The close connection of elements in a system is determined by physical, or rather, natural relations between them, or other fundamental properties of the system, for example, economic, social, characterizing the development of human society.

The depth of such connections depends on the level of the system in the hierarchy of systems related to subject area existence of the subject being studied complex object. Connections include both general relations between the elements of nature and society that make up the system, and private ones relating to a certain limited range of its elements. In connection with the above, these connections are called either general laws nature (fundamental) or private, relating to a limited set of phenomena (empirical laws) or to trends that manifest themselves in the form of some repetitions in mass phenomena and called regularities.

Fundamental connections are called laws. Law is a philosophical category that has the properties of universality in relation to all natural objects, phenomena, events. In this regard, the definition of the law is as follows: a law is an essential, stable, repeating relationship between any phenomena.

The law expresses a certain connection between the systems themselves, constituent elements associations of objects and phenomena, as well as within the objects and phenomena themselves.

Not every connection is law. It can be necessary and accidental, the Law is a necessary connection. It expresses the essential connection between things coexisting in space (material formations, in the general sense).

Everything said above applies to laws of operation(existence natural environment or artificially created by man). There are also laws of development, expressing the trend, direction or order of events in time. All natural laws- are not made by human hands, they exist in the world objectively and express the relationships of things, and are also reflected in human consciousness.

As already mentioned, laws are divided according to the degree of generality. Universal laws are philosophical laws. The fundamental laws of nature, in their generality, are also divided into two large classes. To more general ones, studied by a number, or even an absolute variety of sciences (these include, for example, the laws of conservation of energy and information, etc.). And less general laws, which extend to limited areas, studied by specific sciences (physics, chemistry, biology).

Empirical laws are studied by special sciences, which include all technical sciences. As an example, we can take the discipline of strength of materials. It studies subjects and systems in which all the fundamental laws and empirical laws apply, based on experimental data that relate to the subjects of the discipline only those mechanical bodies, which obey Hooke’s law: the deformation of a body is directly proportional to the force acting on the body (and vice versa).

IN technical sciences there are sections that are based on more specific empirical connections, accepted as axioms.

Some laws express a strict quantitative dependence and are fixed mathematical formulas, while others are not yet amenable to formalization, indicating the obligatory nature of one type of event due to the occurrence of another, for example.

Some laws - determined, that is - that is, they are established on the basis of causality - investigative connections exact quantitative relationships, others - statistical, establishing the probability of the occurrence of an event under certain conditions.

In nature, laws act as a spontaneous force. However, knowing the laws, they can be used purposefully in practical activities(like the force of steam pressure in steam engines, like the force of compressed gas in internal combustion engines).

Social-historical laws are not much different from the laws of nature, but they operate between thinking people. Knowledge of these laws helps better organization economy and society.

Thus, the study of the laws of nature and society is the primary task of humanity. Only knowledge of the laws and the development of measures for their correct use can provide developing and growing humanity with food and the environment of artificially created conditions in which it can exist.

The speed of solving new problems that arise depends on how much reserve scientific knowledge people saved up for this moment and how it was processed and comprehended. Understanding scientific knowledge leads to the formulation scientific problem, the solution of which can lead to the completion of the theory on this range of issues and the use of more rigorous conclusions in practical matters. Scientific problem- not only a philosophical category in the described sense, but also a practical one, on which depends how theoretical science, as well as its practical implementation in people’s lives.

From this explanatory part of the significance of a scientific problem for the completeness of a theory, its definition also follows: a scientific problem is a contradictory situation that appears in the form of opposing positions in the explanation of any phenomena, objects, processes and requires an adequate single theory to resolve it.

An important prerequisite for its successful solution is its correct positioning. See the contradictions in the received empirical knowledge, to pay attention to them and raise the question of eliminating this contradiction means to begin solving a scientific problem and advancing science towards progress. It is not without reason that in science, people who are able to formulate problems are revered even more than researchers who have specifically solved the formulated problem. Formulation of the wrong problems leads to great stagnation in science.

The category “scientific problem” is directly related to the category "hypothesis". Hypotheses, first of all, are used to theoretically eliminate the contradictions of a scientific problem. Such hypotheses (assumptions), if successful, even turn into fundamental theories (Newton’s assumption about the force of attraction between two physical bodies).

Hypotheses are also used in technical sciences, where they are of a particular nature and represent a description of the method of interaction of factors that determine the behavior of the object being studied and its elements. In this case, the hypothesis is called working hypothesis, which, as in scientific problem, can be proven or rejected on the basis of experimental data.

