Relationship. Basic concepts and definitions
Definition 2.1.Ordered pair<x, y> called a collection of two elements x And y, arranged in a certain order.
Two ordered pairs<x, y> and<u, v> are equal to each other if and only if x = u And y= v.
Example 2.1.
<a, b>, <1, 2>, <x, 4> – ordered pairs.
Similarly, we can consider triplets, quadruples, n-ki elements<x 1 , x 2 ,… x n>.
Definition 2.2.Direct(or Cartesian)work two sets A And B is the set of ordered pairs such that the first element of each pair belongs to the set A, and the second – to the set B:
A ´ B = {<a, b>, ç aÎ A And bÏ IN}.
In general, the direct product n sets A 1 ,A 2 ,…A n called a set A 1 A 2 ´…´ A n, consisting of ordered sets of elements<a 1 , a 2 , …,a n> length n, such that i- th a i belongs to the set A i,a i Î A i.
Example 2.2.
Let A = {1, 2}, IN = {2, 3}.
Then A ´ B = {<1, 2>, <1, 3>,<2, 2>,<2, 3>}.
Example 2.3.
Let A= {x ç0 £ x£ 1) and B= {yç2 £ y£3)
Then A ´ B = {<x, y >, ç0 £ x£1&2£ y£3).
Thus, many A ´ B consists of points lying inside and on the border of a rectangle formed by straight lines x= 0 (y-axis), x= 1,y= 2i y = 3.
The French mathematician and philosopher Descartes was the first to propose a coordinate representation of points on a plane. This is historically the first example of a direct product.
Definition 2.3.Binary(or double)ratio r is called the set of ordered pairs.
If a couple<x, y>belongs r, then it is written as follows:<x, y> Î r or, what is the same, xr y.
Example2.4.
r = {<1, 1>, <1, 2>, <2, 3>}
Similarly we can define n-local relation as a set of ordered n-OK.
Since a binary relation is a set, the methods for specifying a binary relation are the same as the methods for specifying a set (see Section 1.1). A binary relation can be specified by listing ordered pairs or by specifying a general property of ordered pairs.
Example 2.5.
1. r = {<1, 2>, <2, 1>, <2, 3>) – the relation is specified by enumerating ordered pairs;
2. r = {<x, y> ç x+ y = 7, x, y– real numbers) – the relation is specified by specifying the property x+ y = 7.
Additionally, a binary relation can be given binary relation matrix. Let A = {a 1 , a 2 , …, a n) is a finite set. Binary relation matrix C is a square matrix of order n, whose elements c ij are defined as follows:
Example 2.6.
A= (1, 2, 3, 4). Let's define a binary relation r in the three listed ways.
1. r = {<1, 2>, <1, 3>, <1, 4>, <2, 3>, <2, 4>, <3, 4>) – the relation is specified by enumerating all ordered pairs.
2. r = {<a i, a j> ç a i < a j; a i, a jÎ A) – the relation is specified by indicating the property “less than” on the set A.
3. – the relation is specified by the binary relation matrix C.
Example 2.7.
Let's look at some binary relationships.
1. Relations on the set of natural numbers.
a) the relation £ holds for pairs<1, 2>, <5, 5>, but does not hold for the pair<4, 3>;
b) the relation “have a common divisor other than one” holds for pairs<3, 6>, <7, 42>, <21, 15>, but does not hold for the pair<3, 28>.
2. Relations on the set of points of the real plane.
a) the relation “to be at the same distance from the point (0, 0)” is satisfied for points (3, 4) and (–2, Ö21), but is not satisfied for points (1, 2) and (5, 3);
b) the relation “to be symmetrical about the axis OY" is performed for all points ( x, y) And (- x, –y).
3. Relationships with many people.
a) the attitude of “living in the same city”;
b) the attitude of “studying in the same group”;
c) the “being older” attitude.
Definition 2.4. The domain of definition of a binary relation r is the set D r = (x çthere is y such that xr y).
Definition 2.5. The range of values of a binary relation r is the set R r = (y çexists x such that xr y).
Definition 2.6. The domain of specifying a binary relation r is called the set M r = D r ÈR r .
Using the concept of direct product, we can write:
rÎ D r´ R r
If D r= R r = A, then we say that the binary relation r defined on the set A.
Example 2.8.
Let r = {<1, 3>, <3, 3>, <4, 2>}.
Then D r ={1, 3, 4}, R r = {3, 2}, M r= {1, 2, 3, 4}.
Operations on relationships
Since relations are sets, all operations on sets are valid for relations.
Example 2.9.
r 1 = {<1, 2>, <2, 3>, <3, 4>}.
r 2 = {<1, 2>, <1, 3>, <2, 4>}.
r 1 È r 2 = {<1, 2>, <1, 3>, <2, 3>, <2, 4>, <3, 4>}.
r 1 Ç r 2 = {<1, 2>}.
r 1 \ r 2 = {<2, 3>, <3, 4>}.
Example 2.10.
