home » Makeup » Does not apply to relational functions. Functions

Does not apply to relational functions. Functions

Communication has always been seen as multifunctional process. Psychologists define the functions of communication according to various criteria: emotional, informational, socializing, connecting, translational, aimed at self-knowledge (A.V. Mudrik), establishing community, self-determination (A.B. Dobrovich), self-expression (A.A. Brudny), unity, etc. Most often in psychology, the functions of communication are considered in accordance with the model of the “person-activity-society” relationship.

We can distinguish five main functions: pragmatic, formative, confirming, organizing and maintaining interpersonal relationships, intrapersonal (Fig. 7).

IN pragmatic function communication is the most important condition for uniting people in the process of any joint activity. The devastating consequences for human activity if this condition is not met is described in the famous biblical story about the construction of the Tower of Babel.

Rice. 7.

A big role belongs formative function communication. Communication between a child and an adult is not just a process of transferring to the former a sum of skills, abilities and knowledge that he mechanically assimilates, but a complex process of mutual influence, enrichment and change. The vital role of communication is clearly demonstrated in the following example. In the 30s XX century In the USA, an experiment was conducted in two clinics in which children were treated for serious, difficult-to-treat diseases. The conditions in both clinics were the same, but with some differences: in one hospital, relatives were not allowed to see the babies for fear of infection, while in the other, at certain hours, parents could talk and play with the child in a specially designated room. After a few months, the treatment effectiveness rates were compared. In the first department, the mortality rate approached one third, despite the efforts of doctors. In the second department, where babies were treated with the same means and methods, not a single child died.

Confirmation function in the process of communication it gives the opportunity to know and assert oneself. Wanting to establish himself in his existence and his value, a person seeks a foothold in another person. Everyday experience of human communication is replete with procedures organized according to the principle of confirmation: rituals of acquaintance, greeting, naming, providing various signs of attention. The famous English psychiatrist R.D. Laing saw non-confirmation as the universal source of many mental illnesses, primarily schizophrenia.

Interpersonal for any person is associated with evaluating people and establishing certain emotional relationships - either positive or negative. Therefore, an emotional attitude towards another person can be expressed in terms of “sympathy - antipathy”, which leaves its mark not only on personal, but also on business communication.

Intrapersonal function is considered as a universal way of human thinking. L. S. Vygotsky noted in this regard that “a person even when alone with himself retains the function of communication.”

So, the leading importance of communication in human life is that it is a means of organizing joint activities of people and a way of satisfying a person’s need for another person, their live contact.

Communication as a socio-psychological phenomenon is contact between people, which is carried out through language and speech, and has different forms of manifestation. Language is a system of verbal signs, a means by which communication between people is carried out. The use of language for the purpose of communicating between people is called speech. Depending on the characteristics of communication, various types are distinguished (Fig. 8).

Based on contact with the interlocutor, communication can be direct or indirect.

Direct communication (direct) – this is natural communication when the subjects of interaction are nearby and communicate through speech, facial expressions and gestures.

Rice. 8.

This type of communication is the most complete, because in the process individuals receive maximum information about each other.

Indirect (indirect) communication carried out in situations where individuals are separated from each other by time or distance. For example: talking on the phone, correspondence. Indirect communication is incomplete psychological contact when feedback is difficult.

Communication can be interpersonal or mass. Mass communication represents multiple contacts of strangers, as well as communication mediated by various types of media. It may be direct And indirect. Direct mass communication observed at rallies, meetings, demonstrations, in all large social groups: crowd, public, audience. Mediated mass communication has a one-sided character and is associated with mass culture and the means of mass communication.

According to the criterion of equality of partners in interpersonal communication (Fig. 9), two types are distinguished: dialogical and monological.

Dialogical communication– equal subject-subject interaction, with the goal of mutual knowledge, the desire to realize the goals of each partner.