Therefore, a hypothesis is an assumption about a probable (possible) pattern of change in a phenomenon, object, event that has not been proven, but seems probable.

The usefulness of the hypothesis is that it mobilizes researchers to formulate problems experimental work in order to prove the correctness of the stated hypothesis. And if a different result is obtained, then the accumulated material will allow us to correct the hypothesis and plan further scientific research work.

In a more general formulation, modeling as a method of scientific methodology consists in the transition from informally meaningful ideas about the object being studied to the use of mathematical models.

The theoretical level of models obtained on the basis of axioms, rules for deriving theorems, and rules of correspondence is further increased on the basis of hypotico-deductive provisions with the formulation of consequences obtained by analyzing the hypotheses put forward. The mathematical apparatus used in this case is only a means of obtaining new knowledge and in no way final goal methodological analysis.

For compilation mathematical model its use follows, the purpose of which is to obtain information that was missing before its creation, i.e. the resulting model must be heuristic. It is this action that turns the methodology into experimental science, allowing verification of its conclusions in practice.

Model and its properties.

Formalization existing knowledge about the system under study (by the compiler of the model) creates a model in order to obtain the necessary properties of the system: consistency; completeness; independence of the axiom system; content. A good example the fulfillment of these properties are the theories of non-Euclidean geometries of Lobachevsky, Gauss, Bolyai in the 19th century. The Italian Beltrami showed that there are real bodies, on the surface of which the laws of Lobachevsky geometry are satisfied.

At the dawn of the theoretical understanding of human knowledge, the development of theories always proceeded from particular cases to the general. Currently, methods for modeling objects have emerged based on the structuring of a mathematical model. The chain of development of such knowledge goes to reverse order. First, an axiomatic mathematical description of the event (object) being studied appears, and on its basis it is formulated conceptual model– paradigm. Along with this, the principles of compliance are also changing. natural processes And theoretical schemes(models). Instead of a simple coincidence of the calculation results according to the model with the experimental data of experiments, we consider comparative characteristics their mathematical algorithms achieving results in other (indirect) parameters. These principles include, for example, the principles simplicity and beauty scientific theories . Moreover, in this case the model is introduced with a new mathematical apparatus along with interpretation, i.e. The starting point in it is a mathematical formalism that is capable of explaining in the language of mathematics a certain essence that manifests itself in experience. It is this step that makes empirical verification difficult, since not only the description equation, but also its interpretation must be verified by experience.

Entered mathematical apparatus in this case, it contains non-constructive elements that can subsequently lead to a discrepancy between theory and experience. It should be noted that this is precisely the specificity of modern scientific research. On the other hand, this feature of modern scientific research threatens the possibility of discarding the proposed promising apparatus. To prevent this from happening, it is necessary to separately address this side of the matter - eliminating discrepancies on the basis of experiment (an example would be the quantum physics and electrodynamics).

Old system classical physics interpretations scientific facts turned into a step-by-step “creation” of an approximate mathematically formed theory real process to the original model. The question arises, what pushes researchers to such an algorithm of actions, i.e. What are the urges for this way of forming a theoretical picture? To this, the methodology of science gives a very definite answer: the intrinsic value of truth; novelty value.

All of the above is achieved using the following research principles: a) prohibition of plagiarism; b) admissibility of critical revision of the grounds scientific research; c) equality of all (including geniuses) in the face of truth; d) ban on falsification and fraud

An example of this is the Einstein-Lorentz connection. The first according to the then unofficial rating was less authoritative at that time, but its elements of the theory of relativity turned into fundamental theory. .

Despite the numerous works on mathematical modeling, some difficulty has emerged in formulating the exact concept mathematical modeling. They (models) and their content are too diverse. In general, it is clear that something more is required from the model than a comparison with reality: the model must necessarily provide information about the properties of the simulated objects and phenomena. Therefore, an acceptable definition of a model should be one that does not include partial uncertainties. For example: a model of a given object is another object that is compared to the original, modeled and certain properties which reflects (saves) the selected properties of the object in a given way.

The model should reflect everything known (sometimes some known characteristics) about an object and predict or shape new information about him in any new conditions of existence. The purpose of the modeling is Thus, - function representation (description) if there is an explanation of the phenomena considered by the model. It is in this case that the model acts as a theory. And, despite this, the sharp opposition between the mathematical (formal) and substantive sides of the model as a whole is untenable. Taking into account the specific side of the formation of the model, we can summarize that mathematics acts as the most important means developing meaningful ideas about the phenomenon being studied throughout the study.