Let R– set of real numbers. Let us consider the following relations on this set:
r 1 – "£"; r 2 – " = "; r 3 – " < "; r 4 – "³"; r 5 – " > ".
r 1 = r 2 È r 3 ;
r 2 = r 1 Ç r 4 ;
r 3 = r 1 \ r 2 ;
r 1 = ;
Let us define two more operations on relations.
Definition 2.7. The relationship is called reverse to attitude r(denoted r – 1), if
r – 1 = {<x, y> ç< y, x> Î r}.
Example 2.11.
r = {<1, 2>, <2, 3>, <3, 4>}.
r – 1 = {<2, 1>, <3, 2>, <4, 3>}.
Example 2.12.
r = {<x, y> ç x – y = 2, x, y Î R}.
r – 1 = {<x, y> ç< y, x> Î r} = r – 1 = {<x, y> ç y–x = 2, x, y Î R} = {<x, y> ç– x+ y = 2, x, y Î R}.
Definition 2.8.Composition of two relations r and s called relation
s r= {<x, z> çthere is such a thing y, What<x, y> Î r And< y, z> Î s}.
Example 2.13.
r = {<x, y> ç y = sinx}.
s= {<x, y> ç y = Ö x}.
s r= {<x, z> çthere is such a thing y, What<x, y> Î r And< y, z> Î s} = {<x, z> çthere is such a thing y, What y = sinx And z= Ö y} = {<x, z> ç z= Ö sinx}.
The definition of the composition of two relations corresponds to the definition of a complex function:
y = f(x), z= g(y) Þ z= g(f(x)).
Example 2.14.
r = {<1, 1>, <1, 2>, <1, 3>, <3, 1>}.
s = {<1, 2>, <1, 3>, <2, 2>, <3, 2>, <3, 3>}.
Finding process s r in accordance with the definition of composition, it is convenient to depict it in a table in which all possible values are enumerated x, y, z. for each pair<x, y> Î r we need to consider all possible pairs< y, z> Î s(Table 2.1).
Table 2.1
| <x, y> Î r | < y, z> Î s | <x, z> Î s r |
| <1, 1> <1, 1> <1, 2> <1, 3> <1, 3> <3, 1> <3, 1> | <1, 2> <1, 3> <2, 2> <3, 2> <3, 3> <1, 2> <1, 3> | <1, 2> <1, 3> <1, 2> <1, 2> <1, 3> <3, 2> <3, 3> |
Note that the first, third and fourth, as well as the second and fifth rows of the last column of the table contain identical pairs. Therefore we get:
s r= {<1, 2>, <1, 3>, <3, 2>, <3, 3>}.
Properties of relationships
Definition 2.9. Attitude r called reflective on a set X, if for any xÎ X performed xr x.
From the definition it follows that every element<x,x > Î r.
Example 2.15.
a) Let X– finite set, X= (1, 2, 3) and r = {<1, 1>, <1, 2>, <2, 2>, <3, 1>, <3, 3>). Attitude r reflectively. If X is a finite set, then the main diagonal of the reflexive relation matrix contains only ones. For our example
b) Let X r relation of equality. This attitude is reflexive, because every number is equal to itself.
c) Let X- a lot of people and r"live in the same city" attitude. This attitude is reflexive, because everyone lives in the same city with themselves.
Definition 2.10. Attitude r called symmetrical on a set X, if for any x, yÎ X from xry should yr x.
It's obvious that r symmetrical if and only if r = r – 1 .
Example 2.16.
a) Let X– finite set, X= (1, 2, 3) and r = {<1, 1>, <1, 2>, <1, 3>, <2, 1>, <3, 1>, <3, 3>). Attitude r symmetrically. If X is a finite set, then the symmetric relation matrix is symmetric with respect to the main diagonal. For our example
b) Let X– set of real numbers and r relation of equality. This relationship is symmetrical, because If x equals y, then y equals x.
c) Let X– many students and r"study in the same group" attitude. This relationship is symmetrical, because If x studies in the same group as y, then y studies in the same group as x.
Definition 2.11. Attitude r called transitive on a set X, if for any x, y,zÎ X from xry And yr z should xr z.
Simultaneous fulfillment of conditions xry, yr z, xr z means that the pair<x,z> belongs to the composition r r. Therefore for transitivity r it is necessary and sufficient for the set r r was a subset r, i.e. r rÍ r.
Example 2.17.
a) Let X– finite set, X= (1, 2, 3) and r = {<1, 1>, <1, 2>, <2, 3>, <1, 3>). Attitude r transitive, because along with pairs<x,y>and<y,z>have a couple<x,z>. For example, along with pairs<1, 2>, And<2, 3>there is a pair<1, 3>.
b) Let X– set of real numbers and r ratio £ (less than or equal to). This relation is transitive, because If x£ y And y£ z, That x£ z.
c) Let X- a lot of people and r"being older" attitude. This relation is transitive, because If x older y And y older z, That x older z.
Definition 2.12. Attitude r called equivalence relation on a set X, if it is reflexive, symmetric and transitive on the set X.
Example 2.18.
a) Let X– finite set, X= (1, 2, 3) and r = {<1, 1>, <2, 2>, <3, 3>). Attitude r is an equivalence relation.
b) Let X– set of real numbers and r relation of equality. This is an equivalence relation.
c) Let X– many students and r"study in the same group" attitude. This is an equivalence relation.
