Leonhard Euler (1707-1783) - famous Swiss and Russian mathematician, member of the St. Petersburg Academy of Sciences, lived most of his life in Russia. The most famous in mathematical analysis, statistics, computer science and logic is the Euler circle (Euler-Venn diagram), used to denote the scope 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 the concept in logic means a form of thinking that reflects the essential features of a class of homogeneous objects. They are denoted by one or a group of words: “world map”, “dominant fifth-seventh chord”, “Monday”, etc.
In the case when the elements of the scope of one concept fully or partially belong to the scope of another, one speaks of compatible concepts. If, however, no element of the scope of a certain concept belongs to the scope of another, we have incompatible concepts.
In turn, each type of concept has its own set of possible relationships. For compatible concepts, these are:
- identity (equivalence) of volumes;
- crossing (partial match)volumes;
- subordination (subordination).
For incompatible:
- subordination (coordination);
- opposite (contrarality);
- contradiction (contradiction).
Schematically, relations between concepts in logic are usually denoted using Euler-Venn circles.
Equivalent relations
In this case, the concepts mean the same subject. Accordingly, the volumes of these concepts are completely the same. For example:
A - Sigmund Freud;
B is the founder of psychoanalysis.
Or:
A is a square;
B is an equilateral rectangle;
C is an equiangular rhombus.
Completely coinciding Euler circles are used for designation.
Intersection (partial match)
This category includes concepts that have common elements that are in relation to crossing. That is, the volume of one of the concepts is partially included in the volume of the other:
A - teacher;
B is a 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 the concept A is, for example, "city dweller", and the concept B is "driver".
Subordination (subordination)
Schematically denoted as Euler circles of different scales. Relationsbetween concepts in this case are characterized by the fact that the subordinate concept (smaller in volume) is completely included in the subordinate (larger in volume). At the same time, the subordinate concept does not completely exhaust the subordinate one.
For example:
A - tree;
B - pine.
Concept B will be subordinate to concept A. Since pine belongs to trees, concept A in this example becomes subordinate, "absorbing" the scope of concept B.
Coordination (coordination)
Relation characterizes two or more concepts that exclude each other, but at the same time 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 in relation to each other, however, they all belong to the category of musical instruments (the concept D).
Opposite (contrary)
Opposite relationships between concepts imply that these concepts belong to the same genus. At the same time, one of the concepts has certain properties (features), while the other denies them, replacing them with opposite ones in character. Thus, we are dealing with antonyms. For example:
A is a dwarf;
B is a giant.
Euler circle with opposite relations between conceptsis 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 (contradiction)
In this case, both concepts are species of the same genus. As in the previous example, one of the concepts indicates certain qualities (features), while the other denies them. However, in contrast to the relation of opposites, the second, opposite concept does not replace the denied properties with other, alternative ones. For example:
A is a difficult task;
B is an easy task (not-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, the concepts are also antonyms. At the same time, one of them (A) becomes positive (affirming some feature), and the second (B or non-A) becomes negative (negating the corresponding feature): “white paper” - “not white paper”, “national 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
It is also necessary to distinguish between the concepts of elements and sets, the volume of which is displayed by 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 set of objects. The objects themselves areelements of this set. “Many is many thought as one” (Georg Kantor, founder of set theory).
Sets are designated in capital letters: A, B, C, D… etc., elements of sets are designated in lower case: a, b, c, d… etc. Examples of a set can be students who are in the same classrooms, books on a certain shelf (or, for example, all the books in a certain library), pages in a diary, berries in a forest clearing, etc.
In turn, if a certain set does not contain a single element, 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 to the equation x2=-5.
Problem solving
Euler circles are actively used to solve a large number of problems. Examples in logic clearly demonstrate the connection between logical operations and set theory. In this case, truth tables of concepts are used. For example, the circle labeled A represents the truth region. So the area outside the circle will represent false. To determine the area of the diagram for a logical operation, you should shade the areas that define the Euler circle, in which its values for elements A and B will be true.
The use of Euler circles has found wide practical application in various industries. For example, in a situation with a professional choice. If the subject is concerned about the choice of a future profession, he can be guided by the following criteria:
W – what do I like to do?
D – what am I doing?
P– how can I make good money?
Let's draw this as a diagram: Euler circles (examples in logic - intersection relation):
The result will be those professions that will be at the intersection of all three circles.
Euler-Venn circles occupy a separate place in mathematics (set theory) when calculating combinations and properties. The 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 with each other, according to the conditions of the problem (in the most general case). Also, these figures should be labeled accordingly. The elements of the sets under consideration can be points located inside different segments of the diagram. Based on it, you can shade specific areas, thereby designating the newly formed sets.
With these sets it is possible to perform basic mathematical operations: addition (sum of sets of elements), subtraction (difference), multiplication (product). In addition, thanks to the Euler-Venn diagrams, it is possible to compare sets by the number of elements included in them, not counting them.