To build mathematical theory We need not only the elements themselves, but also the relationships between them. For numbers, the concept of equality makes sense: a = b. If the numbers a and b are different, huh? b, then it is possible either a > b, or a

Two straight planes can be perpendicular, parallel, or intersect at a certain angle.

All these relations concern two objects. That's why they are called binary relations.

To study the relationships between objects in mathematics, the theory of binary relations was created.

When we consider certain relations, we are always dealing with ordered pairs formed from the elements of a given set. For example, for the relation “greater by 4”, which is considered on the set X = (2, 6, 10, 14), these will be ordered pairs (2, 6), (6, 10), (10, 14), and for relations “divided” - (6, 2), (10, 2), (14, 2).

It can be noted that the set of pairs that define the relations “greater than by 4”, “divisible”, are subsets of the Cartesian product

X ´ X =((2, 2), (2, 6), (2, 10), (2, 14), (6, 2), (6, 6), (6, 10), (6, 14), (10, 2), (10, 6), (10, 10), (10, 14), (14, 2), (14, 6), (14, 10), (14, 24) ).

Definition 1. A binary relation between elements of a set X or a relation on a set X is any subset of the Cartesian product X ´ X.

Binary relations are usually denoted in capital letters Latin alphabet: P, T, S, R, Q, etc. So, if P is a relation on the set X, then P Ì X ´ X. Various special symbols are often used to write relations, for example, =, >, ~, ½½ , ^, etc. The set of all first elements of pairs from P is called the domain of definition of the relation P. The set of values ​​of the relation P is the set of all second elements of pairs from P.

For clarity, binary relations are depicted graphically using a special graph drawing. Elements of the set X are represented by dots. If (x, y) Î Р(хРу) holds, then an arrow is drawn from point x to point y. Such a drawing is called a relation graph P, and the points representing the elements of the set X are the vertices of the graph. arrows as edges of the graph.

Example. Let the relation P: “the number x is a divisor of the number y” given on the set

X = (5, 10, 20, 30, 40), shown in Figure 25.

Arrows of a graph whose beginning and end are the same point are called loops. If you change the directions of all arrows on the relation graph P to the opposite, you will get a new relation, which is called the inverse for P. It is denoted P–1. Note that xРу Û уР–1х.

Methods for specifying binary relations.

Since the relation R between the elements of the set X is a set whose elements are ordered pairs, it can be specified in the same ways as any set.

1. Most often, the relation R on the set X is specified using characteristic property pairs of elements that are in relation R. This property is formulated as a sentence with two variables.

For example, among the relations on the set X = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10), we can consider the following: “number x less number y is 2 times”, “the number x is a divisor of the number y”, “the number x is greater than the number y” and others.

2. The relation R on the set X can also be defined by listing all pairs of elements of the set X, connected by relationship R.

For example, if we write down a set of pairs (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4), then on the set X = (1, 2, 3, 4) we will define some relation R. The same relation R can also be given

3. using a graph (Fig. 26).

Properties of binary relations.

Definition 2. A relation R on a set X is called reflexive if each element from the set X is in this relation with itself.

In short: R is reflexive on X Û xRx for any x О X.

or, what is the same: at each vertex of the relation graph there is a loop. The converse is also true: if not every vertex of a relation graph has a loop, then it is a reflexive relation.

Example. Reflexive relations: “to be equal on the set of all triangles of the plane”, “? and £ on the set of all real numbers."

Note that there are relations that do not have the property of reflexivity (give an example “x is greater than y”)

Definition 3. A binary relation R on a set X is called anti-reflexive on X if for each x from X (x, x) Ï R, i.e. for each x of X the condition xRx is not satisfied.

If a relation R is anti-reflexive, then no vertex of its graph has a loop. Conversely: if no vertex of the graph has a loop, then the graph represents an anti-reflexive relation.

Examples of anti-reflexive relationships: “to be older”, “to be smaller”, “to be a daughter”, etc.

Definition 4. A relation R on a set X is called symmetric if, for any elements x, Î X the condition is satisfied: if x and y are in a relation R, then y and x are also in this relation.

In short: R is symmetric on X Û xRу Û yRx.

A symmetric relation graph has the property: if there is an arrow connecting a pair of elements, then there is necessarily a second one that connects the same elements, but goes in the opposite direction. The converse is also true.