Let r X.
Definition 2.13. Let r– equivalence relation on the set X And xÎ X. Equivalence class, generated by the element x, is called a subset of the set X, consisting of those elements yÎ X, for which xry. Equivalence class generated by element x, denoted by [ x].
Thus, [ x] = {yÎ X|xry}.
The equivalence classes form partition sets X, i.e., a system of non-empty pairwise disjoint subsets of it, the union of which coincides with the entire set X.
Example 2.19.
a) The equality relation on the set of integers generates the following equivalence classes: for any element x from this set [ x] = {x), i.e. each equivalence class consists of one element.
b) The equivalence class generated by the pair<x, y> is determined by the relation:
[<x, y>] = .
Each equivalence class generated by a pair<x, y>, defines one rational number.
c) For the relation of belonging to one student group, the equivalence class is the set of students of the same group.
Definition 2.14. Attitude r called antisymmetric on a set X, if for any x, yÎ X from xry And yr x should x = y.
From the definition of antisymmetry it follows that whenever a pair<x,y>owned at the same time r And r – 1 , the equality must be satisfied x = y. In other words, r Ç r – 1 consists only of pairs of the form<x,x >.
Example 2.20.
a) Let X– finite set, X= (1, 2, 3) and r = {<1, 1>, <1, 2>, <1, 3>, <2, 2>, <2, 3>, <3, 3>). Attitude r antisymmetric.
Attitude s= {<1, 1>, <1, 2>, <1, 3>, <2, 1>, <2, 3>, <3, 3>) is non-antisymmetric. For example,<1, 2> Î s, And<2, 1> Î s, but 1¹2.
b) Let X– set of real numbers and r ratio £ (less than or equal to). This relationship is antisymmetric, because If x £ y, And y £ x, That x = y.
Definition 2.15. Attitude r called partial order relation(or just a partial order) on the set X, if it is reflexive, antisymmetric and transitive on the set X. A bunch of X in this case it is called partially ordered and the specified relation is often denoted by the symbol £, if this does not lead to misunderstandings.
The inverse of the partial order relation will obviously be a partial order relation.
Example 2.21.
a) Let X– finite set, X= (1, 2, 3) and r = {<1, 1>, <1, 2>, <1, 3>, <2, 2>, <2, 3>, <3, 3>). Attitude r
b) Attitude AÍ IN on the set of subsets of some set U there is a partial order relation.
c) The divisibility relation on the set of natural numbers is a partial order relation.
Functions. Basic concepts and definitions
In mathematical analysis, the following definition of a function is accepted.
Variable y called a function of a variable x, if according to some rule or law each value x corresponds to one specific value y = f(x). Variable change area x is called the domain of definition of a function, and the domain of change of a variable y– range of function values. If one value x corresponds to several (and even infinitely many values) y), then the function is called multivalued. However, in the course on the analysis of functions of real variables, multi-valued functions are avoided and single-valued functions are considered.
Let's consider another definition of function in terms of relationships.
Definition 2.16. Function is any binary relation that does not contain two pairs with equal first components and different second ones.
This property of a relationship is called unambiguity or functionality.
Example 2.22.
A) (<1, 2>, <3, 4>, <4, 4>, <5, 6>) – function.
b) (<x, y>: x, y Î R, y = x 2) – function.
V) (<1, 2>, <1, 4>, <4, 4>, <5, 6>) is a relation, but not a function.
Definition 2.17. If f– function, then D f – domain, A R f – range functions f.
Example 2.23.
For example 2.22 a) D f – {1, 3, 4, 5}; R f – {2, 4, 6}.
For example 2.22 b) D f = R f = (–¥, ¥).
Each element x D f function matches the only one element y R f. This is denoted by the well-known notation y = f(x). Element x called function argument or element preimage y with function f, and the element y function value f on x or element image x at f.
So, from all relations, functions stand out in that each element from the domain of definition has the only one image.
Definition 2.18. If D f = X And R f = Y, then they say that the function f determined on X and takes its values at Y, A f called mapping the set X to Y(X ® Y).
Definition 2.19. Functions f And g are equal if their domain is the same set D, and for anyone x Î D equality is true f(x) = g(x).
This definition does not contradict the definition of equality of functions as equality of sets (after all, we defined a function as a relation, i.e., a set): sets f And g are equal if and only if they consist of the same elements.
Definition 2.20. Function (display) f called surjective or simply surjection, if for any element y Y there is an element x Î X, such that y = f(x).
So every function f is a surjective mapping (surjection) D f® R f.
If f is a surjection, and X And Y are finite sets, then ³ .
Definition 2.21. Function (display) f called injective or simply injection or one-to-one, if from f(a) = f(b) should a = b.
Definition 2.22. Function (display) f called bijective or simply bijection, if it is both injective and surjective.
If f is a bijection, and X And Y are finite sets, then = .
Definition 2.23. If the range of the function D f consists of one element, then f called constant function.
Example 2.24.
