Binary relationships and their properties Solution Examples. Binary relationship. Examples of binary relationships. Binary relationships and their properties

for the relationship of non-neat order:

h.X. - True (reflexiveness);

hu and wow x \u003d- (antisymmetry);

x y and z x x z- (transitivity).

Lots of X.called ordered if any two elements h.and w.this set is comparable, i.e., if one of the conditions is performed for them: h.< u, H.= u, W.< x.

Ordered set are called a tuple. In the general case, the tuple is a sequence of elements, i.e., the set of elements in which each element occupies a completely definite place. Elements of the ordered set are called the components of the tuple. Examples of the cortex can be an ordered sequence of arithmetic or geometric progressions, a sequence of technological operations in the manufacture of a radio-electronic product, an ordered sequence of the installation positions of the printed circuit board for fixing structural elements.

In all these sets, the place of each element is fully defined and cannot arbitrarily change.

When processing design information on computers, dominance ratios often use. They say that xxdominates over uxi.e. x \u003e\u003e y,if item h.in something superior (has a priority) element w.of the same set. For example, under h.you can understand one of the data lists, which should be received for the processing first. When analyzing several REA structures, some of them should be given priority, since this design has the best, from our point of view, properties than others, i.e. design h.dominates design y

The property of transitivity does not have space. Indeed, if, for example, the design h.for any one parameters preferred designs y,and design w.according to any other parameters, Z designs preferred, then it does not yet follow that the designs h.must be given preference compared to the design g.

Display sets. One of the basic concepts of set theory is the concept of display. If two non-empty sets are specified X.and Y,then the law according to which each element x X.put in conformity of the element , called unambiguous mapping X.in Y.or a function defined on x and the receiving value on Y.

In practice, it is necessary to deal with multi-valued mappings of sets X.on the set Y,which define the law according to which each element xxput in line with some subset , called the way items. Cases are possible when Gh \u003d 0.

Let some subset be given AX.For anyone haway h.is a subset . A combination of all elements Y,are images for all x in aclaim a way of set BUTand we will denote Ha.In this case

Binary relationship.

Let a and b be arbitrary sets. Take one element from each set, and from A, B from B and write them like this: (First, the element of the first set, then the element of the second set - that is, we are important to the order in which the elements are taken). Such an object will be called ordered pair. Equal We will consider only those pairs that have elements with the same numbers are equal. = if a \u003d c and b \u003d d. Obviously, if a ≠ b, then ≠ .

Cartesian work Arbitrary sets a and b (denoted: AB) called a set consisting of all possible ordered steam, the first element of which belongs to A, and the second belongs to B. By definition: AB \u003d ( | AA and BB). Obviously, if a ≠ b, then AB ≠ BA. Cartesovo works of set A itself called n times cartesian degree A (denotes: a n).

Example 5. Let a \u003d (x, y) and b \u003d (1, 2, 3).

AB \u003d ( , , , , , }.

Ba \u003d (<1, x>, <2, x>, <3, x>, <1, y>, <2, y>, <3, y>}.

Aa \u003d a 2 \u003d ( , , , }.

Bb \u003d b 2 \u003d (<1, 1>, <1, 2>, <1, 3>, <2, 1>, <2, 2>, <2, 3>, <3, 1>, <3, 2>, <3, 3>}.

Binary attitude On the set M, a variety of some ordered pairs of elements of the set M. If R is binary attitude and steam belongs to this relationship, then write: R or X R y. Obviously, R í m 2.

Example 6. Set (<1, 2>, <2, 2>, <3, 4>, <5, 2>, <2, 4>) is a binary attitude on the set (1, 2, 3, 4, 5).

Example 7. The ratio ³ on a plurality of integers is a binary attitude. This is an infinite set of ordered couples where x ³ y, x and y are integers. This relationship belongs, for example, pairs<5, 3>, <2, 2>, <324, -23> and do not belong to pairs<5, 7>, <-3, 2>.