Examples of symmetrical relations are the relations: “to be mutually perpendicular on the set of all straight lines of the plane”, “to be similar on the set of all rectangles of the plane”.

Definition 5. If for no elements x and y from the set X it can happen that both xRy and yRx occur simultaneously, then the relation R on the set X is called asymmetric.

An example of an asymmetrical relation: “to be the father” (if x is the father of y, then y cannot be the father of x).

Definition 6. A relation R on a set X is called antisymmetric if for different elements x, y О X From the fact that element x is in relation R with element y, it follows that element y is not in relation R with element x.

In short: R is antisymmetric on X Û xRу and x? y? .

For example, the relation "less than" on the set of integers is antisymmetric.

An antisymmetric relation graph has a special feature: if two vertices of the graph are connected by an arrow, then there is only one arrow. The opposite statement is also true.

Note that there are relations that have neither the property of symmetry nor the property of antisymmetry.

Definition7. A relation R on a set X is called transitive if for any elements x, y, z О X the following condition is satisfied: if x is in the relation R with y and y is in the relation R with z, then the element x is in the relation R with the element z.

In short: R is transitive on X Û xRу and уRz? xRz.

For example, the relation “line x is parallel to line y,” defined on the set of lines in a plane, is transitive.

The transitive relation graph has the peculiarity that for every pair of arrows going from x to y and from y to z, it also contains an arrow going from x to z. The converse is also true.

Note that there are relations that do not have the property of transitivity. For example, the relation “standing next to each other on a shelf” is not transitive.

All general properties relationships can be divided into three groups:

reflexivity (every relationship is reflexive or anti-reflexive),

symmetry (the relationship is always either symmetrical, asymmetrical, or antisymmetrical),

transitivity (every relation is transitive or non-transitive). Relationships that have a certain set properties are given special names.

The word “conformity” is used quite often in Russian; it means a relationship between something, expressing consistency, equality in some respect ( Dictionary Ozhegova).

In life you often hear: “This textbook corresponds to this program, but this textbook does not correspond (but may correspond to another program); This apple corresponds to the highest grade, but this is only the first.” We say that this answer in the exam corresponds to an “excellent” grade, while this answer corresponds to a “good” grade. We say that this person fits (in the sense of fits) clothes of size 46. In accordance with the instructions, you should do this and not otherwise. There is a correspondence between the number sunny days per year and crop yield.

If you try to analyze these examples, you will notice that in all cases we're talking about about two classes of objects, and between objects from the same class it is established by certain rules some connection with objects of another class. For example, in the case of matching clothes to a certain size, one class of objects is people, and another class of objects is some natural numbers that play the role of clothing sizes. We can set the rule by which compliance is established, for example, using a natural algorithm - trying on a specific suit or determining its suitability “by eye”.

We will consider correspondences for which the classes of objects between which the correspondence is established and the rule for establishing the correspondence are completely defined. Numerous examples of such correspondences were studied at school. First of all, these are, of course, functions. Any function is an example of correspondence. Indeed, consider, for example, the function at = X+ 3. If it is not specifically said about the domain of definition of the function, then it is considered that each numerical value of the argument X corresponds to a numeric value at, which is found according to the rule: to X you need to add 3. In this case, correspondence is established between the sets R And R real numbers.

Note that establishing connections between two sets X And Y associated with the consideration of pairs of objects formed from elements of the set X and the corresponding elements of the set Y.

Definition. Compliance between sets X And Y call any non-empty subset of a Cartesian product X ´ Y.

A bunch of X called departure area matches, set Y – arrival area compliance.

Correspondences between sets are usually denoted in capital letters Latin alphabet, for example, R, S, T. If R– some correspondence between sets X And Y, then, according to the definition of correspondence, RÍ X´ Y And R≠ Æ. Times correspondence between sets X And Y is every subset of the Cartesian product X ´ Y, i.e. is a set of ordered pairs, then the methods for specifying correspondences are essentially the same as the methods for specifying sets. So, matching R between sets X And Y you can set:

a) listing all pairs of elements ( x, y) Î R;

b) indicating the characteristic property that all pairs have ( x, y) sets R and no pair that is not its element possesses it.

EXAMPLES.

1) Compliance R between sets X= (20, 25) and Y= (4, 5, 6) is specified by indicating the characteristic property: “ X multiple at»,
X Î X, at Î Y. Then many R = {(20, 4), (20, 5),(25, 5)}.