A) f(x) = x 2 is a mapping from the set of real numbers to the set of non-negative real numbers. Because f(–a) = f(a), And a ¹ – a, then this function is not an injection.
b) For everyone x R= (– , ) function f(x) = 5 – constant function. It displays many R to set (5). This function is surjective, but not injective.
V) f(x) = 2x+ 1 is an injection and a bijection, because out of 2 x 1 +1 = 2x 2 +1 follows x 1 = x 2 .
Definition 2.24. Function that implements the display X 1 X 2 ´...´ X n ® Y called n-local function.
Example 2.25.
a) Addition, subtraction, multiplication and division are two-place functions on a set R real numbers, i.e. functions like R 2® R.
b) f(x, y) = is a two-place function that implements the mapping R ´ ( R \ )® R. This function is not an injection, because f(1, 2) = f(2, 4).
c) The lottery winnings table specifies a two-place function that establishes a correspondence between pairs of N 2 (N– a set of natural numbers) and a set of winnings.
Since functions are binary relations, it is possible to find inverse functions and apply the composition operation. The composition of any two functions is a function, but not for every function f attitude f–1 is a function.
Example 2.26.
A) f = {<1, 2>, <2, 3>, <3, 4>, <4, 2>) – function.
Attitude f –1 = {<2, 1>, <3, 2>, <4, 3>, <2, 4>) is not a function.
b) g = {<1, a>, <2, b>, <3, c>, <4, D>) is a function.
g -1 = {<a, 1>, <b, 2>, <c, 3>, <D, 4>) is also a function.
c) Find the composition of functions f from example a) and g-1 from example b). We have g -1f = {<a, 2>, <b, 3>, <c, 4>, <d, 2>}.
fg-1 = Æ.
Notice, that ( g -1f)(a) = f(g -1 (a)) = f(1) = 2; (g -1f)(c) = f(g -1 (c)) = f(3) = 4.
An elementary function in mathematical analysis is any function f, which is a composition of a finite number of arithmetic functions, as well as the following functions:
1) Fractional-rational functions, i.e. functions of the form
a 0 + a 1 x + ... + a n x n
b 0 + b 1 x + ... + b m x m.
2) Power function f(x) = x m, Where m– any constant real number.
3) Exponential function f(x) = e x.
4) logarithmic function f(x) = log a x, a >0, a 1.
5) Trigonometric functions sin, cos, tg, ctg, sec, csc.
6) Hyperbolic functions sh, ch, th, cth.
7) Inverse trigonometric functions arcsin, arccos etc.
For example, the function log 2 (x 3 +sincos 3x) is elementary, because it is a composition of functions cosx, sinx, x 3 , x 1 + x 2 , logx, x 2 .
An expression describing the composition of functions is called a formula.
For a multiplace function, the following important result is valid, obtained by A. N. Kolmogorov and V. I. Arnold in 1957 and which is a solution to Hilbert’s 13th problem:
Theorem. Any continuous function n variables can be represented as a composition of continuous functions of two variables.
Methods for specifying functions
1. The simplest way to specify functions is through tables (Table 2.2):
Table 2.2
However, functions defined on finite sets can be defined in this way.
If a function defined on an infinite set (segment, interval) is given at a finite number of points, for example, in the form of trigonometric tables, tables of special functions, etc., then interpolation rules are used to calculate the values of functions at intermediate points.
2. A function can be specified as a formula that describes the function as a composition of other functions. The formula specifies the sequence for calculating the function.
Example 2.28.
f(x) = sin(x + Ö x) is a composition of the following functions:
g(y) = Ö y; h(u, v) = u+ v; w(z) = sinz.
3. The function can be specified as recursive procedure. The recursive procedure specifies a function defined on the set of natural numbers, i.e. f(n), n= 1, 2,... as follows: a) set the value f(1) (or f(0)); b) value f(n+ 1) determined through composition f(n) and other known functions. The simplest example of a recursive procedure is the calculation n!: a) 0! = 1; b) ( n + 1)! = n!(n+ 1). Many numerical methods procedures are recursive procedures.
4. There are possible ways to specify a function that do not contain a method for calculating the function, but only describe it. For example:
f M(x) =
Function f M(x) – characteristic function of the set M.
So, according to the meaning of our definition, set the function f– means to set the display X ® Y, i.e. define a set X´ Y, so the question comes down to specifying a certain set. However, it is possible to define the concept of a function without using the language of set theory, namely: a function is considered given if a computational procedure is given that, given the value of the argument, finds the corresponding value of the function. A function defined this way is called computable.
Example 2.29.
Determination procedure Fibonacci numbers, is given by the relation
Fn= Fn- 1 + Fn- 2 (n³ 2) (2.1)
with initial values F 0 = 1, F 1 = 1.
Formula (2.1) together with the initial values determines the following series of Fibonacci numbers:
| n | 0 1 2 3 4 5 6 7 8 9 10 11 … |
| Fn | 1 1 2 3 5 8 13 21 34 55 89 144 … |
The computational procedure for determining the value of a function from a given argument value is nothing more than algorithm.
