Not related to relational functions. Section i

Exercises.

1) Using Newton's binomial formula for a = 1, b = i calculate +++…, +++…, +++…, +++…

2) Using the Moivre formula, calculate orally sin 4j and cos 5j .

Lecture 3

1. COMPLIANCE. FUNCTIONS. RELATIONS. EQUIVALENCE RATIO

Definition. We will say that on the set X given binary relation R, if " x, y О X we can determine (by some rule) these elements are in relation R or not.

Let us define the notion of relation more strictly.

We introduce the concept Cartesian (direct) product A´B arbitrary sets A and b.

A-priory A´B = ( (a, b), a О A , bО B). The Cartesian product of 3, 4 and an arbitrary number of sets is defined similarly. A-priory A´A´ …´A = A n .

Definitions.

1. S from many A into the multitude B called a subset S Í A´B. The fact that the elements aО A, bО B are in line S, we will write in the form (a, b) О S or in the form aSb.

2. In a natural way for correspondences S1 and S2 determined S1∩S2 and S 1 U S 2– as the intersection and union of subsets. As for any subsets, the notion of inclusion of correspondences is defined S1 Í S2. So S1 Í S2 Û

from a S 1 b Þ a S 2 b.

3. For matches S 1 H A´B and S 2 H B´C define composition correspondences S 1 *S 2 Í A´С. We will assume that for the elements aО A, сО С a-priory a S 1 *S 2 with Û $ bО B such that a S 1 b and b S 2 s.

4. To match S Í A´B define correspondence

S -1 Í B´A so: by definition bS -1 a Û a S b.

5. Let, by definition, the correspondence D A Í A´A,

D A =((a,a), aн A).

6. Compliance F from many A into the multitude B called function, determined on A, with values ​​in B(or mapping from A in B), if " aО A $! bО B such that aFb. In this case, we will also write aF = b or, more commonly, Fa = b. In this definition, a function is identified with its graph. In our notation aF 1 *F 2 s can be written in the form c \u003d (aF 1)F 2. Composition F 2 F 1 functions means by definition that (F 2 F 1)(a)= F 2 (F 1 (a)). Thus, F 2 F 1 \u003d F 1 * F 2.

7. For display F from A in B way subsets A 1 H A

called a subset F(A 1)= (F(a)| aн A 1 ) Н B, a prototype subsets B 1 H B called a subset

F -1 (B 1)= ( aн A | F(a) О B 1 ) Н A .

8. Display F from A in B called injection if from

a 1 ¹ a 2 Þ Fa 1 ¹ Fa 2.

9. Display F from A in B called surjection, if

" bО B $ aО A such that Fa = b.

10. Display F from A in B called bijection or one-to-one mapping, if F– injection and surjection at the same time.

11. A bijection of a finite (and sometimes infinite) set is called substitution.

12. binary relation on the set X called a subset R Í X´X. The fact that the elements x, y О X are in relation R, we will write in the form (x, y) О R or in the form xRy.

Human beings have an inherent need for communication and interaction with other people. Satisfying this need, he manifests and realizes his capabilities.

Human life throughout its duration is manifested primarily in communication. And all the diversity of life is reflected in an equally endless variety of communication: in the family, at school, at work, at home, 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 need for joint activities.

Communication performs a number of basic functions:

• Information - the function of receiving, 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 excitation in the partner of the necessary emotions, experiences, feelings, involves emotional exchange, a change in the emotional state;
• The function of establishing relationships is the awareness and fixation of one's social status, social role in a particular social community.
• The function of exerting influence is a change in state, behavior, intentions, ideas, attitudes, opinions, decisions, needs, actions, etc.

Along with the functions, the main kinds communication.

By number of participants:

• interpersonal;
• group.

By way of communication:

• verbal;
• non-verbal.

According to the position of the speakers:

• contact;
• distant.

According to the terms of communication:

• official;
• unofficial.