Monologue communication is implemented when the partners have unequal positions and represents a subject-object relationship. It can be imperative and manipulative. Imperative communication– an authoritarian, directive form of interaction with a partner in order to achieve control over his behavior, attitudes, thoughts and coercion to certain actions or decisions. Moreover, this goal is not veiled. Manipulative communication– a form of interpersonal communication in which influence on a communication partner is carried out covertly in order to achieve one’s intentions.

Rice. 9.

There are two types of communications – role and personal. IN role communication people act based on their status. For example, role-playing communication will be between a teacher and students, a shop manager and workers, etc. Role communication is regulated by the rules accepted in society and the specifics of treatment. Personal communication depends on the individual characteristics of people and the relationships between them.

Communication can be short-term or long-term depending on the goals, content of the activity, individual characteristics of the interlocutors, their likes, dislikes, etc.

Information exchange can occur through verbal and non-verbal interaction. Verbal communication occurs through speech nonverbal– using paralinguistic means of transmitting information (speech volume, voice timbre, gestures, facial expressions, postures).

Communication takes place at different levels. Levels of communication are determined by the general culture of interacting objects, their individual and personal characteristics, characteristics of the situation, social control, value orientations of those communicating, and their attitude towards each other (Fig. 10).

Rice. 10.

The most primitive level of communication is phatic(from Latin fatuus - stupid). It involves a simple exchange of remarks to maintain a conversation, and has no deep meaning. Such communication is necessary in standardized conditions or is determined by etiquette norms.

Informational The level of communication involves the exchange of new information that is interesting to the interlocutors, which is a source of emotional, mental, and behavioral activity of a person.

Personal the level of communication characterizes such interaction in which subjects are capable of deep self-disclosure and comprehension of the essence of another person, themselves and the world around them. It is built on a positive attitude towards yourself, other people and the world around you in general. This is the highest spiritual level of communication.

Display f of a set X into a set Y is considered given if each element x of X is associated with exactly one element y of Y, denoted f(x).

The set X is called domain of definition mapping f, and the set Y is range of values. Set of ordered pairs

Г f = ((x, y) | x∈X, y∈Y, y = f(x))

called display graph f. It follows directly from the definition that the graph of f is a subset of the Cartesian product X×Y:

Strictly speaking, a map is a triple of sets (X, Y, G) such that G⊂ X×Y, and each element x of X is the first element of exactly one pair (x, y) of G. Denoting the second element of such a pair by f(x), we obtain a mapping f of the set X into the set Y. Moreover, G=Г f. If y=f(x), we will write f:x→y and say that element x goes or maps to element y; the element f(x) is called the image of the element x with respect to the mapping f. To denote mappings we will use notations of the form f: X→Y.

Let f: X→Y be a mapping from set X to set Y, and A and B are subsets of sets X and Y, respectively. The set f(A)=(y| y=f(x) for some x∈A) is called way set A. Set f − 1 (B)=(x| f(x) ∈B)

called prototype set B. A mapping f: A→Y such that x→f(x) for all x∈A is called narrowing mapping f to the set A; the narrowing will be denoted by f| A.

Let there be mappings f: X→Y and g: Y→Z. The mapping X→Z under which x goes to g(f(x)) is called composition mappings f and g and is denoted by fg.

A mapping of a set X into X, in which each element goes into itself, x→x, is called identical and is denoted by id X .

For an arbitrary mapping f: X→Y we have id X ⋅f = f⋅id Y .

The mapping f: X→Y is called injective, if for any elements from and it follows that . The mapping f: X→Y is called surjective, if every element y from Y is the image of some element x from X, that is, f(x)=y. The mapping f: X→Y is called bijective, if it is both injective and surjective. The bijective map f: X→Y is invertible. This means that there is a mapping g: Y→X called reverse to a map f such that g(f(x))=x and f(g(y))=y for any x∈X, y∈Y. The inverse of f is denoted by f − 1 .

