Quest Source: Task 10_20. USE 2018 Social studies. Solution

Task 20. Read the text below, in which a number of words (phrases) are missing. Choose from the proposed list of words (phrases) that you want to insert in place of the gaps.

“The quality of life depends on many factors, ranging from where a person lives to the general socio-economic and (A) situation, as well as the state of political affairs in the country. The quality of life to one degree or another can be influenced by the demographic situation, living and working conditions, the volume and quality of _____ (B), etc. Depending on the degree of satisfaction of needs in the economy, it is customary to distinguish different levels of living of the population: (B) ensuring the all-round development of a person; a normal level of _____ (G) according to scientifically based standards, providing a person with the restoration of his physical and intellectual strength; poverty - the consumption of goods at the level of maintaining working capacity as the lower limit of reproduction _____ (D); poverty is the consumption of a set of goods and services that is minimally acceptable according to biological criteria, which only allow maintaining human viability.

The population, adapting to market conditions, uses various additional sources of income, including income from personal subsidiary plots, profit from _____ (E)”.

Words (phrases) in the list are given in the nominative case. Each word (phrase) can only be used once.

Choose sequentially one word (phrase) after another, mentally filling in each gap. Please note that there are more words (phrases) in the list than you need to fill in the gaps.

List of terms:

1) capital

2) ecological

3) rational consumption

4) consumer goods

5) means of production

7) labor force

8) entrepreneurial activity

9) social mobility

Solution.

Let's insert the terms into the text.

“The quality of life depends on many factors, ranging from where a person lives to the general socio-economic and environmental (2) (A) situation, as well as the state of political affairs in the country. The quality of life can be affected to some extent by the demographic situation, living and working conditions, the volume and quality of consumer goods (4) (B), etc. Depending on the degree of satisfaction of needs in the economy, it is customary to distinguish different levels of living of the population : prosperity - the use of benefits (6) (B) that ensure the comprehensive development of a person; a normal level of rational consumption (3) (D) according to scientifically based standards, providing a person with the restoration of his physical and intellectual strength; poverty - consumption of goods at the level of maintaining working capacity as the lower limit of labor force reproduction (7) (E); poverty is the consumption of a set of goods and services that is minimally acceptable according to biological criteria, which only allow maintaining human viability.

Let R be a binary relation on a set X. The relation R is called reflective , if (x, x) О R for all x О X; symmetrical – if (x, y) О R implies (y, x) О R; the transitive number 23 corresponds to the variant 24 if (x, y) Î R and (y, z) Î R imply (x, z) Î R.

Example 1

We will say that x н X has in common with element y н X if the set
x З y is not empty. The relation to have in common will be reflexive and symmetrical, but not transitive.

Equivalence relation on X is called a reflexive, transitive, and symmetric relation. It is easy to see that R Н X ´ X will be an equivalence relation if and only if the inclusions take place:

Id X Í R (reflexivity),

R -1 Í R (symmetry),

R ° R Í R (transitivity).

In fact, these three conditions are equivalent to the following:

Id X Í R, R -1 = R, R ° R = R.

splitting set X is a set A of pairwise disjoint subsets a н X such that UA = X. With each partition of A, we can associate an equivalence relation ~ on X by setting x ~ y if x and y are elements of some a н A.

To each equivalence relation ~ on X there corresponds a partition A whose elements are subsets, each of which consists of those in the relation ~. These subsets are called equivalence classes . This partition A is called the factor set of the set X with respect to ~ and is denoted: X/~.

Let us define the relation ~ on the set w of natural numbers by setting x ~ y if the remainders after dividing x and y by 3 are equal. Then w/~ consists of three equivalence classes corresponding to remainders 0, 1, and 2.

Order relation

A binary relation R on a set X is called antisymmetric , if from x R y and y R x follows: x = y. A binary relation R on a set X is called order relation , if it is reflexive, antisymmetric and transitive. It is easy to see that this is equivalent to the following conditions:

1) Id X Í R (reflexivity),

2) R Ç R -1 (antisymmetry),

3) R ° R Í R (transitivity).