2) Compliance R between sets X= (2, 4, 6, 8) and

Y= (1, 3, 5) given by a set of pairs R = {(4, 1), (6, 3), (8, 5)}.

If R– correspondence between two numerical sets X And Y, then, depicting all pairs of numbers that correspond R on coordinate plane, we get a figure called a correspondence graph R. Conversely, any subset of points on the coordinate plane is considered a graph of some correspondence between numerical sets X And Y.

Matching graph

To visually display correspondences between finite sets, in addition to graphs, graphs are used. (From Greek word“grapho” – I write, compare: graph, telegraph).

To construct a correspondence graph between sets X And Y elements of each of the sets are depicted as points on the plane, then arrows are drawn from X Î X To at Î Y, if pair ( x, y) belongs to this correspondence. The result is a drawing consisting of dots and arrows.

EXAMPLE Correspondence R between sets X= (2, 3, 4, 5) and Y= (4, 9) is given by listing the pairs R = {(2, 4), (4, 4), (3, 9)}.

In the same way you can write 4 R 4, 3R 9. And in general, if a couple
(x, y) Î R, then they say that the element X Î X matches element at Î Y and write down xRу. Element 2 О X called the inverse image of the element
4 Î Y subject to compliance R and is designated 4 R-1 2. Similarly, you can write 4 R -1 4, 9R -1 3.

The concept of compliance. Methods for specifying correspondences

Initially, algebra was the study of solving equations. Over many centuries of its development, algebra has turned into a science that studies operations and relationships on different sets. Therefore, it is no coincidence that already in primary school children become familiar with algebraic concepts such as expression (numeric and variable), numerical equality, numerical inequality, the equation. They study various properties arithmetic operations over numbers that allow you to rationally perform calculations. And, of course, in initial course mathematics is their acquaintance with various dependencies, relationships, but to use them for development purposes mental activity children, the teacher must master some general concepts of modern algebra - the concept of correspondence, relationship, algebraic operation etc. In addition, by mastering the mathematical language used in algebra, the teacher will be able to better understand the essence of mathematical modeling real phenomena and processes.

Studying the world around us, mathematics considers not only its objects, but also mainly the connections between them. These connections are called dependencies, correspondences, relationships, functions. For example, when calculating the lengths of objects, correspondences are established between objects and numbers, which are the values ​​of their lengths; when solving motion problems, a relationship is established between the distance traveled and time if the speed of movement is constant.

Specific dependencies, correspondences, and relationships between objects in mathematics have been studied since its inception. But the question of what the most different correspondences have in common, what is the essence of any correspondence, was posed in late XIX- the beginning of the 20th century, and the answer to it was found within the framework of set theory.

In the initial mathematics course, various relationships between elements of one, two or more sets are studied. Therefore, the teacher needs to understand their essence, which will help him ensure unity in the methodology for studying these relationships.

Let's look at three examples of correspondences studied in an initial mathematics course.

In the first case, we establish a correspondence between given expressions and their numerical values. In the second, we find out what number corresponds to each of these figures, characterizing its area. In the third we are looking for a number that is a solution to the equation.

What do these correspondences have in common?

We see that in all cases we have two sets: in the first it is a set of three numerical expressions and set N natural numbers(the values ​​of these expressions belong to him), in the second - this is a set of three geometric shapes and the set N of natural numbers; in the third it is a set of three equations and a set of N natural numbers.

By completing the proposed tasks, we establish a connection (correspondence) between the elements of these sets. It can be represented visually using graphs (Fig. 1).

You can specify these matches by listing all pairs of elements that are in a given match:

I. ((in 1, 4), (in 3, 20));

II. ((F 1, 4), (F 2, 10), (F 3, 10));

III. ((y 1, 4), (y 2, 11), (y 3, 4)).

The resulting sets show that any correspondence between two sets X and Y can be considered as set of ordered pairs , formed from their elements. And since ordered pairs are elements of a Cartesian product, we arrive at the following definition general concept compliance.

Definition. A correspondence between elements of the set X and Y is any subset of the Cartesian product of these sets.

Correspondences are usually denoted by the letters P, S, T, R, etc. If S is a correspondence between elements of the sets X and Y, then, according to the definition, S X x Y.