Test questions for topic 2
1. Indicate ways to define a binary relation.
2. The main diagonal of the matrix of which relation contains only ones?
3. For what relationship? r the condition is always met r = r – 1 ?
4. For what attitude r the condition is always met r rÍ r.
5. Introduce equivalence relations and partial order on the set of all lines in the plane.
6. Specify ways to specify functions.
7. Which of the following statements is true?
a) Every binary relation is a function.
b) Every function is a binary relation.
Topic 3. GRAPHS
Euler's first work on graph theory appeared in 1736. In the beginning, this theory was associated with mathematical puzzles and games. However, subsequently graph theory began to be used in topology, algebra, and number theory. Nowadays, graph theory is used in a wide variety of areas of science, technology and practical activity. It is used in the design of electrical networks, transportation planning, and the construction of molecular circuits. Graph theory is also used in economics, psychology, sociology, and biology.
function ". Let's start with a special but important case of functions acting from to .
If we understand what a relation is, then understanding what a function is is quite simple. A function is a special case of a relation. Every function is a relation, but not every relation is a function. What relations are functions? What additional condition must be met for a relation to be a function?
Let's return to the consideration of the relation operating from the domain of definition to the domain of values. Consider an element from . This element corresponds to an element such that the pair belongs to , which is often written in the form: (for example, ). The relation may also contain other pairs, the first element of which may be the element . This situation is not possible for functions.
A function is a relation in which an element from the domain of definition corresponds to a single element from the domain of values.
The relation “having a brother”, presented in Fig. 1, is not a function. Two arcs go from a point in the domain of definition to different points in the domain of values, therefore this relation is not a function. Content-wise, Elena has two brothers, so there is no one-to-one correspondence between the element from and the element from.
If we consider the relation “to have an older brother” on the same sets, then such a relation is a function. Each person can have many brothers, but only one of them is the elder brother. Functions include such family relationships as “father” and “mother”.
Usually, when talking about functions, the letter , and not, as in the case of relations, is used to generally designate a function, and the general notation has the usual form: .
Consider the well-known function 
. The domain of definition of this function is the entire real axis: 
. The range of values of the function is a closed interval on the real axis: 
. The graph of this function is a sinusoid; each point on the axis corresponds to a single point on the graph .
One-to-one function
Let the relation define the function. What can be said about the reverse relationship? Is it also a function? Not at all necessary. Let's look at examples of relations that are functions.
For the relation “has an older brother,” the inverse relation is the relation “has a brother or sister.” Of course, this relationship is not a function. An older brother may have many sisters and brothers.
For the "father" and "mother" relationships, the inverse relationship is the "son or daughter" relationship, which is also not a function, since there can be many children.
If we consider the function 
, then the inverse relation 
is not a function, since one value corresponds to as many values as desired. To consider
As for the functions (from the Latin Functio - execution, implementation) of communication, they are understood as the external manifestation of the properties of communication, the roles and tasks that it performs in the process of an individual’s life in society.
There are various approaches to the classification of communication functions. Some researchers consider communication in the context of its organic unity with the life of society as a whole and with direct contacts of people and the inner spiritual life of a person.
The listed functions, taking into account their integral nature, are those factors that show a significantly more significant role of communication for a person than simply transmitting information. And knowledge of these integral functions that communication performs in the process of individual human development makes it possible to identify the causes of deviations, disruptions in the interaction process, defective structure and form of communication in which a person has been involved throughout his life. The inadequacy of a person’s forms of communication in the past significantly affects his personal development and determines the problems that confront him today.
The following functions are distinguished:
communication is a form of existence and manifestation of human essence, it plays a communicative and connecting role in the collective activities of people;
represents the most important vital need of a person, a condition for his prosperous existence, has a psychotherapeutic, confirmatory meaning (confirmation of one’s own “I” by another person) in the life of an individual of any age.
A significant part of researchers highlight the functions of communication related to the exchange of information, interaction and perception of each other by people.
Thus, B. Lomov identifies three functions in communication: information-communicative (consists in any exchange of information), regulatory-communicative (regulation of behavior and regulation of joint activities in the process of interaction, and affective-communicative (regulation of the emotional sphere of a person.
The information and communication function covers the processes of generating, transmitting and receiving information; its implementation has several levels: at the first level, differences in the initial awareness of people who come into psychological contact are equalized; the second level involves the transfer of information and decision-making (here communication realizes the goals of information, training, etc.); the third level is associated with a person’s desire to understand others (communication aimed at forming assessments of achieved results).
The second function - regulatory-communicative - is to regulate behavior. Thanks to communication, a person regulates not only his own behavior, but also the behavior of other people, and reacts to their actions, that is, a process of mutual adjustment of actions occurs.
Under such conditions, phenomena characteristic of joint activity appear, in particular, the compatibility of people, their teamwork, mutual stimulation and correction of behavior. This function is performed by such phenomena as imitation, suggestion, etc.
The third function - affective-communicative - characterizes the emotional sphere of a person, in which the individual’s attitude to the environment, including social, is revealed.