An ordered pair (X, R) consisting of a set X and an order relation R on X is called partially ordered set .

Example 1

Let X = (0, 1, 2, 3), R = ((0, 0), (0, 1), (0, 2), (0, 3), (1, 1), (1, 2 ), (1, 3), (2, 2), (3, 3)).

Since R satisfies conditions 1–3, then (X, R) is a partially ordered set. For elements x = 2, y = 3, neither x R y nor y R x is true. Such elements are called incomparable . Usually the order relation is denoted by £. In the example above, 0 £ 1 and 2 £ 2, but it is not true that 2 £ 3.

Example 2

Let< – бинарное отношение строгого неравенства на множестве w натуральных чисел, рассмотренное в разд. 1.2. Тогда объединение отношений = и < является отношением порядка £ на w и превращает w в частично упорядоченное множество.

Elements x, y О X of a partially ordered set (X, £) are called comparable , if x £ y or y £ x.

The partially ordered set (X, £) is called linearly ordered or chain if any two of its elements are comparable. The set in Example 2 will be linearly ordered, but the set in Example 1 will not.

A subset A Í X of a partially ordered set (X, £) is called bounded from above , if there exists an element x н X such that a £ x for all a н A. An element x н X is called greatest in X if y £ x for all y О X. An element x О X is called maximal if there are no elements y О X different from x for which x £ y. In example 1, elements 2 and 3 will be the maximum, but not the largest. The bottom constraint subsets, least and minimum elements. In example 1, element 0 would be both the least and the minimum. In example 2, 0 also has these properties, but (w, t) has neither the greatest nor the maximum element.

Let (X, £) be a partially ordered set, A Í X a subset. A relation on A consisting of pairs (a, b) of elements a, b Î A, for which a £ b, will be an order relation on A. This relation is denoted by the same symbol: £. Thus, (A, £) is a partially ordered set. If it is linearly ordered, then we say that A is chain in (X, £).

Maximum principle

Some mathematical statements cannot be proved without the axiom of choice. These statements are said to be depend on the axiom of choice or valid in ZFC theory , in practice, instead of the axiom of choice, one usually uses for proof either Zermelo's axiom, or the Kuratovsky-Zorn lemma, or any other statement equivalent to the axiom of choice.

Lemma of Kuratowski-Zorn. If every chain in a partially ordered set(X, £) bounded from above, then X there is at least one maximum element.

This lemma is equivalent to the axiom of choice and can therefore be taken as an axiom.

Theorem.For any partially ordered set(X, £) there is a relation containing the relation£ and transforming X into a linearly ordered set.

Proof. The set of all order relations containing the relation £ is ordered by the inclusion relation U. Since the union of a chain of order relations is an order relation, then by the Kuratowski-Zorn lemma there exists a maximum relation R such that x £ y implies x R y. Let us prove that R is a relation linearly ordering X. Assume the contrary: let there exist a, b н X such that neither (a, b) nor (b, a) belong to R. Consider the relation:

R¢ = R È ((x, y): x R a and b R y).

It is obtained by adding the pair (a, b) to R and the pairs (x, y), which must be added to R¢ from the condition that R¢ is an order relation. It is easy to see that R¢ is reflexive, antisymmetric and transitive. We get R Ì R¢, which contradicts the maximality of R, therefore, R is the desired linear order relation.

A linearly ordered set X is called well-ordered if any of its non-empty subsets A н X contains the least element a н A. The Kuratowski-Zorn lemma and the axiom of choice are also equivalent to the following statement:

Zermelo's axiom. For every set there is an order relation that turns it into a well-ordered set.

For example, the set w of natural numbers is well ordered. The principle of inductance is summarized as follows:

Transfinite induction. If a(X, £) is a well-ordered set and F(x) is a property of its elements, true for the smallest element x 0 н X and such that from the truth of F(y) for all y < z следует истинность F(z), то F(x) true for everyone x О X .