Example 8. The ratio of equality on the set A is a binary ratio: I A \u003d ( | x î a). I A called diagonal Sets A.

Since binary relationships are sets, they are applicable to the operations of the association, intersection, additions and differences.

Definition area The binary ratio R is called the set D (R) \u003d (x | There is such a Y that XRY). Area of \u200b\u200bvalues The binary ratio R is called the set R (R) \u003d (Y | There is such a X that XRY).

Relation inverse to the binary ratio R í m 2, is called binary ratio R -1 \u003d ( | Î R). Obviously, D (R -1) \u003d R (R), R (R -1) \u003d D (R), R - 1 í m 2.

Composition Binary relations R 1 and R 2 specified on the set M are called binary ratio R 2 O R 1 \u003d ( | There is y such that Î R 1 and Í R 2). Obviously, R 2 O R 1 í m 2.

Example 9. Let the binary ratio of R set on the set M \u003d (A, B, C, D), R \u003d ( , , , ). Then D (R) \u003d (A, C), R (R) \u003d (B, C, D), R -1 \u003d ( , , , ), R O R \u003d ( , , , ), R -1 O R \u003d ( , , , ), R O R -1 \u003d ( , , , , , , }.

Let R be a binary attitude on the set M. The ratio R is called reflectiveif X R x for any X î M. The ratio R is called symmetricif with each pair it contains a couple . The ratio R is called transitiveif from the fact that x r y and y r z follows that x r z. The ratio R is called antisymmetricif it does not contain a pair at the same time and Different elements x ¹ y set M.

We indicate the criteria for performing these properties.

Binary ratio R on the set M reflexively then and only if I m í r.

The binary ratio R is symmetrically then and only if R \u003d R -1.

The binary ratio R on the set M is antisymmetric if and only if R ç R -1 \u003d i m.

The binary ratio R is transitively if and only if R o r í r.

Example 10. The ratio of Example 6 is antisymmetric, but is not symmetrical, reflexive and transitive. The ratio of Example 7 is reflexive, antisymmetric and transitive, but is not symmetric. The ratio I A has all the four properties in question. The ratios R -1 O R and R O R -1 are symmetric, transitive, but are not antisymmetric and reflexive.

Relation equivalence On the set M is called transitive, symmetric and reflexive on M binary attitude.

Relation partial order On the set M is called transitive, antisymmetric and reflexive on M binary ratio R.

Example 11. The ratio of Example 7 is a partial order ratio. The ratio I A is the ratio of equivalence and partial order. The ratio of parallelism on the set of direct is equivalence ratio.

Properties of relations:

1) reflexivity;

2) symmetry;

3) transitivity.

4) Linking.

Attitude R. On the set H. called reflexive If about each element of the set H. We can say that it is in relation to R. With myself: h.RX. If the ratio is reflexive, then in each vertex there is a loop. And back, the graph, each vertex of which contains a loop, is a graph of a reflexive relationship.

Examples of reflexive relationships are and the ratio of "multiple" on the set of natural numbers (every number of multiple himself), and the attitude of the likeness of the triangles (each triangle is similar to itself), and the attitude of "equality" (every number equally) and others.

There are relations that do not have the property of reflexivity, for example, the ratio of perpendicularity of segments: aB, BA. (There is not a single segment that can be said that he is perpendicular to himself) . Therefore, there is no loop on the column of this relationship.

Does not have the property of reflexivity and the ratio "longer" for segments, "more by 2" for natural numbers, etc.

Attitude R. On the set H.called antireflemissiveif for any element from the set H.always false h.RX: .

There are ratings that are neither reflexive or antireflems. An example of such a relationship is the attitude "point h. Symmetrical point w.related l.", Specified on the set of points of the plane. Indeed, all points are direct l. symmetrical ourselves, and points that do not lie on a straight l, ourselves are not symmetrical.