AT structure There are three closely interrelated, interdependent aspects of communication:

• The perceptual side of communication is the process of perception of each other.
• The communicative side of communication involves the transfer of information. At the same time, it must be taken 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 (coherence 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 falls out, the effectiveness of communication is sharply reduced and there is a possibility of not achieving the goals that were set when organizing communication. The ability to effectively achieve 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 called a set consisting of ordered pairs: A´ B= {(a,b): (aÎ A) & (bÎ B)}.

For sets A 1, …, A n the Cartesian product is defined 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® A i , what is for everyone iÎ I right f(i)Î A i .

Theorem 1

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

Proof

Let be A = (a 1 , …,am), 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

With the help of induction on n. Let the formula be true for n. Then

Relations

Let be n³1 is a positive integer and A 1, …, A n are arbitrary sets. The relationship between the elements of the 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 mapping is called a triple consisting of sets A and B and subsets fÍ A´ B(function graph) satisfying the following two conditions;

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

2) if (x,y)Î f and (x,z)Î f, then y=z.

It is easy to see that fÍ A´ B will define a function if and only if 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', takes place 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

For a function to be a bijection, it is necessary and sufficient that there exists a function such that fg =ID B and gf =ID A.

Proof

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

By function construction g, there are equalities f(g(y)) = y and g(f(x)) = x. So that's right fg =ID B and gf =ID A. The converse is obvious: if fg =ID B and gf =ID A, then f– surjection into force f(g(y)) = y, for everybody yÎ B. In this case, from will follow , which means . Hence, f- injection. Hence it follows that f- bijection.

Image and prototype

Let be a function. way 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.

Relationships and Graphs

Binary relationships can be visualized using directed graphs.

Definition 2

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

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 be 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 arrive at the following definition:

Definition 3

Simple (undirected) graph G = (V,e) is called a pair consisting of a set V and many E, consisting of some unordered pairs ( v 1 ,v2) elements v 1 ,v2Î V such that v1¹ v2. These couples are called ribs, and elements from V – peaks.

Rice. 1.3. Simple undirected graph K 4

A bunch of E defines a binary symmetric antireflexive relation consisting of pairs ( v 1 ,v2), for which ( v 1 ,v2} Î E. The vertices of a simple graph are shown as points, and the edges as line segments. On 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 called an arbitrary subset RÍ A´ B. Recording aRb(at aÎ A, bÎ B) means that (a,b)Î R.

The following relational operations 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);

· R n =R°(Rn-1);

Let be Id A = ((a,a):aÎ A)- identical relationship. Attitude R Í X´ X called:

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

2) anti-reflexive, if (a,a)Ï R for all aÎ X;

3) symmetrical if for all a,bÎ X the implication is correct aRbÞ bRa;

4) antisymmetric, if aRb &bRaÞ a=b;

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

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

Denote ID A through ID. It is easy to see that the following holds.

Suggestion 1

Attitude RÍ X´ X:

1) reflexively Û IDÍ R;

2) antireflexively Û RÇ Id=Æ ;

3) symmetrically Û R=R-1;

4) antisymmetric Û RÇ R-1Í ID;

5) transitively Û R° RÍ R;

6) linearly Û RÈ IDÈ R-1 = X´ X.

binary relation matrix

Let be 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 be A is a finite set, | A| = n and B= A. Consider the algorithm for calculating the composition matrix T= R° S relations R, S Í A´ A. Denote the coefficients of the relationship matrices R, S and T respectively through rij, sij and tij.

Since the property ( a i,a k)Î T is tantamount 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 rij= 1 and sjk= 1. In other cases tik equals 0. Therefore, tik= 1 if and only if .

This implies that to find the composition matrix of relations, it is necessary to multiply these matrices and, in the resulting product of matrices, replace non-zero coefficients with units. The following example shows how the composition matrix is ​​calculated in this way.

Example 2

Consider a binary relation on A = (1,2,3) equal to R = ((1,2),(2,3)). Let's write the relation matrix R. By definition, it consists of the coefficients r 12 = 1, r23 = 1 and others rij= 0. Hence the relation matrix R is equal to:

Let's find the relation R° R. To this end, we multiply the ratio matrix R to myself:

.

We get the relation matrix:

Hence, R° R= {(1,2),(1,3),(2,3)}.