Here y< z означает, что у £ z, но y ¹ z. Действительно, в противном случае среди x Î X, не обладающих свойством F(x), можно выбрать наименьший элемент x 1 , и выполнение F(y) для всех y < x 1 приводит к выполнению F(x 1), противоречащему предположению.

The concept of power

Let f: X à Y and g: Y à Z be set mappings. Since f and g are relations, their composition g ° f(x) = g(f(x)) is defined. If h: Z à T is a set mapping, then h ° (g ° f) = (h ° g) ° f. The relations Id X and Id Y are functions, so the compositions Id Y ° f = f ° Id x = f are defined. For X = Y, we define f 2 = f ° f, f 3 = f 2 ° f, ..., f n+1 = f n ° f.

The mapping f: X àY is called injection , if f(x 1) ¹ f(x 2) is true for any elements x 1 ¹ x 2 of the set X. The mapping f is called surjection , if for each y нY there exists x н X such that f(x) = y. If f is both a surjection and an injection, then f is called bijection . It is easy to see that f is a bijection if and only if the inverse relation f -1 н Y ´ X is a function.

We will say that the equality |X| = |Y| if there is a bijection between X and Y. Put |X| £ |Y| if there is an injection f: X à Y.

Cantor-Schroeder-Bernstein theorem. If a|X| £ |Y| and|Y| £ |X| , then|X| = |Y|.

Proof. By assumption, there are injections f: X à Y and g: Y à X. Let A = g¢¢Y = Img be the image of Y with respect to g. Then

(X \ A) Ç (gf)¢¢(X \ A) = Æ,

(gf)¢¢(X \ A) Ç (gf) 2 ¢¢(X \ A) = Æ, …,

(gf) n ¢¢(X \ A) Ç (gf) n+1 ¢¢(X \ A) = Æ, …

Consider a mapping j: X à A defined as j(x) = gf(x) with

x н (X \ A) È (gf)¢¢(X \ A) È (gf) 2 ¢¢(X \ A) È …, and j(x) = x otherwise. It is easy to see that j is a bijection. The desired bijection between X and Y will be equal to g -1 ° j.

Antinomy of Cantor

Let us set |X|< |Y|, если |X| £ |Y| и не существует биекции между X и Y.

Cantor's theorem. For any set X, |X|< |P(X)|, где P(X) – множество всех подмножеств множества X.

The following theorems can be proved.

Theorem 1.4. A function f has an inverse function f -1 if and only if f is bijective.

Theorem 1.5. The composition of bijective functions is a bijective function.

Rice. 1.12 show different relationships, all but the first one are functions.

attitude, but

injection, but

surjection, but

not a function

not a surjection

not an injection

Let f : A→ B be a function, and sets A and B be finite sets, let A = n , B = m . Dirichlet's principle states that if n > m, then at least one value of f occurs more than once. In other words, there is a pair of elements a i ≠ a j , a i , a j A for which f(a i )= f(a j ).

The Dirichlet principle is easy to prove, so we leave it to the reader as a trivial exercise. Consider an example. Let there be more than 12 students in the group. Then it is obvious that at least two of them have a birthday in the same month.

§ 7. Equivalence relation. Factor set

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

The relation of equality on the set of numbers has the indicated properties, therefore it is an equivalence relation.

The similarity relation of triangles is obviously an equivalence relation.

The relation of non-strict inequality (≤ ) on the set of real numbers will not be an equivalence relation, because it is not symmetric: from 3 ≤ 5 it does not follow that 5 ≤ 3.

An equivalence class (coset) generated by an element a for a given equivalence relation R is the subset of those x A that are in relation R with a. The specified equivalence class is denoted by [a] R, therefore, we have:

[a] R = (x A: a, x R).

Consider an example. A similarity relation is introduced on the set of triangles. It is clear that all equilateral triangles fall into one coset, since each of them is similar, for example, to a triangle, all sides of which have unit length.

Theorem 1.6. Let R be an equivalence relation on a set A and [a] R be a coset, i.e. [a] R = (x A: a, x R), then:

1) for any a A : [a] R ≠ , in particular, a [a] R ;

2) different cosets do not intersect;

3) the union of all cosets coincides with the entire set A;

4) the set of different cosets form a partition of the set A.