Attitude R.on the set H. called symmetric, If condition is satisfied: from what the element h. is in relation to the element y., it follows that the element y. Located in Right R. with element x:xryyrx.

The graph of a symmetric relationship has the following feature: together with each arrow coming from h. to y., the graph contains an arrow coming from y. to h. (Fig. 35).

Examples of symmetric relations can be the following: the ratio of "parallelism" of segments, the ratio of "perpendicularity" of segments, the ratio of "equality" of segments, the ratio of the similarity of triangles, the ratio of "equality" fractions, etc.

There are relations that do not have a symmetry property.

Indeed, if the segment h. Long cut w., then cut w. can not be longer segment h.. The graph of this relationship has a feature: the arrow connecting the vertices is directed only in one direction.

Attitude R. Call antisymmetricif for any elements h. and y.from truth xRYfalse follows yRX :: Xryyrx.

In addition to the relation "longer" on the set of segments there are other antisymmetric relationships. For example, the ratio "more" for numbers (if h. more w.T. w. can not be more h.), the ratio "more on" and others.

There are relations that do not have a symmetry property nor the property of antisymmetry.

Ratio R on the set H.call transitive If from the fact that the element h. Located in Right R. with element y, And element y. Located in Right R. with element z., It follows that the element h. Located in Right R. with element z.: xRY and yrz.xRZ.

Count of transitive relationship with each pair of arrows coming from h. to y. and from y. to z., It contains an arrow coming from h.to z.

The relationship of transitivity has the ratio "longer" on the set of segments: if the segment but Long cut b., section b.long cut from, then cut butlong cut from. The ratio of "equality" on the set of segments also has the property of transitivity: (A \u003d.b, b \u003d c) (a \u003d c).

There are relations that do not have the property of transitivity. Such an attitude is, for example, the attitude of perpendicularity: if the segment but Perpendicular to the segment b., and cut b. Perpendicular to the segment from, then segments but and from Not perpendicular!

There is another property of relationship, which is called the property of connectedness, and the attitude possessing them is called connected.

Attitude R. On the set H. called associated If for any elements h. and y. A condition is satisfied from this set: if h. and y. different, then either h. Located in Right R. with element y.or element y. Located in Right R. with element h.. With the help of characters it can be written as: xY. XRY or yRX.

For example, the relationship property has the ratio of "more" for natural numbers: for any different numbers x and y, it can be argued or x\u003e Y.either y\u003e x.

On the column of the associated relationship, any two vertices are connected by an arrow. Fair and reverse statement.

There are relations that do not have the property of connectedness. Such an attitude, for example, is the relation of divisibility on a set of natural numbers: you can call such numbers x and y.none number h.is not a divider number y.nor a number y. is not a divider number h.(Numbers 17 and 11 , 3 and 10 etc.) .

Consider several examples. On the set X \u003d (1, 2, 4, 8, 12) The ratio "Number h.paint number y." We construct the graph of this relationship and formulate its properties.

The ratio of equality of fractions speak, it is equivalence ratio.

Attitude R. On the set H. called equivalence relation If it simultaneously has the property of reflexivity, symmetry and transitivity.

Examples of equivalence relationships include: the relationship of geometric figures, the ratio of direct parallelism (provided that the coinciding straight lines are considered parallel).

In the ratio of "equality of fractions", many H.broke into three subsets: ( ; ; }, {; } , (). These subsets do not intersect, and their association coincides with many H.. We have a splitting of many classes.

So, if the equivalence ratio is specified on the set x, it generates the splitting of this set on pairwise disseminating subsets - equivalence classes.

So, we found that the relation of equality on the set
H.\u003d (;;;;;) corresponds to the partition of this set on the equivalence classes, each of which consists of equal fractions.

The principle of splitting the set on classes with some equivalence relationship is an important principle of mathematics. Why?