You can give another, slightly similar to the previous, classification - a four-element model (A. Rean), in which communication forms: cognitive-informational (reception and transmission of information), regulatory-behavioral (focuses attention on the characteristics of the behavior of subjects, on the mutual regulation of their actions ), affective-empathic (describes communication as a process of exchange and regulation at the emotional level) and social-perceptual components (the process of mutual perception, understanding and cognition of subjects).
A number of researchers are trying to expand the number of communication functions by clarifying them. In particular, A. Brudny distinguishes the instrumental function necessary for the exchange of information in the process of management and collaboration; syndicative, which is reflected in the cohesion of small and large groups; translational, necessary for training, transfer of knowledge, methods of activity, evaluation criteria; function of self-expression, focused on searching and achieving mutual understanding.
L. Karpenko, according to the “goal of communication” criterion, identifies eight more functions that are implemented in any interaction process and ensure the achievement of certain goals in it:
contact - establishing contact as a state of mutual readiness to receive and transmit messages and maintain communication during interaction in the form of constant mutual orientation;
informational - exchange of messages (information, opinions, decisions, plans, states), i.e. reception - transmission of what data in response to a request received from a partner;
incentive - stimulating the activity of the communication partner, which directs him to perform certain actions;
coordination - mutual orientation and coordination of actions to organize joint activities;
understanding - not only adequate perception and understanding of the essence of the message, but also the partners’ understanding of each other;
amotivational - inducing the necessary emotional experiences and states from a communication partner, changing one’s own experiences and states with his help;
establishing relationships - awareness and fixation of one’s place in the system of role, status, business, interpersonal and other connections in which the individual will act;
implementation of influence - a change in the state, behavior, personal and meaningful formations of the partner (aspirations, opinions, decisions, actions, activity needs, norms and standards of behavior, etc.).
Among the functions of communication, scientists also highlight social ones. The main one is related to the management of social and labor processes, the other is related to the establishment of human relations.
The formation of a community is another function of communication, which is aimed at supporting socio-psychological unity in groups and is associated with communicative activities (the essence of the activity is in creating and maintaining a specific relationship between people in groups); it allows for the information exchange of knowledge, relationships and feelings between people, i.e. .e. has the goal of transmitting and perceiving social experience by the individual. Among the social functions of communication, the functions of imitation of experience and personality change are important (the latter is carried out on the basis of mechanisms of perception, imitation, persuasion, infection).
Studying the specifics of socio-political activity allows us to identify the following main functions of communication in this area of knowledge (A. Derkach, N. Kuzmina):
Socio-psychological reflection. Communication arises as a result and as a form of conscious reflection by partners of the peculiarities of the course of interaction. The socio-psychological nature of this reflection is manifested in the fact that, first of all, through linguistic and other forms of signaling, elements of the interaction situation, perceived and processed by an individual, become really valid for his partners. Communication becomes less an exchange of information and more a process of joint interaction and influence. Depending on the nature of this mutual influence, coordination, clarification, mutual complementation of the substantive and quantitative aspects of the “individual” display occurs with the formation of group thought, as a form of collective thinking of people, or, conversely, a clash of opinions, their neutralization, containment, as happens in interpersonal conflicts and inadequate mutual influences (cessation of communication);
Regulatory. In the process of communication, direct or indirect influence is exerted on a group member in order to change or maintain at the same level his behavior, actions, state, general activity, characteristics of perception, value system and relationships. The regulatory function allows you to organize joint actions, plan and coordinate, coordinate and optimize group interaction of team members. Regulation of behavior and activity is the goal of interpersonal communication as a component of objective activity and its final result. It is the implementation of this important function of communication that allows us to evaluate the effect of communication, its productivity or unproductivity;
Cognitive. The named function is that as a result of systematic contacts in the course of joint activities, group members acquire various knowledge about themselves, their friends, and ways to most rationally solve the tasks assigned to them. Mastering the relevant skills and abilities, it is possible to compensate for insufficient knowledge of individual group members and their achievement of the necessary mutual understanding is ensured precisely by the cognitive function of communication in combination with the function of socio-psychological reflection;
Expressive. Various forms of verbal and nonverbal communication are indicators of the emotional state and experience of a group member, often contrary to the logic and requirements of joint activity. This is a kind of manifestation of one’s attitude to what is happening through an appeal to another member of the group. Sometimes a discrepancy in the methods of emotional regulation can lead to alienation of partners, disruption of their interpersonal relationships and even conflicts;
Social control. Methods for solving problems, certain forms of behavior, emotional reactions and relationships are normative in nature; their regulation through group and social norms ensures the necessary integrity and organization of the team, the consistency of joint actions. Various forms of social control are used to maintain consistency and organization in group activities. Interpersonal communication mainly acts as negative (condemnation) or positive (approval) sanctions. It should be noted, however, that not only approval or condemnation is perceived by participants in joint activities as punishment or reward. Often, the lack of communication can be perceived as one or another sanction;
Socialization. This function is one of the most important in the work of the subjects of activity. By engaging in joint activities and communication, group members master communication skills, which allows them to interact effectively with other people. Although the ability to quickly assess an interlocutor, navigate situations of communication and interaction, listen and speak play an important role in a person’s interpersonal adaptation, the ability to act in the interests of the group, a friendly, interested and patient attitude towards other group members are even more important.