Proof. 1) Due to the reflexivity of R, we get that for any a, a A, we have a, a R , therefore a [ a] R and [ a] R ≠ ;

2) suppose that [a] R ∩ [b] R ≠ , i.e. there is an element c from A and c [a] R ∩ [b] R . Then from (cRa)&(cRb), due to the symmetry of R, we obtain (aR c)&(cRb), and from the transitivity of R we have aRb.

For any х [а] R we have: (хRa)&(аRb) , then due to the transitivity of R we obtain хRb, i.e. x[b]R, so [a]R[b]R. Similarly, for any y, y [b] R , we have: (уRb)&(аRb) , and due to the symmetry of R we get that (уRb)&(bR а), then, due to the transitivity of R, we get that уR а , i.e. y[a]r and

so [b] R [a] R . From [a] R [b] R and [b] R [a] R we get [a] R = [b] R, i.e. if cosets intersect, then they coincide;

3) for any a, a A, as proved, we have a [ a] R , then it is obvious that the union of all cosets coincides with the set A.

Assertion 4) of Theorem 1.6 follows from 1)–3). The theorem has been proven. We can prove the following theorem.

Theorem 1.7. Different equivalence relations on a set A give rise to different partitions of A.

Theorem 1.8. Each partition of the set A generates an equivalence relation on the set A, and different partitions generate different equivalence relations.

Proof. Let a partition В= (B i ) of the set A be given. Let's define the relation R : a,b R if and only if there exists a B i such that a and b both belong to this B i . It is obvious that the introduced relation is reflexive, symmetric and transitive, hence R is an equivalence relation. It can be shown that if the partitions are different, then the equivalence relations generated by them are also different.

The set of all cosets of a set A with respect to a given equivalence relation R is called a quotient set and is denoted by A/R . The elements of the factor set are cosets. The coset class [ a ] ​​R , as you know, consists of elements A that are in relation to each other R .

Consider an example of an equivalence relation on the set of integers Z = (…, -3, -2, -1, 0, 1, 2, 3, …).

Two integers a and b are called comparable (congruent) modulo m if m is a divisor of the number a-b, i.e. if we have:

a=b+km , k=…, -3, -2, -1, 0, 1, 2, 3, ….

In this case, write a≡ b(mod m) .

Theorem 1.9. For any numbers a , b , c and m>0 we have:

1) a ≡ a(mod m) ;

2) if a ≡ b(mod m), then b ≡ a(mod m);

3) if a ≡ b(mod m) and b ≡ c(mod m), then a ≡ c(mod m).

Proof. Statements 1) and 2) are obvious. Let us prove 3). Let a=b+k 1 m , b=c+k 2 m , then a=c+(k 1 +k 2 )m , i.e. a ≡ c(mod m) . The theorem has been proven.

Thus, the relation of comparability modulo m is an equivalence relation and divides the set of integers into non-overlapping classes of numbers.

Let us construct an infinitely unwinding spiral, which in Fig. 1.13 is depicted with a solid line, and an infinitely twisting spiral, depicted with a dashed line. Let a non-negative integer m be given. We place all integers (elements from the set Z ) at the intersection points of these spirals with m rays, as shown in Fig. 1.13.

For the relation of comparability modulo m (in particular, for m = 8) the equivalence class is the numbers lying on the ray. Obviously, each number falls into one and only one class. It can be obtained that for m= 8 we have:

[ 0] ={…, -8, 0, 8, 16, …};
[ 1] ={…, -7, 1, 9, 17, …};
[ 2] ={…, -6, 2, 10, 18, …};
[ 7] ={…, -9, -1, 7, 15, …}.

The set factor of a set Z with respect to comparison modulo m is denoted as Z/m or as Z m . For the case under consideration, m =8

we obtain that Z/8 = Z8 = ( , , , …, ) .