First, equivalent is equivalent, interchangeable. Therefore, elements of a single class of equivalence are interchangeable. So, the fraction, which was in one equivalence class (;;), indistinguishable in terms of relations of equality and fraction may be replaced by another, for example . And this replacement will not change the result of the calculations.

Secondly, since in the equivalence class it is elements that are indistinguishable from the point of view of some relationship, they believe that the equivalence class is determined by any representative, i.e. An arbitrary element of the class. So, any class of equal fractions can be set, indicating any fraction belonging to this class. An equivalence class for one representative allows instead of all elements of the set to explore the set of representatives from equivalence classes. For example, the equivalence ratio of "have the same number of vertices" specified on the set of polygons, generates the partition of this set on the classes of triangles, quadrangles, pentagons, etc. Properties inherent in some class are considered on one of its representative.

Thirdly, the splitting of the set to classes using equivalence ratio is used to introduce new concepts. For example, the concept of "beam of direct" can be determined as common, which has parallel straightaries.

Another important type of relationship is the relationship of order. Consider the task. On the set H.={3, 4, 5, 6, 7, 8, 9, 10 ) The ratio is set to "have the same residue when dividing on 3 " This attitude creates the splitting of the set H. to classes: one will fall into one numbers, when dividing which on 3 it turns out in the rest 0 (These are numbers 3, 6, 9 ). In the second - the number, when dividing which on 3 In the residue it turns out 1 (These are numbers 4, 7, 10 ). In the third, all numbers will fall, when dividing which on 3 In the residue it turns out 2 (These are numbers 5, 8 ). Indeed, the resulting sets do not intersect and their association coincides with the set H.. Therefore, the attitude "to have the same residue in dividing on 3 "Set on a set H.is equivalence relationship.

Take another example: a variety of class students can be arranged by growth or age. Note that this ratio has the properties of antisymmetry and transitivity. Or everyone knows the order of the letters in the alphabet. It provides the relation "follow."

Attitude R.on the set H. called relation of strict orderif it simultaneously has antisymmetry and transitivity properties. For example, the relation " h.< y.».

If the relationship has the properties of reflexiveness, antisymmetry and transitivity, then it will be the attitude of the non-strict order. For example, the relation " h.y.».

Examples of the relationship of the order can be: the ratio "less" on the set of natural numbers, the ratio "shorter" on the set of segments. If the order ratio has also a property of connectedness, they say that it is relation linear order. For example, the ratio "less" on the set of natural numbers.

Lots of H. called ordered If the order ratio is specified.

For example, the set X \u003d{2, 8, 12, 32 ) You can streamline with the help of the "less" ratio (Fig. 41), and you can do it with the help of a "multiple" relationship (Fig. 42). But, being an attitude of order, the relationship "less" and "more paint" arrange many natural numbers in different ways. The ratio "less" allows you to compare two any numbers from the set H.And the ratio of "multiple" does not possess such a property. So, a couple of numbers 8 and 12 The ratio is "multiple" is not related: it is impossible to say that 8 edge 12 or 12 edge 8.

It should not be thought that all relationships are divided into equivalence relationship and relationship relationship. There is a huge number of non-equivalence relationships or relationships of order.

Basics of discrete mathematics.

The concept of set. The relationship between sets.

The set is a set of objects with a specific property combined into a single whole.

Object components are called elements sets. In order for some sets of objects to be called a set, the following conditions must be performed:

· There must be a rule for which mono determine whether the element is belong to this set.

· There should be a rule by which items can be distinguished from each other.

Sets are indicated by capital letters, and its elements are small. Methods for setting sets:

· List the elements of the set. - For finite sets.

· Indication of the characteristic property .

Empty set - Called a set that does not contain any element (Ø).

Two sets are called equal, if they consist of the same elements. . A \u003d B.

Lots of B. called a subset of the set BUT (, then and only when all elements of the set B. belong to set A..

For example: , B. =>

Property:

Note: Usually consider a subset of one and that E set, which is called universal (U). The universal set contains all the elements.

