, we are dealing with antonyms. For example:

A - the dwarf;

B is a giant.

The Euler circle, with the opposite relationship between concepts, is divided into three segments, the first of which corresponds to the concept of A, the second to the concept of B, and the third to all other possible concepts.

Contradiction

In this case, both concepts are species of the same genus. As in the previous example, one of the concepts indicates certain qualities (attributes), while the other denies them. However, unlike the relation of the opposite, the second, the opposite concept, does not replace the negated properties by others, alternative. For example:

A is a complicated problem;

B is an uncomplicated task (not-A).

Expressing the scope of concepts of this kind, the Euler 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 some attribute), and the second (B or not-A) - negative (denying the corresponding sign): "white paper" - "not white paper", "domestic history" - "foreign history", etc.

Thus, the ratio of the volume of concepts in relation to each other is a key characteristic that determines the Euler circles.

Relations between sets

Also, it is necessary to distinguish the concepts of elements and sets, the volume of which reflects the Euler circles. The notion of set is borrowed from mathematical science and has a rather wide meaning. Examples in logic and mathematics display it as a collection of objects. The objects themselves are elements of a given set. "Many are many, conceivable as one" (Georg Kantor, founder of set theory).

The designations of sets are carried out in capital letters: A, B, C, D ..., etc., elements of sets - lower case: a, b, c, d ... etc. Examples of the set can be students in the same class, books standing On a certain shelf (or, for example, all books in a particular library), pages in the diary, berries in a forest glade, etc.

In turn, if a certain set does not contain any elements, it is called empty and is denoted by the sign of Ø. For example, the set of points of intersection of parallel lines, the solution set of the equation x ² = -5.

Problem Solving

To solve a large number of problems, Euler circles are actively used. Examples in logic clearly demonstrate the relationship of logical operations to set theory. In this case, truth tables of concepts are used. For example, the circle indicated by the name A is a truth area. Thus, the area outside the circle will be a lie. To determine the area of the diagram for a logical operation, it is necessary to shade the areas defining the Euler circle in which its values for elements A and B are true.

The use of Euler circles has found wide practical application in different branches. For example, in a situation with a professional choice. If the subject is preoccupied with choosing a future profession, he can be guided by the following criteria:

W - what do I like to do?

D - what do I get?

P - how can I make good money?

We depict this in the form of a diagram: Euler circles (examples in logic are the intersection ratio):

The result will be those professions that will be at the intersection of all three circles.

The Euler-Venn circles occupy a separate place in mathematics (set theory) in the calculation of combinations and properties. Euler circles of the set of elements are enclosed in the image of the rectangle denoting the universal set (U). Instead of circles, other closed figures can also be used, but the essence of this 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. As the elements of the sets under consideration, points located within different segments of the diagram can act. On its basis, it is possible to shade specific areas, thus denoting newly formed sets.

With these sets it is possible to perform the basic mathematical operations: addition (the sum of sets of elements), subtraction (difference), multiplication (product). In addition, thanks to Euler-Venn diagrams, it is possible to perform operations of comparing sets by the number of elements included in them, not counting them.