The invertible mapping f: X→Y sets one-to-one correspondence between elements of the sets X and Y. The injective mapping f: X→Y establishes a one-to-one correspondence between the set X and the set f(X).

Examples. 1) The function f:R→R >0, f (x)=e x, establishes a one-to-one correspondence between the set of all real numbers R and the set of positive real numbers R >0. The inverse of the mapping f is the mapping g:R >0 →R, g(x)=ln x.

2) The mapping f:R→R ≥ 0, f(x)=x 2, the set of all real R onto the set of non-negative numbers R ≥ 0 is surjective, but not injective, and therefore is not bijective.

Function properties:

1. The composition of two functions is a function, i.e. if , then .

2. The composition of two bijective functions is a bijective function, if , then .

3. A mapping has an inverse mapping then and

if and only if f is a bijection, i.e. if , then .

Definition. n – local relation, or n – local predicate P, on the sets A 1 ; A 2 ;…; And n is any subset of the Cartesian product.

Designation n - local relation P(x 1 ;x 2 ;…;x n). When n=1 the relation P is called unary and is a subset of the set A 1 . Binary(binary for n=2) relation is a set of ordered pairs.

Definition. For any set A, the relation is called the identical relation, or diagonal, and - the complete relation, or the complete square.

Let P be some binary relation. Then domain of definition of a binary relation P is called a set for some y), and range of values– a set for some x). Reverse a set is called a relation to P.

The relation P is called reflective, if it contains all pairs of the form (x,x) for any x from X. The relation P is called anti-reflective, if it does not contain any pairs of the form (x,x). For example, the relation x≤y is reflexive, and the relation x

The relation P is called symmetrical, if along with each pair (x,y) it also contains a pair (y,x). The symmetry of the relationship P means that P = P –1.

The relation P is called antisymmetric, if (x;y) and (y;x), then x=y.

The relation R is called transitive, if, together with any pairs (x,y) and (y,z), it also contains the pair (x,z), that is, from xPy and yPz follows xPz.

Properties of binary relations:

Example. Let A=(x/x – Arabic numeral); Р=((x;y)/x,yA,x-y=5). Find D;R;P -1 .

Solution. The relation P can be written in the form P=((5;0);(6;1);(7;2);(8;3);(9;4)), then for it we have D=(5;6 ;7;8;9); E=(0;1;2;3;4); P -1 =((0;5);(1;6);(2;7);(3;8);(4;9)).

Consider two finite sets and a binary relation. Let us introduce the matrix of the binary relation P as follows: .

The matrix of any binary relation has properties:

1. If and , then , and the addition of matrix elements is carried out according to the rules 0+0=0; 1+1=1; 1+0=0+1=1, and multiplication is termwise in the usual way, i.e. according to the rules 1*0=0*1=0; 1*1=1.

2. If , then , and the matrices are multiplied according to the usual rule for matrix multiplication, but the product and sum of elements when multiplying matrices is found according to the rules of step 1.

4. If , then and

Example. The binary relation is shown in Fig. 2. Its matrix has the form .

Solution. Let, then;

Let P be a binary relation on the set A, . The relation P on the set A is called reflective, if , where asterisks indicate zeros or ones. The relation P is called irreflexive, If . The relation P on the set A is called symmetrical, if for and for it follows from the condition that . It means that . The relation P is called antisymmetric, if it follows from the conditions that x=y, i.e. or . This property leads to the fact that all elements of the matrix outside the main diagonal will be zero (there can also be zeros on the main diagonal). The relation P is called transitive, if from and it follows that , i.e. .

Example. The relation P and . Here on the main diagonal of the matrix are all units, therefore, P is reflexive. The matrix is asymmetrical, then the ratio P is asymmetrical

Because not all elements located outside the main diagonal are zero, then the relation P is not antisymmetric.

Those. , therefore the relation P is intransitive.