Operations on sets.

2.Intersection 2 sets are called a new set consisting of elements, simultaneously belong to the first and second set.

Nr: ,,

Property: Combining and intersection operations.

· Communtation.

· Associativity. ;

· Distribution. ;

Binary relations and their properties.

Let be BUT and IN These are a variety of derivative of nature, consider an ordered pair of elements. (A, B) A ε A, in ε inyou can consider ordered "Enki".

(A 1, and 2, and 3, ... and n)where but 1 ε a 1; but 2 ε a 2; ...; but N. ε and n;

Cartesian (straight) A 1, and 2, ..., and nis called mn in, which consists of an ordered n k of the species.

Nr: M.= {1,2,3}

M × m \u003d m 2= {(1,1);(1,2);(1,3); (2,1);(2,2);(2,3); (3,1);(3,2);(3,3)}.

Subsets of the decartian works called the ratio of degree n. or an enar relation. If a n.\u003d 2, then consider binary relations. What do they say that a 1, and 2 are in binary terms R.when a 1 R a 2.

Binary attitude on the set M. called a subset of the direct product of the set n. by itself.

M × m \u003d m 2= {(a, B.)| a, b ε m) In the previous example, the ratio is less on the set M. It gives rise to the following set: ((1,2); (1,3); (2.3))

Binary relations have various properties including:

· Reflexivity: .

· Antireflexivity (irreflexusion) :.

· Symmetry:.

· Antisymmetry:.

· Transitivity :.

· Asymmetry:.

Types of relationships.

· Equivalence ratio;

· The ratio of order.

v Reflexive transitive relationship is called the ratio of quasi-arms.

v Reflexive symmetric transitive relationship is called equivalence ratio.

v The reflexive antisymmetric transitive relationship is called the ratio of (partial) order.

v An antireflexive antisymmetric transitive relationship is called a ratio of a strict order.

Obviously, arbitrary binary relations to study in general terms is not particularly interesting, we can say very little about them. However, if relations satisfy some additional conditions, more substantive statements can be made relative to them. In this section, we will look at some of the basic properties of binary relationships.

• 1. The binary attitude on the set X is called reflexive, if a condition A is satisfied for any element AX:
  • (AX) A * a.

If the ratio is presented using a graph, then the reflexivity of this relationship means that there is no loop in each vertex.

For the relationship given by the help of a militant matrix, its reflexivity is equivalent to the fact that on the main diagonal of this matrix (coming from its upper left corner to the lower right) only characters 1 cost.

2. The binary attitude to X is called antireflems, if none of the AX is satisfied with the condition A * A:

Denote by the i x the ratio on the set X consisting of pairs of the form (A, A), where a x:

I x \u003d ((a, a) | a x).

The ratio of IX is usually called the diagonal of the set X or the identity ratio on X.

Obviously, the attitude on the set x is reflexive if the diagonal I x is a subset of the set:

The ratio of antireflexically, if the diagonal I x and the ratio b do not have any general element:

• 3. The binary attitude on the set X is called symmetrical if from A * B follows B * A:
  • (A, BX) (A * B B * a).

Examples of symmetric relationships are:

the attitude of perpendicularity on the set of straight lines;

touch ratio on a plurality of circles;

the ratio of "be similar" on the set of people;

the ratio "to have the same gender" on the set of animals.

The ratio "X brother y" on the set of all people is not symmetric. At the same time, the ratio "X brother y" on the set of men is symmetrical.

In a graph of a symmetrical ratio for each arc from the top X to the top of Y there is an arc from Y to x. Therefore, symmetric relations can be represented by graphs with non-oriented ribs. In this case, each pair of oriented edges XY and YX is replaced with one non-oriented edge.

Figure 8 shows the attitude

b \u003d ((A, B), (B, A), (B, C), (C, B), (D, C))

using oriented and non-oriented graphs.

