Does not apply to relational functions. Section i. sets, functions, relations. Functions of international economic relations
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.
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:
.
We get the relation matrix:
Hence, R° R= {(1,2),(1,3),(2,3)}.
The following corollary follows from Proposition 1.
Corollary 2
If A= B, then the relation R on A:
1) reflexively if and only if all elements of the main diagonal of the relation matrix R equal to 1;
2) anti-reflexive if and only if all elements of the main diagonal of the relation matrix R equal to 0;
3) symmetric if and only if the relation matrix R symmetrical;
4) transitive if and only if each coefficient of the relation matrix R° R no more than the corresponding ratio matrix coefficient R.
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.
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
Any set of 2-lists or pairs is called a relation. Relationships will be especially helpful when discussing the meaning of programs.
The word "relation" can mean a comparison rule, "equivalence" or "is a subset", etc. Formally, relations, which are sets of 2-lists, can describe these informal rules precisely by including exactly those pairs whose elements are in the desired relationship with each other. For example, the relationship between characters and 1-strings containing these characters is given by the following relationship:
C = (
Since a relation is a set, an empty relation is also possible. For example, the correspondence between even natural numbers and their odd squares does not exist. Moreover, set operations apply to relations. If s and r are relations, then there are
s È r, s – r, s Ç r
since these are sets of ordered pairs of elements.
A special case of a relation is a function, a relation with a special property, characterized in that each first element is paired with a unique second element. The relation r is a function if and only if for any
In this case, each first element can serve as a name for the second in the context of the relationship. For example, the C relation between characters and 1-strings described above is a function.
Set operations also apply to functions. Although the result of an operation on sets of ordered pairs that are functions will necessarily be another set of ordered pairs, and therefore a relation, it is not always a function.
If f, g are functions, then f Ç g, f – g are also functions, but f È g may or may not be a function. For example, let's define the relation head
H = (< Θ y, y>: y - string) = (
And take the relation C described above. Then from the fact that C Í H:
is a function
H - C = (< Θ y, y>: y – string of at least 2 characters)
is a relation, but not a function,
is an empty function, and
is a relation.
The set of the first elements of pairs of a relation or function is called the domain of definition, and the set of the second elements of the pairs is called the range. For relation elements, say
When
reads "y is the r value of x" or, more briefly, "y is the r value of x" (functional form).
Let's set an arbitrary relation r and argument x, then there are three options for their correspondence:
- x Р domain(r), in this case r undefined by x
- x О domain(r), and there are different y, z such that
- x О domain(r), and there is a unique pair
Thus, a function is a relation that is uniquely defined for all elements of its domain of definition.
There are three special functions:
Empty function(), has no arguments or values, that is
domain(()) = (), range(()) = ()
Identity function, function I is,
that if x О domain(r), then I(x) = x.
Constant function, the range of values of which is specified by a 1-set, that is, all arguments correspond to the same value.
Since relations and functions are sets, they can be described by listing elements or specifying rules. For example:
r = (<†ball†, †bat†>, <†ball†, †game†>, <†game†, †ball†>}
is a relation since all its elements are 2-lists
domain(r) = (†ball†, †game†)
range (r) = (†ball†, †game†, †bat†)
However, r is not a function because two different values are paired with the same argument †ball†.
An example of a relationship defined using a rule:
s = (
in the line †this is a relation that is not a function†)
This relationship can also be specified by an enumeration:
s = (<†this†, †is†>, <†is†, †a†>, <†a†, †relation†>, <†relation†, †that†>,
<†that†, †is†>, <†is†, †not†>, <†not†, †a†>, <†a†, †function†>}
The following rule defines the function:
f = (
in the line †this is a relation that is also a function†)
which can also be specified by an enumeration:
f = (<†this†, †is†>, <†is†, †a†>, <†a†, †relation†>,
<†relation†, †that†>, <†that†, †is†>, <†also†, †a†>}
The meaning of programs.
Relationships and functions are vital to descriptions to describe the meaning of programs. Using these concepts, a notation is developed to describe the meaning of programs. For simple programs the meaning will be obvious, but these simple examples will serve to master the theory as a whole.
New ideas: box notation, program and program meaning.
The set of input-output pairs for all possible normal executions of a program is called the program value. The concepts can also be used program function And program attitude. It is important to distinguish between the meaning of a program and the elements of meaning. For a specific input, a Pascal machine controlled by a Pascal program can produce a specific output. But the meaning of a program is much more than a way of expressing the result of one particular execution. It expresses all possible execution of a Pascal program on a Pascal machine.
A program can take input broken into lines and produce output broken into lines. Thus, pairs in a program value can be pairs of lists of character strings.
Box notation.
Any Pascal program is a string of characters passed to the Pascal machine for processing. For example:
P = †PROGRAM PrintHello(INPUT, OUTPUT); BEGIN WRITELN('HELLO') END.†
Represents one of the first programs discussed at the beginning of Part I as a string.
You can also write this line by omitting the line markers, like
P = PROGRAM PrintHello(INPUT, OUTPUT);
WRITELN('HELLO')
The string P represents the syntax of the program, and we will write its value as P. The value of P is a set of 2-lists (ordered pairs) of lists of character strings in which the arguments represent the inputs of the program and the values represent the outputs of the program, that is
P = (
and returns a list of strings M)
Box notation for program meaning retains the syntax and semantics of the program, but clearly distinguishes one from the other. For the PrintHello program above:
P = (
{
Putting the program text in box:
P = PROGRAM PrintHello(INPUT, OUTPUT); BEGIN WRITELN('HELLO') END
Since P is a function,
PROGRAM PrintHello(INPUT, OUTPUT); BEGIN WRITELN('HELLO') END (L) =<†HELLO†>
for any list of strings L.
Box notation hides the way the program controls the Pascal machine and shows only what accompanies execution. The term "black box" is often used to describe a mechanism viewed only from the outside in terms of inputs and outputs. Thus, this notation is suitable for the meaning of a program in terms of input/output. For example, the R program
PROGRAM PrintHelloInSteps(INPUT, OUTPUT);
WRITE('HE');
WRITE('L');
WRITELN('LO')
Has the same meaning as P, that is, R = P.
The R program also has a CFPascal name PrintHelloInSteps. But since the string †PrintHelloInSteps† is part of an R string, it is better not to use PrintHelloInSteps as the name of an R program in box notation.
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