A reflexive, symmetrical and transitive relation is called equivalence relation. It is customary to use the symbol ~ to denote equivalence relations. The conditions of reflexivity, symmetry and transitivity can be written as follows:

Example. 1) Let X be a set of functions defined on the entire number line. We will assume that the functions f and g are related by the relation ~ if they take the same values at point 0, that is, f(x)~g(x), if f(0)=g(0). For example, sinx~x, e x ~cosx. The relation ~ is reflexive (f(0)=f(0) for any function f(x)); symmetrically (from f(0)=g(0) it follows that g(0)=f(0)); transitive (if f(0)=g(0) and g(0)=h(0), then f(0)=h(0)). Therefore, ~ is an equivalence relation.

2) Let ~ be a relation on the set of natural numbers such that x~y, if x and y give the same remainder when divided by 5. For example, 6~11, 2~7, 1~6. It is easy to see that this relation is reflexive, symmetrical and transitive and, therefore, is an equivalence relation.

Partial order relation A binary relation on a set is called if it is reflexive, antisymmetric, transitive, i.e.

1. - reflexivity;

2. - antisymmetry;

3. - transitivity.

A relationship of strict order A binary relation on a set is called if it is anti-reflexive, antisymmetric, transitive. Both of these relationships are called order relations. A set on which an order relation is specified, can be: a completely ordered set or partially ordered. Partial order is important in cases where we want to somehow characterize precedence, i.e. decide under what conditions to consider one element of the set to be superior to another. A partially ordered set is called linearly ordered, if there are no incomparable elements in it, i.e. one of the conditions or is satisfied. For example, sets with a natural order on them are linearly ordered.

Essence and classification of economic relations

From the moment of his separation from the world of wild nature, man develops as a biosocial being. This determines the conditions for its development and formation. The main stimulus for the development of man and society is needs. To satisfy these needs, a person must work.

Labor is the conscious activity of a person to create goods in order to satisfy needs or obtain benefits.

The more the needs increased, the more complex the labor process became. It required ever greater expenditures of resources and ever more coordinated actions of all members of society. Thanks to work, both the main features of the external appearance of modern man and the characteristics of man as a social being were formed. Labor moved into the phase of economic activity.

Economic activity refers to human activity in the creation, redistribution, exchange and use of material and spiritual goods.

Economic activity involves the need to enter into some kind of relationship between all participants in this process. These relations are called economic.

Definition 1

Economic relations are the system of relationships between individuals and legal entities formed in the production process. redistribution, exchange and consumption of any goods.

These relationships have different forms and durations. Therefore, there are several options for their classification. It all depends on the criterion chosen. The criterion may be time, frequency (regularity), degree of benefit, characteristics of the participants in this relationship, etc. The most frequently mentioned types of economic relations are:

international and domestic;
mutually beneficial and discriminatory (benefiting one party and infringing on the interests of the other);
voluntary and forced;
stable regular and episodic (short-term);
credit, financial and investment;
purchase and sale relations;
proprietary relations, etc.

In the process of economic activity, each of the participants in the relationship can act in several roles. Conventionally, three groups of carriers of economic relations are distinguished. These are:

producers and consumers of economic goods;
sellers and buyers of economic goods;
owners and users of goods.

Sometimes a separate category of intermediaries is distinguished. But on the other hand, intermediaries simply exist in several forms at the same time. Therefore, the system of economic relations is characterized by a wide variety of forms and manifestations.

There is another classification of economic relations. The criterion is the characteristics of the ongoing processes and goals of each type of relationship. These types are the organization of labor activity, the organization of economic activity and the management of economic activity.

The basis for the formation of economic relations of all levels and types is the right of ownership of resources and means of production. They determine the ownership of the goods produced. The next system-forming factor is the principles of distribution of produced goods. These two points formed the basis for the formation of types of economic systems.

Functions of organizational and economic relations

Definition 2

Organizational-economic relations are relationships to create conditions for the most efficient use of resources and reduce costs through the organization of forms of production.