Theorem 1.10. For any integers a, b, a * , b * , k and m :

1) if a ≡ b(mod m), then ka ≡ kb(mod m);

2) if a ≡ b(mod m) and a* ≡ b* (mod m), then:

a) a + a * ≡ b + b * (mod m); b) aa * ≡ bb* (mod m).

We present the proof for case 2b). Let a ≡ b(mod m) and a * ≡ b * (mod m) , then a=b+sm and a * =b * +tm for some integers s and t . Multiplying,

we get: aa* =bb* + btm+ b* sm+ stm2 =bb* +(bt+ b* s+ stm)m. Consequently,

aa* ≡ bb* (mod m).

Thus, modulo comparisons can be added and multiplied term by term, i.e. operate in exactly the same way as with equalities. For example,

Let R be a binary relation on a set X. The relation R is called reflective , if (x, x) О R for all x О X; symmetrical – if (x, y) О R implies (y, x) О R; the transitive number 23 corresponds to the variant 24 if (x, y) Î R and (y, z) Î R imply (x, z) Î R.

Example 1

We will say that x н X has in common with element y н X if the set
x З y is not empty. The relation to have in common will be reflexive and symmetrical, but not transitive.

Equivalence relation on X is called a reflexive, transitive, and symmetric relation. It is easy to see that R Н X ´ X will be an equivalence relation if and only if the inclusions take place:

Id X Í R (reflexivity),

R -1 Í R (symmetry),

R ° R Í R (transitivity).

In fact, these three conditions are equivalent to the following:

Id X Í R, R -1 = R, R ° R = R.

splitting set X is a set A of pairwise disjoint subsets a н X such that UA = X. With each partition of A, we can associate an equivalence relation ~ on X by setting x ~ y if x and y are elements of some a н A.

To each equivalence relation ~ on X there corresponds a partition A whose elements are subsets, each of which consists of those in the relation ~. These subsets are called equivalence classes . This partition A is called the factor set of the set X with respect to ~ and is denoted: X/~.

Let us define the relation ~ on the set w of natural numbers by setting x ~ y if the remainders after dividing x and y by 3 are equal. Then w/~ consists of three equivalence classes corresponding to remainders 0, 1, and 2.

Order relation

A binary relation R on a set X is called antisymmetric , if from x R y and y R x follows: x = y. A binary relation R on a set X is called order relation , if it is reflexive, antisymmetric and transitive. It is easy to see that this is equivalent to the following conditions:

1) Id X Í R (reflexivity),

2) R Ç R -1 (antisymmetry),

3) R ° R Í R (transitivity).

An ordered pair (X, R) consisting of a set X and an order relation R on X is called partially ordered set .

Example 1

Let X = (0, 1, 2, 3), R = ((0, 0), (0, 1), (0, 2), (0, 3), (1, 1), (1, 2 ), (1, 3), (2, 2), (3, 3)).

Since R satisfies conditions 1–3, then (X, R) is a partially ordered set. For elements x = 2, y = 3, neither x R y nor y R x is true. Such elements are called incomparable . Usually the order relation is denoted by £. In the example above, 0 £ 1 and 2 £ 2, but it is not true that 2 £ 3.

Example 2

Let< – бинарное отношение строгого неравенства на множестве w натуральных чисел, рассмотренное в разд. 1.2. Тогда объединение отношений = и < является отношением порядка £ на w и превращает w в частично упорядоченное множество.

Elements x, y О X of a partially ordered set (X, £) are called comparable , if x £ y or y £ x.

The partially ordered set (X, £) is called linearly ordered or chain if any two of its elements are comparable. The set in Example 2 will be linearly ordered, but the set in Example 1 will not.

A subset A Í X of a partially ordered set (X, £) is called bounded from above , if there exists an element x н X such that a £ x for all a н A. An element x н X is called greatest in X if y £ x for all y О X. An element x О X is called maximal if there are no elements y О X different from x for which x £ y. In example 1, elements 2 and 3 will be the maximum, but not the largest. The bottom constraint subsets, least and minimum elements. In example 1, element 0 would be both the least and the minimum. In example 2, 0 also has these properties, but (w, t) has neither the greatest nor the maximum element.