Let us now find out how to define correspondences between two sets. Since correspondence is a subset, it can be specified as any set, i.e. either by listing all pairs of elements that are in a given correspondence, or by indicating a characteristic property of the elements of this subset. Thus, the correspondence between the sets X = (1, 2, 4, 6) and Y = (3, 5) can be specified:

1) using a sentence with two variables: a< b при условии, что а X, b Y;

2) listing pairs of numbers belonging to a subset of the Cartesian product XxY: ((1, 3), (1, 5), (2, 3), (2, 5), (4, 5)). This method of assignment also includes assignment of correspondence using a graph (Fig. 2) and a graph (Fig. 3)

Rice. 2 Fig. 3

Often, when studying correspondences between elements of the sets X and Y, one has to consider the correspondence that is its opposite. Let, for example,

S - “more than 2” correspondence between elements of sets

X = (4,5,8, 10) and Y= (2,3,6). Then S=((4, 2), (5,3), (8, 6)) and its graph will be the same as in Figure 4a.

The inverse of the match given is the match "less than 2". It is considered between the elements of the sets Y and X, and to present it clearly, it is enough to change the direction of the arrows on the relation graph S to the opposite (Fig. 4b). If the correspondence “less by 2” is denoted by S -1, then S -1 = ((2.4), (3.5), (6.8)).

Let us agree to write the sentence “the element x is in accordance with the element y” as follows: xSy. The entry xSy can be considered as a generalization of the entries for specific correspondences: x = 2y; x > 3y+1, etc.

Let us use the introduced notation to define the concept of correspondence inverse to the given one.

Definition. Let S be a correspondence between elements of the sets X and Y. A correspondence S -1 between elements of the sets Y and X is said to be its inverse if yS -x if and only if xSy .

The correspondences S and S -1 are called mutually inverse. Let's find out the features of their graphs.

Let's construct a correspondence graph S = ((4, 2), (5, 3), (8, 6)) (Fig. 5a). When constructing a correspondence graph S -1 = ((2, 4), (3, 5), (6, 8)), we must select the first component from the set Y = (2, 3, 6), and the second from the set X = (4, 5, 8, 10). As a result, the correspondence graph S -1 will coincide with the correspondence graph S. To distinguish between the correspondence graphs S and S -1 ,

agreed to consider the first component of the correspondence pair S -1 as the abscissa, and the second as the ordinate. For example, if (5, 3) S, then (3, 5) S -1. Points with coordinates (5, 3) and (3, 5), and in general case(x, y) and (y, x) are symmetrical with respect to the bisector of the 1st and 3rd coordinate angles. Consequently, the graphs of mutually inverse correspondences S and S -1 are symmetrical with respect to the bisector of the 1st and 3rd coordinate angles.

To construct a correspondence graph S -1, it is enough to depict points on the coordinate plane that are symmetrical to the points of the graph S relative to the bisector of the 1st and 3rd coordinate angles.

Topic 8. Relationships and correspondences

The concept of a binary relationship between elements of a set

In everyday life, we constantly talk about the relationship between two objects. For example, x works for management, x is a father, x and y are friends - these are relationships between people. Numbers more number m, a number is divisible by y, numbers and y when divided by 3 gives the same remainder - these are the relationships between numbers.

Any mathematical theory deals with a set of some objects or elements. To build a mathematical theory, you need not only the elements themselves, but also the relationships between them. For numbers, the concept of relationships makes sense: a = b, ilia > b, ilia< b. Две прямые плоскости могут быть параллельными или пересекаться.

All these relations concern two objects. That's why they are called binary relations.

When we consider certain relations, we are always dealing with ordered pairs formed from the elements of a given set. For example, for the relation “the number x is 4 greater than the number y”, which is considered on the set X = (2, 6, 10, 14), these will be ordered pairs (6,2), (10, 6), (14, 10 ). They are a subset of the Cartesian product X X .

Definition. A binary relation between elements of a set X or a relation on a set X is any subset of the Cartesian product X X.

Binary relations are usually denoted in capital letters of the Latin alphabet: P, T, S, R, Q, etc. So, if P is a relation on a set X, then P X X. The set of all first elements of pairs from P is called the domain of definition of the relation P. The set of values ​​of the relation P is the set of all second elements of pairs from P.

In many cases it is convenient to use graphic image binary relation.

Elements of the set X are represented by points, and arrows connect the corresponding elements so that if (x,y)P(xPy) occurs, then the arrow is drawn from points to points. The resulting drawing is called a relation graph P, and the points representing the elements of the set X

vertices of the graph.

For example, the graph of the relation P: “number - divisor of numbery”, defined on the set X = (5, 10, 20, 30,40), is shown in Fig. 54.

