Leonhard Euler (1707-1783) - a famous Swiss and Russian mathematician, member of the St. Petersburg Academy of Sciences, most of his life he lived in Russia. The most famous in mathematical analysis, statistics, computer science and logic is considered the Euler circle (Euler-Venn diagram), used to indicate the volume 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
Under a concept in logic is meant a form of thinking that reflects the essential features of a class of homogeneous objects. They are indicated by one or a group of words: “world map”, “dominant quintseptaccord”, “Monday”, etc.
In the case when the elements of the volume of one concept completely or partially belong to the volume of another, they talk about compatible concepts. If no element of the volume of a certain concept belongs to the volume of another, 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, these are the following:
- identity (equivalence) of volumes;
- intersection (partial coincidence) of volumes;
- submission (subordination).
For incompatible:
- subordination (coordination);
- the opposite (counterattack);
- contradiction (counter-narrativeness).
Schematically, the relationship between concepts in logic is usually denoted by Euler-Venn circles.
Equivalence relationships
In this case, the concepts mean the same thing. Accordingly, the volumes of these concepts completely coincide. For instance:
A - Sigmund Freud;
B is the founder of psychoanalysis.
Or:
A is the square;
B is an equilateral rectangle;
C is an equiangular rhombus.
For designation, completely overlapping Euler circles are used.
Intersection (partial match)
This category includes concepts that have common elements that are related to crossover. That is, the volume of one of the concepts is partially included in the volume of the other:
A - teacher;
In - music lover.
As can be seen from this example, the volumes of concepts partially coincide: 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, for example, “city dweller” acts as a concept , and “car driver” acts as B.
Submission (subordination)
Schematically designated as Euler circles of different scale. The relations between the concepts in this case are characterized by the fact that the subordinate concept (smaller in volume) is fully included in the subordinate (larger in volume). Moreover, the subordinate concept does not completely exhaust the subordinate.
For instance:
A is a tree;
In - a pine.
Concept B will be subordinate to concept A. Since pine refers to trees, concept A becomes in this example a subordinate, “absorbing” the volume of concept B.
Subordination (coordination)
The relationship characterizes two or more concepts that are mutually exclusive, but at the same time belong to a certain common tribal circle. For instance:
A - clarinet;
In - a guitar;
C is the violin;
D is a musical instrument.
Concepts A, B, C are not intersecting in relation to each other, however, they all belong to the category of musical instruments (concept D).
Opposition (counteraction)
Opposite relations between concepts mean that these concepts are related to the same genus. In this case, one of the concepts has certain properties (signs), while the other denies them, replacing the opposite in nature. Thus, we are dealing with antonyms. For instance:
A is a dwarf;
B is a giant.
In the opposite relationship between concepts, the Euler 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.
Contradiction (counter-narrativeness)
In this case, both concepts are species of the same kind. As in the previous example, one of the concepts indicates certain qualities (signs), while the other negates them. However, in contrast to the relationship of opposites, the second, opposite concept does not replace the denied properties with other alternative ones. For instance:
A is a difficult task;
B is a simple task (non-A).
Expressing the volume 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. At the same time, one of them (A) becomes positive (confirming any sign), and the second (B or non-A) becomes negative (denying the corresponding sign): “white paper” - “not white paper”, “Russian history” - “foreign history”, etc.
Thus, the ratio of the volumes of concepts in relation to each other is a key characteristic that defines Euler circles.
Relationships between sets
You should also distinguish between the concepts of elements and sets, the volume of which is reflected in 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 certain collection of objects. The objects themselves are elements of this set. “Plenty is much that can be thought of as one” (Georg Cantor, founder of set theory).
The designation of sets is carried out in capital letters: A, B, C, D ... etc., elements of sets - lowercase: a, b, c, d ... etc. Examples of sets can be students in the same classroom, books standing on a specific shelf (or, for example, all 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 elements, then it is called empty and denoted by the sign Ø. For example, the set of intersection points of parallel lines, the set of solutions 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 connection of logical operations with set theory. In this case, truth tables of concepts are used. For example, the circle denoted by the name A represents a region of truth. Thus, the area outside the circle will represent a lie. To determine the area of the diagram for a logical operation, 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 various industries. For example, in a professional choice situation. If the subject is concerned about the choice of a future profession, he may 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?
Let us depict this in the form of a diagram: Euler circles (examples in logic are the intersection relation):
The result will be those professions that find themselves at the intersection of all three circles.
Euler-Venn circles occupy a separate place in mathematics (set theory) when calculating combinations and properties. Euler circles of the set of elements are enclosed in the image of a 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 elements of the considered sets, points located inside various segments of the diagram can act. On its basis, specific regions can be hatched, thereby designating newly formed sets.
It is permissible to perform basic mathematical operations with these sets: addition (sum of the sets of elements), subtraction (difference), multiplication (product). In addition, thanks to Euler-Venn diagrams, it is possible to carry out operations of comparing sets by the number of elements included in them, not counting them.