Fig. eight.

The matrix of a symmetric relationship is symmetric relative to the main diagonal.

Theorem: Association and intersection of any family of symmetric relationships are again symmetric relations.

Definition. The binary attitude on the set X is called antisymmetric, if for any different elements a and b conditions A * B and B * A are not performed simultaneously:

(A, BX) (A * B & B * A A \u003d B).

For example, the ratio "shares" on the set of natural numbers is antisymmetric, since it follows from A B and B a, that a \u003d b. However, on a plurality of integers, the ratio "shares" is not antisymmetric, since (-2) 2 and 2 (-2), but -22.

Relationship "above", "heavier", "older" antisymmetric on a variety of people. The ratio "to be sister" on the set of all people is not antisymmetric.

In the graph of antisymmetric relationship, two different vertices can be connected by no more than one arc.

Definition 3.5. The binary ratio A on the set X is called transitive, if for any three elements a, b, c x from A * B and B * C follows the A * C:

(A, B, C x) (A * B & B * C A * C).

Examples of transitive relationships serve:

the ratio "shares" on the set of valid numbers;

the ratio "more" on the set of valid numbers;

the ratio of "older" on the set of people toys;

the ratio "to have the same color" on the set of children's toys;

e) the attitude "to be a descendant" on a variety of people.

The feudal attitude "to be vassal" is not transitive. This in particular is emphasized in some history textbooks: "My Vassal Vassal is not my vassal."

The ratio of "look similar" on the set of people does not have the property of transitivity.

For an arbitrary relationship, you can find the minimum transitive relationship such that AB. Such an attitude is the transitive closure of the relationship.

Example 3.1. The transitive closure of the binary relationship on the set of people "to be a child" is the ratio of "to be a descendant".

Fair theorem.

Theorem 3.2. For any relationship, transitive closure is equal to the intersection of all transitive relationships, including a subset.

Definition 3.6. The binary attitude on the set X is called connected if for any two different elements a and b take place a * b, or b * a:

(A, B, C X) (AB A * B B * A).

An example of a coherent relationship is the ratio of "more" on the set of valid numbers. The ratio is "sharing" on a plurality of integers is not connected.

4. Invariance of relationships

In this paragraph, we will list some cases when certain properties of relations are saved when performing operations over them.

Theorem 4.4. In order for the product of symmetric relationships, it is symmetrically, it is necessary and enough for the relationship and commute.

Equivalence ratio

An important type of binary relationship is equivalence ratio.

Definition 1. The binary attitude on the set X is called the equivalence ratio to x, if reflexive, symmetrically and transitively.

The equivalence ratio is often denoted by symbols ~,.

Examples of equivalence ratio serve:

the identity ratio I x \u003d ((A, A) | AX) on a non-empty set x;

the ratio of parallelism on the set of direct plane;

the ratio of similarity on the set of plane shapes;

the ratio of equivagivity on the set of equations;

attitude "to have the same residues in dividing to a fixed natural number M" on a plurality of integers. This ratio in mathematics is called the ratio of comparability by module M and denote AB (MOD M);

the ratio "belongs to one type" on the set of animals;

the ratio of "be relatives" on the set of people;

the ratio of "to be one growth" on a variety of people;

attitude "to live in the same house" on a variety of people.

The relationship "to live on one street", "to be similar" on the set of people are not equivalence relationships, since they do not have the property of transitivity.

Of the above properties of binary relationships, it follows that the intersection of equivalence relationship is equivalence ratio.

Equivalence classes

With the equivalence attitude, the splitting of the set per classes is closely connected.

Definition 1. The system of non-empty subsets

(M 1, m 2, ...)

multiple M is called the splitting of this set if

The sets M 1, M 2, ... are called the classes of this partition.