Arrows of a graph whose beginning and end are the same point are called loops. If on the relation graph P change the directions of all arrows to

opposite, then a new relation will be obtained, which is called the inverse for P. It is denoted P -1. Note that xPy yP -1 x.

Methods for specifying binary relations, their properties

Since the relation R between the elements of the set X is a set whose elements are ordered pairs, it can be specified in the same ways as any set.

Most often, the relation R on the set X is specified using the characteristic property of pairs of elements that are in the relation R. This property is formulated as a sentence with two variables. For example, among the relations on the set X = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) we can consider the following: “number is 2 times less than number y”, “number is a divisor of numbery”, etc. .

A relation R on a set X can also be defined by listing all pairs of elements taken from the set X and related by the relation R.

For example, if we write down a set of pairs (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3,

4), then on the set	X = (1, 2, 3, 4) we will set some
attitude
R = ((x, y)| x X, y	X, x< y} .

The same relation R can be specified using a graph (Fig.). Let's highlight the most important properties binary relationships.

Definition 1. A relation R on a set X is called reflexive if each element from the set X is in this relation with itself.

Briefly speaking this definition can be written as follows: R is reflexive on X xRx for any x X.

Obviously, if a relation R on a set X is reflexive, then there is a loop at each vertex of the relation graph. The opposite statement is also true.

Examples of reflexive relations are the relations: “to be equal on the set of all triangles of the plane”, “x ≤ y”.

Note that there are relations that do not have the property of reflexivity, for example, the relation of perpendicularity of lines.

Definition 2. A relation R on a set X is called symmetric if for any elements of X the following condition is satisfied: if x and y are in relation R, then y are also in this relation.

In short: R is symmetrical on X xRy yRx.

A symmetric relation graph has the property: if there is an arrow connecting a pair of elements, then there is necessarily a second one that connects the same elements, but goes in the opposite direction. The converse is also true.

Examples of symmetrical relations are the relations: “to be mutually perpendicular on the set of all straight lines of the plane”, “to be similar on the set of all rectangles of the plane”.

Definition 3. If for no elements and y from the set X it can happen that both xRy and yRx are present at the same time, then the relation R on the set X is called asymmetric. An example of an asymmetrical relationship: “to be a father” (if ih - to a father, then you cannot be a father).

Definition 4. The relation R on the set X is called antisym-

For example, the relation "less than" on the set of integers is antisymmetric.

An antisymmetric relation graph has a special feature: if two vertices of the graph are connected by an arrow, then there is only one arrow. The opposite statement is also true. The property of asymmetry is a combination of the property of antisymmetry and lack of reflexivity.

Definition 5. A relation R on a set X is called transitive if for any elements x, y, z X the following condition is satisfied: if x is in the relation R and y is in the relation R cz, then the element x is in the relation R with the element z.

In short: R is transitive on X xRy and yRz xRz.

For example, the relation “a line x is parallel to a line,” defined on the set of lines in a plane, is transitive.

The transitive relation graph has a special feature: with every pair of arrows going from x to ky and oty to z, it also contains an arrow going from x to z. The converse is also true.

Note that there are relations that do not have the property of transitivity. For example, the relation “standing next to each other on a shelf” is not transitive.

Equivalence relation

Let X be a set of people. On this set we define a binary relation R using the law: aRb, if a and b were born in the same year.

It is easy to verify that the relation R has the properties of reflexivity, symmetry and transitivity. The relation R is said to be an equivalence relation.

Definition 1. A binary relation R on a set X is called an equivalence relation if it is reflexive, symmetric and transitive.

Let us return again to the relation R, defined on a set of people by the law: aRb, if a and b were born in the same year.

Together with each person a, consider the set of people K a who were born in the same year sa. Two sets K a and K b either do not have common elements, or coincide completely.

The set of sets K a represents a partition of the set of all people into classes, since from its construction it follows that two conditions are met: each person is included in some class and each person is included in only one class. Note that each class consists of people born in the same year.

Thus, the equivalence relation R generates a partition of the set X into classes (equivalence classes). The opposite is also true.

Theorem. Each equivalence relation on the set X corresponds to a partition of the set X into classes (equivalence classes). Each partition of sets corresponds to an equivalence relation on the set X.

We accept this theorem without proof.

It follows from the theorem that each class obtained as a result of partitioning a set into classes is determined by any (one) of its representatives, which makes it possible, instead of studying all the elements of a given set, to study only the collection individual representatives each class.