An analysis of the features of communication in the field of business relationships also indicates its multifunctionality (A. Panfilova, E. Rudensky):
the instrumental function characterizes communication as a social control mechanism, which makes it possible to receive and transmit information necessary to carry out a certain action, make a decision, etc.;
integrative - used as a means of uniting business partners for a joint communication process;
the function of self-expression helps to assert oneself, demonstrate personal intelligence and psychological potential;
broadcast - serves to convey specific methods of activity, assessments, opinions, etc.;
the function of social control is designed to regulate the behavior, activities, and sometimes (when it comes to trade secrets) the language actions of participants in business interaction;
the socialization function contributes to the development of business communication culture skills; With the help of the expressive function, business partners try to express and understand each other’s emotional experiences.
V. Panferov believes that the main functions of communication are often characterized without resorting to an analysis of the functions of a person as a subject of interaction with other people in joint life activities, which leads to the loss of the objective basis for their classification. Analyzing the classification of communication functions proposed by B. Lomov, the researcher poses the question: “Are the series of functions exhaustive in terms of their number? How many such rows can there be? What main classification can we talk about? How are the different bases related to each other?
Taking this opportunity, let us recall that B. Lomov identified two series of communication functions with different bases. The first of them includes three classes of already known functions - information-communicative, regulatory-communicative and affective-communicative, and the second (according to a different system of bases) - covers the organization of joint activities, people’s knowledge of each other, the formation and development of interpersonal relationships.
Answering the first question posed, V. Panferov identifies six among the main functions of communication: communicative, informational, cognitive (cognitive), emotive (that which causes emotional experiences), conative (regulation, coordination of interaction), creative (transformative).
All of the above functions are transformed into one main function of communication - regulatory, which manifests itself in the interaction of an individual with other people. And in this sense, communication is a mechanism of social-psychological regulation of people’s behavior in their joint activities. The identified functions, according to the researcher, should be considered as one of the grounds for classifying all other functions of a person as a subject of communication.
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.
Let r Í X X Y.
Functional relation- this is such a binary relationship r, in which each element corresponds exactly one such that the pair belongs to the relation or such doesn't exist at all: or.
Functional relation – it's such a binary relationship r, for which the following is executed: .
Everywhere a certain attitude– binary relation r, for which D r =X("there are no lonely X").
Surjective relation– binary relation r, for which J r = Y("there are no lonely y").
Injective attitude– a binary relation in which different X correspond different at.
Bijection– functional, everywhere defined, injective, surjective relation, defines a one-to-one correspondence of sets.
For example:
Let r= ( (x, y) О R 2 | y 2 + x 2 = 1, y > 0 ).
Attitude r- functional,
not defined everywhere ("there are lonely X"),
not injective (there are different X, at),
not surjective ("there are lonely at"),
not a bijection.
For example:
Let Ã= ((x,y) О R 2 | y = x+1)
The relation à is functional,
The relation Ã- is defined everywhere (“there are no lonely X"),
The relation Ã- is injective (there are no different X, which correspond to the same at),
The relation Ã- is surjective (“there are no lonely at"),
The relation à is bijective, mutually homogeneous correspondence.
For example:
Let j=((1,2), (2,3), (1,3), (3,4), (2,4), (1,4)) be defined on the set N 4.
The relation j is not functional, x=1 corresponds to three y: (1,2), (1,3), (1,4)
Relation j is not definite everywhere D j =(1,2,3)¹ N 4
The relation j is not surjective I j =(1,2,3)¹ N 4
The relation j is not injective; different x correspond to the same y, for example (2.3) and (1.3).
Laboratory assignment
1. Sets are given N1 And N2. Calculate sets:
(N1 X N2) Ç (N2 X N1);
(N1 X N2) È (N2 X N1);
(N1 Ç N2) x (N1 Ç N2);
(N1 È N2) x (N1 È N2),
Where N1 = ( digits of the record book number, the last three };
N2 = ( digits of date and month of birth }.
2. Relationships r And g are given on the set N 6 =(1,2,3,4,5,6).
Describe the relationship r,g,r -1 , r ○g, r - 1 ○g list of pairs
Find relationship matrices r And g.
For each relationship, determine the domain of definition and the domain of values.
Determine the properties of relationships.
Identify equivalence relations and construct equivalence classes.
Identify order relations and classify them.