Examples of parties serve:

decomposition of all polygons into groups in the number of vertices - triangles, quadrangles, pentagons, etc.;

partitioning of all triangles according to the properties of angles (acute-angled, rectangular, stupid);

the partition of all triangles according to the properties of the parties (versatile, equal, equilateral);

the partition of all triangles on the classes of similar triangles;

selling a variety of all students in this school class.

The widespread use of equivalence relations in modern science is due to the fact that any equivalence relation carry out the setting of the set in which it is defined, the classes usually taken for new objects. In other words, with the help of equivalence relationships, new objects are generated, concepts.

Thus, for example, the ratio of the radiation cooler breaks the set of all the rays of the plane or space on the classes of the coated rays. Each of these classes of rays is called the direction. Thus, the intuitive concept of the direction receives an accurate mathematical description as a class of partitioning a set of rays by equivalence ratio.

About such figures are usually indicated that they have the same shape. But what is a form of a geometric shape? It is intuitive that this is general that unites such figures. Using equivalence ratio, this intuitive concept is managed to accurate mathematical. The ratio of similarity, being an equivalence ratio, breaks the many figures on the classes of such figures. Each such class can be called the form. Then the expression "two identical figures have the same form" has the following accurate meaning "two similar figures belong to one form."

Equivalence relationships are found everywhere where sets of sets on classes. We often use them without noticing it.

We give an elementary example. When children play with many multi-colored toys (for example, with Dielesh blocks) and decide to decompose toys in colors, then they enjoy the relationship "to have one color". Received as a result of the classes of monochrome figures are perceived by children as new concepts: red, yellow, blue, etc.

Similarly, as a result of solving the problem of decomposition of blocks in shape, children receive classes, each of which is perceived as a form: rectangular, round, triangular, etc.

The relationship between equivalence relations defined on the set M, and the partitions of the set M into classes are described in the following two theorems.

Theorem 1 Any partitioning of a non-empty set M into classes determines (induces) on this set equivalence ratio such that:

all two elements of the same class are in relation to;

all two elements of different classes are not in relation to. Evidence. Let there be some partitioning of a non-empty set M. Determine the binary ratio as follows: xay (k) (xk & yk).

That is, the two elements X and Y A for the set M are associated with the ratio in that and only if there is such a class K, which simultaneously belongs the elements x and y.

So a certain relation is obviously reflexive and symmetrically. We prove the transitivity of the relationship. Let x * y and x * z be. Then, by definition, there are classes K 1 and K 2 such as X, YK 1 and Y, ZK 2. Since various classes in partitions do not have common elements, then k 1 \u003d k 2, that is, x, z k 1. Therefore, x * z, which was required to prove.

Theorem 2. Any equivalence ratio in a non-empty set M generates the partition of this set on the equivalence classes such that all sorts of two elements of the same class are in relation to;

all two elements of different classes are not in relation to.

Evidence. Let b be some equivalence ratio on the set M. Each element X from put into line with a subset [x] of the set M consisting of all elements Y, which are in relation to the X element:

The subset system [x], forms the splitting of the set M. Indeed, first, each subset [x] O, since due to the reflexivity of the ratio x [x].

Secondly, two different subsets [x] and [y] do not have common elements. Arguing from the other, let's say the existence of an element z is such that Z [x] and z [y]. Then ZAX and Zay. Therefore, for any element A [x] from a * x, z * x and z * y, due to symmetry and transitivity, A * Y follows, that is, A [Y]. Consequently, [x] [y]. Similarly, we obtain that [y] [x]. The two inclusions obtained entertain the equality [x] \u003d [y], which contradicts the assumption of the mismatch of subsets [x] and [y]. Thus, [x] y] \u003d O.

Thirdly, the merging of all subsets [x] coincides with the set M, for for any element Xm, the X [x] condition is performed.

So, the system of subsets [x], forms the splitting of the set M. It is easy to show that the constructed partition satisfies the conditions of the theorem. The splitting of the set M, which has the properties specified in the theorem, is called a set of set M with respect and designated M / b.