Order relation

We constantly use order relations in Everyday life. Definition 1. Every antisymmetric and transitive relation R on

some set X is called an order relation.

A set X on which an order relation is given is called ordered.

Let's take the set X = (2, 4, 10, 24). It is ordered by the relation “x is greater” (Fig. 63).

Let us now consider on it another relation of the order “x divides

y" (Fig. 64).

The result of this review may seem strange. The relations “x is greater” and “x divides” order the set X in different ways. The x-greater relation allows you to compare any two numbers from

set X. As for the relation “x divides”, it does not have such a property. So the pair of numbers 10 and 24 is not related by this relationship.

Definition 2. An order relation R on some set X is called a relation linear order, if it has the following property: for any elements u

the set X is eitherxRy or yRx.

A set X on which a linear order relation is given is called linearly ordered.

Linearly ordered sets have a number of properties. Let a, b, c be elements of the set X on which the linear order relation R is specified. If aRb and bRc, then we say that element b lies between elements a and .

A linearly ordered set X is called discrete if between any two of its elements there lies only a finite set of elements.

If for any two various elements linearly ordered set X there is an element of the set lying between them, then the set X is called dense.

The concept of correspondence between sets. Methods for specifying correspondences

Let two sets X and Y be given. If for each element x X is specified to the element Y with which it is matched, then a correspondence is said to be established between the sets X and Y.

In other words, the correspondence between elements of the sets X and Y is any subset G of the Cartesian product X and Y of these sets: G X Y .

Since a match is a set, it can be specified in the same ways as any set: by listing all pairs (x, y), where

When the sets X and Y are finite, then the correspondence between the elements can be specified in a table where the elements of the set X are written in the left column, and the elements of the set Y are written in the top row. Pairs of elements that match G will be at the intersection of the corresponding columns and rows.

The correspondence between two finite sets can also be shown using a graph. The sets X and Y are shown as ovals, the elements of the sets X and Y are designated by dots, and the corresponding elements are connected by arrows so that if (x,y) G occurs, then the arrow is drawn from points to points.

For example, the graph shown in Fig. 16, sets the correspondence “Writer x wrote the work.”

When the sets and Y are numeric, then it is possible to construct a correspondence graph of G on the coordinate plane.

Correspondence is the inverse of the given one. One-to-one correspondences

Let R be the correspondence “The number is five times less than the numbery” between the elements of the sets X = (1, 2, 4, 5, 6) and

Y = (10, 5, 20, 13, 25).

The graph of this correspondence will be as in Fig. 23. If you change the direction of the arrows of this graph to

the opposite, then we get a graph (Fig. 22) of the new correspondence “The number y is five times greater than the number x”, ​​considered

between sets Y and X.

This correspondence is called inverse correspondence

correspondence to R, and is denoted by R -1.
Definition. Let	R - compliance
elements of the sets X and Y. Compliance R-1
elements of the sets Y and X is called the inverse of the given one,
when (y, x) R -1 if and only if (x,		y) R.
The correspondences R and R -1 are called mutually inverse.
If the sets X and Y are numeric, then the graph
correspondence R -1 , the inverse of correspondence R, consists of
points, symmetrical points R matching graphics
relative to the bisector of the first and	third

coordinate angles.

Let's imagine a situation: in the auditorium there is a spectator in every seat and there is a place for each spectator. In this case they say that between the set

seats in the auditorium and the multitude of spectators have established a one-to-one correspondence.

Definition. Let two sets X and Y be given. The correspondence between elements of sets X and Y, in which each element of set X corresponds to a single element of set Y, and each element of set Y corresponds to only one element from set X, is called one-to-one.

Let's look at examples of one-to-one correspondences. Example 1. In every school, every class

corresponds to a cool magazine. This correspondence is one-to-one.

Example 2. Given triangle ABC (Fig. 25).A 1 C 1 middle line of the triangle. Let X be the set of points on the segment A 1 C 1, Y the set of points on AC.

We connect an arbitrary point x of the segment A 1 C 1 to the vertex B of the triangle with a straight line segment and

Let's continue it until it intersects with AC at pointy. Let us match the points with the point constructed in this way. In this case, a one-to-one correspondence will be established between the sets X and Y.

Definition. Sets X and Y are called equivalent, or equally powerful, if a one-to-one correspondence can be established between them in some way. The equivalence of two sets is denoted as follows: X ~ Y.

The concept of power is a generalization of the concept of quantity. This is an extension of the concept of quantity to infinite sets.