The function of this form of economic relations is the maximum use of relative economic advantages and the rational use of obvious opportunities. The main forms of organizational and economic relations include concentration (consolidation) of production, combination (combination of production from different industries in one enterprise), specialization and cooperation (to increase productivity). The formation of territorial production complexes is considered the completed form of organizational and economic relations. An additional economic effect is obtained due to the favorable territorial location of enterprises and the rational use of infrastructure.

Soviet Russian economists and economic geographers in the middle of the twentieth century developed the theory of energy production cycles (EPC). They proposed organizing production processes in a certain area in such a way as to use a single flow of raw materials and energy to produce a whole range of products. This would dramatically reduce production costs and reduce production waste. Organizational and economic relations are directly related to economic management.

Functions of socio-economic relations

Definition 3

Socio-economic relations are the relations between economic agents, which are based on property rights.

Property is a system of relations between people, manifested in their attitude towards things - the right to dispose of them.

The function of socio-economic relations is to streamline property relations in accordance with the norms of a given society. After all, legal relations are built, on the one hand, on the basis of property rights, and on the other, on the basis of volitional property relations. These interactions between the two parties take the form of both moral norms and legislative (legally enshrined) norms.

Socio-economic relations depend on the social formation in which they develop. They serve the interests of the ruling class in that particular society. Socio-economic relations ensure the transfer of ownership from one person to another (exchange, purchase and sale, etc.).

Functions of international economic relations

International economic relations perform the function of coordinating the economic activities of countries around the world. They bear the character of all three main forms of economic relations - economic management, organizational-economic and socio-economic. This is especially relevant nowadays due to the variety of models of a mixed economic system.

The organizational and economic side of international relations is responsible for expanding international cooperation based on integration processes. The socio-economic aspect of international relations is the desire for a general increase in the level of well-being of the population of all countries of the world and a reduction in social tension in the world economy. Management of the global economy is aimed at reducing contradictions between national economies and reducing the impact of global inflation and crisis phenomena.

Humans have an inherent need to communicate and interact with other people. By satisfying this need, he manifests and realizes his capabilities.

Human life throughout its entire duration manifests itself, first of all, in communication. And all the diversity of life is reflected in the equally endless variety of communication: in the family, school, at work, in everyday life, in companies, etc.

Communication- one of the universal forms of personality activity, manifested in the establishment and development of contacts between people, in the formation of interpersonal relationships and generated by the needs for joint activities.

Communication fulfills a number of basic functions:

Information - the function of receiving and transmitting information;
Contact - establishing contact as a state of mutual readiness of people to receive and transmit information;
Incentive - the function of stimulating activity to action;
Coordination - the function of mutual orientation and coordination of actions;
Understanding - involves not only the reception of information, but also the understanding of this information by each other;
Amotive - the function of arousing the necessary emotions, experiences, feelings in a partner, involves emotional exchange, a change in the emotional state;
The function of establishing relationships is awareness and fixation of one’s social status, social role in a specific social community.
The function of influencing is changing the state, behavior, intentions, ideas, attitudes, opinions, decisions, needs, actions, etc.

Along with the functions, the main ones are identified kinds communication.

By number of participants:

interpersonal;
group.

By way of communication:

verbal;
non-verbal.

According to the position of those communicating:

contact;
distant.

According to the terms of communication:

official;
unofficial.

IN structure communication is distinguished by three closely interconnected, interdependent aspects:

The perceptual side of communication is the process of perceiving each other.
The communicative side of communication involves the transfer of information. It is necessary to take into account that a person expresses 80% of what he wants to say, the listener perceives 70% and understands 60% of what is said.
The interactive side of communication involves the organization of interaction (coordination of actions, distribution of functions, etc.).

When organizing communication, it is necessary to take into account that it goes through a number of stages, each of which affects its effectiveness.

If one of the stages of communication is missed, the effectiveness of communication decreases sharply and there is a possibility of not achieving the goals that were set when organizing communication. The ability to effectively achieve set goals in communication is called sociability, communicative competence, social intelligence.