Proposition 1 implies the following corollary.

Consequence 2

If a A= B, then the ratio R on the A:

1) reflexively if and only if all elements of the main diagonal of the relation matrix R are equal to 1;

2) antireflexively if and only if all elements of the main diagonal of the relation matrix R are 0;

3) symmetrical if and only if the relation matrix R symmetrical;

4) transitively if and only if each coefficient of the relation matrix R° R not greater than the corresponding coefficient of the ratio matrix R.

function ". Let's start with a particular but important case of functions acting from to .

If we understand what a relation is, then it is quite easy to understand what a function is. A function is a special case of a relation. Every function is a relation, but not every relation is a function. What relationships are functions? What additional condition must be met for a relation to be a function?

Let us return to the consideration of the relation acting 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 as: (for example, ). Other pairs can also belong to the relation, the first element of which can be the element . For functions, this situation is impossible.

A function is a relationship in which an element from the domain of definition corresponds to a single element from the domain of values.

The relationship "to have a brother", shown in Figure 1, is not a function. From a point in the domain of definition, two arcs go to different points in the domain of values, therefore this relation is not a function. Essentially, 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 are also such kindred relations as "father" and "mother".

Usually, when it comes to functions, the letter is used for the general designation of the function, and not, as in the case of relations, and the general notation has the usual form: .

Consider the well-known function . The scope of this function is the entire real axis: . The range 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? Is it also a function? Not at all necessary. Consider examples of relations that are functions.

For the "has an older brother" relationship, the inverse relationship is the "has a brother or sister" relationship. Of course, this relation is not a function. An older brother may have many sisters and brothers.

For the relationship "father" and "mother", the inverse relationship is the relationship "son or daughter", 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 arbitrarily many values. To consider

Essence and classification of economic relations

From the moment of its separation from the world of wild nature, a person develops as a biosocial being. This determines the conditions for its development and formation. Needs are the main stimulus for the development of man and society. To meet 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 difficult the labor process became. It required more and more resources and more and more coordinated actions of all members of society. Thanks to labor, both the main features of the external appearance of modern man and the features of man as a social being were formed. Labor has entered the phase of economic activity.

Economic activity is called human activity in the creation, redistribution, exchange and use of material and spiritual wealth.

Economic activity is associated with the need to enter into some kind of relationship between all participants in this process. These relationships are called economic.

Definition 1

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

These relationships have different forms and duration. Therefore, there are several options for their classification. It all depends on the chosen criterion. The criterion can be time, periodicity (regularity), degree of benefit, characteristics of the participants in these relations, etc. the following types of economic relations are most often mentioned:

• international and domestic;
• mutually beneficial and discriminatory (beneficial to one side and infringing on the interests of the other);
• voluntary and compulsory;
• 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 happen simultaneously in several guises. 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 features of 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 at all levels and types is the 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 and economic relations are called relations to create conditions for the most efficient use of resources and reduce costs through the organization of production forms.

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 (enlargement) of production, combination (combination of production of different industries at one enterprise), specialization and cooperation (to increase productivity). The final form of organizational and economic relations is the formation of territorial-productive complexes. An additional economic effect is obtained due to the successful territorial location of enterprises and the rational use of infrastructure.

Soviet Russian economists and economic geographers in the middle of the $20th century developed the theory of energy production cycles (EPC). They proposed to organize production processes in a certain territory in such a way as to use a single flow of raw materials and energy for the production of a whole range of products. This would dramatically reduce the cost of production and reduce waste production. Organizational and economic relations are directly related to the management of the economy.

Functions of socio-economic relations

Definition 3

Socio-economic relations are called relations between economic agents, which are based on the right of ownership.

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

The function of socio-economic relations is the regulation of 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 hand, on the basis of volitional property relations. These interactions between the two parties take the form of both moral norms and legislative (legally enshrined).

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 activity of the countries of 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 at the present time 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 on the basis of 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 decrease in social tension in the world economy. Management of the world economy is aimed at reducing contradictions between national economies and reducing the impact of global inflationary and crisis phenomena.