1) r= { (m,n) | m > n)
g= { (m,n) | comparison modulo 2 }
2) r= { (m,n) | (m - n) divisible by 2 }
g= { (m,n) | m divider n)
3) r= { (m,n) | m< n }
g= { (m,n) | comparison modulo 3 }
4) r= { (m,n) | (m + n)- even }
g= { (m,n) | m 2 =n)
5) r= { (m,n) | m/n- degree 2 }
g= { (m,n) | m = n)
6) r= { (m,n) | m/n- even }
g = ((m,n) | m³ n)
7) r= { (m,n) | m/n- odd }
g= { (m,n) | comparison modulo 4 }
8) r= { (m,n) | m * n - even }
g= { (m,n) | m£ n)
9) r= { (m,n) | comparison modulo 5 }
g= { (m,n) | m divided by n)
10) r= { (m,n) | m- even, n- even }
g= { (m,n) | m divider n)
11) r= { (m,n) | m = n)
g= { (m,n) | (m + n)£ 5 }
12) r={ (m,n) | m And n have the same remainder when divided by 3 }
g= { (m,n) | (m-n)³2 }
13) r= { (m,n) | (m + n) is divisible by 2 }
g = ((m,n) | £2 (m-n)£4 }
14) r= { (m,n) | (m + n) divisible by 3 }
g= { (m,n) | m¹ n)
15) r= { (m,n) | m And n have a common divisor }
g= { (m,n) | m 2£ n)
16) r= { (m,n) | (m - n) is divisible by 2 }
g= { (m,n) | m< n +2 }
17) r= { (m,n) | comparison modulo 4 }
g= { (m,n) | m£ n)
18) r= { (m,n) | m divisible by n)
g= { (m,n) | m¹ n, m- even }
19) r= { (m,n) | comparison modulo 3 }
g= { (m,n) | £1 (m-n)£3 }
20) r= { (m,n) | (m - n) divisible by 4 }
g= { (m,n) | m¹ n)
21) r= { (m,n) | m- odd, n- odd }
g= { (m,n) | m£ n, n- even }
22) r= { (m,n) | m And n have an odd remainder when divided by 3 }
g= { (m,n) | (m-n)³1 }
23) r= { (m,n) | m * n - odd }
g= { (m,n) | comparison modulo 2 }
24) r= { (m,n) | m * n - even }
g= { (m,n) | £1 (m-n)£3 }
25) r= { (m,n) | (m+ n) - even }
g= { (m,n) | m is not completely divisible n)
26) r= { (m,n) | m = n)
g= { (m,n) | m divisible by n)
27) r= { (m,n) | (m-n)- even }
g= { (m,n) | m divider n)
28) r= { (m,n) | (m-n)³2 }
g= { (m,n) | m divisible by n)
29) r= { (m,n) | m 2³ n)
g= { (m,n) | m / n- odd }
30) r= { (m,n) | m³ n, m - even }
g= { (m,n) | m And n have a common divisor other than 1 }
3. Determine whether the given relation is f- functional, everywhere defined, injective, surjective, bijection ( R- set of real numbers). Construct a relationship graph, determine the domain of definition and the range of values.
Do the same task for relationships r And g from point 3 of the laboratory work.
1) f=( (x, y) Î R 2 | y=1/x +7x )
2) f=( (x, y) Î R 2 | x³ y)
3) f=( (x, y) Î R 2 | y³ x)
4) f=( (x, y) Î R 2 | y³ x, x³ 0 }
5) f=( (x, y) Î R 2 | y 2 + x 2 = 1)
6) f=( (x, y) Î R 2 | 2 | y | + | x | = 1)
7) f=( (x, y) Î R 2 | x+y£ 1 }
8) f=( (x, y) Î R 2 | x = y 2 )
9) f=( (x, y) Î R 2 | y = x 3 + 1)
10) f=( (x, y) Î R 2 | y = -x 2 )
11) f=( (x, y) Î R 2 | | y | + | x | = 1)
12) f=( (x, y) Î R 2 | x = y -2 )
13) f=( (x, y) Î R 2 | y2 + x2³ 1, y> 0 }
14) f=( (x, y) Î R 2 | y 2 + x 2 = 1, x> 0 }
15) f=( (x, y) Î R 2 | y2 + x2£ 1.x> 0 }
16) f=( (x, y) Î R 2 | x = y 2 ,x³ 0 }
17) f=( (x, y) Î R 2 | y = sin(3x + p) )
18) f=( (x, y) Î R 2 | y = 1 /cos x )
19) f=( (x, y) Î R 2 | y = 2| x | + 3)
20) f=( (x, y) Î R 2 | y = | 2x + 1| )
21) f=( (x, y) Î R 2 | y = 3x)
22) f=( (x, y) Î R 2 | y = e -x )
23) f =( (x, y)Î R 2 | y = e | x | )
24) f=( (x, y) Î R 2 | y = cos(3x) - 2 )
25) f=( (x, y) Î R 2 | y = 3x 2 - 2 )
26) f=( (x, y) Î R 2 | y = 1 / (x + 2) )
27) f=( (x, y) Î R 2 | y = ln(2x) - 2 )
28) f=( (x, y) Î R 2 | y = | 4x -1| + 2)
29) f=( (x, y) Î R 2 | y = 1 / (x 2 +2x-5))
30) f=( (x, y) Î R 2 | x = y 3, y³ - 2 }.
Control questions
2. Definition of a binary relation.
3. Methods of describing binary relations.
4.Domain of definition and range of values.
5.Properties of binary relations.
6.Equivalence relations and equivalence classes.
7. Relations of order: strict and non-strict, complete and partial.
8. Classes of residues modulo m.
9.Functional relationships.
10. Injection, surjection, bijection.
Laboratory work No. 3