In this subsection we introduce Cartesian products, relations, functions and graphs. We study the properties of these mathematical models and the connections between them.

Cartesian product and enumeration of its elements

Cartesian product sets A And B is a set consisting of ordered pairs: A´ B= {(a,b): (aÎ A) & (bÎ B)}.

For sets A 1, …, A n the Cartesian product is determined by induction:

In the case of an arbitrary set of indices I Cartesian product families sets ( A i} i Î I is defined as a set consisting of such functions f:I® Ai, that's for everyone iÎ I right f(i)Î A i .

Theorem 1

Let A andB are finite sets. Then |A´ B| = |A|×| B|.

Proof

Let A = (a 1 , …,a m), B = (b 1 , …,bn). The elements of a Cartesian product can be arranged using a table

(a 1 ,b 1), (a 1 ,b 2), …, (a 1 ,b n);

(a 2 ,b 1), (a 2 ,b 2), …, (a 2 ,b n);

(a m ,b 1), (a m ,b 2),…, (a m ,b n),

consisting of n columns, each of which consists of m elements. From here | A´ B|=mn.

Corollary 1

Proof

Using induction on n. Let the formula be true for n. Then

Relationship

Let n³1 is a positive integer and A 1, …, A n– arbitrary sets. Relationship between elements of sets A 1, …, A n or n-ary relation is called an arbitrary subset.

Binary relations and functions

Binary relation between elements of sets A And B(or, for short, between A And B) is called a subset RÍ A´ B.

Definition 1

Function or display is called a triple consisting of sets A And B and subsets fÍ A´ B(function graphics), satisfying the following two conditions;

1) for anyone xÎ A there is such yÎ f, What (x,y)Î f;

2) if (x,y)Î f And (x,z)Î f, That y=z.

It's easy to see that fÍ A´ B will then and only define a function when for any xÎ A there is only one yÎ f, What ( x,y) Î f. This y denote by f(x).

The function is called injection, if for any x,x'Î A, such What x¹ x', occurs f(x)¹ f(x'). The function is called surjection, if for each yÎ B there is such xÎ A, What f(x) = y. If a function is an injection and a surjection, then it is called bijection.

Theorem 2

In order for a function to be a bijection, it is necessary and sufficient for the existence of a function such that fg =ID B And gf =ID A.

Proof

Let f– bijection. Due to surjectivity f for each yÎ B you can select an element xÎ A, for which f(x) = y. Due to injectivity f, this element will be the only one, and we will denote it by g(y) = x. Let's get the function.

By constructing the function g, the equalities hold f(g(y)) = y And g(f(x)) = x. So it's true fg =ID B And gf =ID A. The opposite is obvious: if fg =ID B And gf =ID A, That f– surjection in force f(g(y)) = y, for each yÎ B. In this case it will follow , and that means . Hence, f– injection. It follows from this that f– bijection.

Image and prototype

Let be a function. In a manner subsets XÍ A called a subset f(X) = (f(x):xÎ X)Í B. For YÍ B subset f - -1 (Y) =(xÎ A:f(x)Î Y) called prototype subsetsY.

Relations and graphs

Binary relationships can be visualized using directed graphs.

Definition 2

Directed graph called a pair of sets (E,V) along with a couple of mappings s,t:E® V. Elements of the set V are represented by points on a plane and are called peaks. Elements from E are called directed edges or arrows. Each element eÎ E depicted as an arrow (possibly curvilinear) connecting the vertex s(e) with top t(e).

To an arbitrary binary relation RÍ V´ V corresponds to a directed graph with vertices vÎ V, whose arrows are ordered pairs (u,v)Î R. Displays s,t:R® V are determined by the formulas:

s(u,v) =u And t(u,v) =v.

Example 1

Let V = (1,2,3,4).

Consider the relation

R = ((1,1), (1,3), (1.4), (2,2), (2,3), (2,4), (3,3), (4,4)).

It will correspond to a directed graph (Fig. 1.2). The arrows of this graph will be pairs (i,j)Î R.

Rice. 1.2. Directed binary relation graph

In the resulting directed graph, any pair of vertices is connected by at most one arrow. Such directed graphs are called simple. If we do not consider the direction of the arrows, then we come to the following definition:

Definition 3

A simple (undirected) graph G = (V,E) a pair consisting of a set is called V and many E, consisting of some unordered pairs ( v 1,v 2) elements v 1,v 2Î V such that v 1¹ v 2. These pairs are called ribs, and the elements from V – peaks.

Rice. 1.3. Simple undirected graph K 4

A bunch of E defines a binary symmetric anti-reflexive relation consisting of pairs ( v 1,v 2), for which ( v 1,v 2} Î E. The vertices of a simple graph are depicted as points, and the edges as segments. In Fig. 1.3 shows a simple graph with many vertices

V={1, 2, 3, 4}

and many ribs

E= {{1,2}, {1,3},{1,4}, {2,3}, {2,4}, {3, 4}}.

Operations on binary relations

Binary relation between elements of sets A And B an arbitrary subset is called RÍ A´ B. Record aRb(at aÎ A, bÎ B) means that (a,b)Î R.

The following operations on relations are defined RÍ A´ A:

· R -1= ((a,b): (b,a)Î R);

· R° S = ((a,b): ($ xÎ A)(a,x)Î R&(x,b)Î R);

· Rn=R°(R n -1);

Let Id A = ((a,a):aÎ A)– identical relation. Attitude R Í X´ X called:

1) reflective, If (a,a)Î R for all aÎ X;

2) anti-reflective, If (a,a)Ï R for all aÎ X;

3) symmetrical, if for everyone a,bÎ X the implication is true aRbÞ bRa;

4) antisymmetric, If aRb &bRaÞ a=b;

5) transitive, if for everyone a,b,cÎ X the implication is true aRb &bRcÞ aRc;

6) linear, for all a,bÎ X the implication is true a¹ bÞ aRbÚ bRa.

Let's denote ID A through ID. It is easy to see that the following takes place.

Sentence 1

Attitude RÍ X´ X:

1) reflexively Û IDÍ R;

2) anti-reflexive Û RÇ Id=Æ ;

3) symmetrically Û R = R -1;

4) antisymmetric Û RÇ R -1Í ID;

5) transitive Û R° RÍ R;

6) linear Û RÈ IDÈ R -1 = X´ X.

Binary relation matrix

Let A= {a 1, a 2, …, a m) And B= {b 1, b 2, …, b n) are finite sets. Binary relation matrix R Í A ´ B is called a matrix with coefficients:

Let A– finite set, | A| = n And B= A. Let's consider the algorithm for calculating the composition matrix T= R° S relations R, S Í A´ A. Let us denote the coefficients of the relationship matrices R, S And T accordingly through r ij, s ij And t ij.

Since the property ( a i,a k)Î T is equivalent to the existence of such a jÎ A, What ( a i,a j)Î R And ( a j,a k) Î S, then the coefficient tik will be equal to 1 if and only if such an index exists j, What r ij= 1 and sjk= 1. In other cases tik equals 0. Therefore, tik= 1 if and only if .

It follows from this that to find the matrix of the composition of relations it is necessary to multiply these matrices and in the resulting product of matrices the non-zero coefficients are replaced by ones. The following example shows how the composition matrix is calculated in this way.

Example 2

Consider the binary relation on A = (1,2,3), equal R = ((1,2),(2,3)). Let's write the relation matrix R. According to definition, it consists of coefficients r 12 = 1, r 23 = 1 and the rest r ij= 0. Hence the relation matrix R is equal to:

Let's find a relationship R° R. For this purpose, we multiply the relation matrix R to myself: