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The structure of some number sets. Continuum (set theory) Zeno's continuum paradoxes and their solution by Aristotle

There are infinite sets whose elements cannot be renumbered. Such sets are called uncountable.

Cantor's theorem. The set of all points on a segment is uncountable.

Proof.

Let the set of points of the segment be countable. This means that these points can be renumbered, i.e. arranged in a sequence x 1 , x 2 … x n, … .

Let's divide the segment into three equal parts. Wherever the point is x 1, it cannot belong to all segments , , . Therefore, among them there is a segment D 1 that does not contain the point x 1 (Fig. 1.7). Let's take this segment D 1 and divide it into three equal parts. Among them there is always a segment D 2 that does not contain a point x 2. Let's divide this segment into three equal parts, etc. We obtain a sequence of segments D 1 É D 2 É D 3 É…ÉD nÉ… . By virtue of Cantor's axiom, converges to a certain point x at n® ¥. By construction this point x belongs to each segment D 1, D 2, D 3,…, D n, ..., i.e. it cannot coincide with any of the points x 1 , x 2 ,… x n, ..., i.e. the sequence x 1 , x 2 … x n, ...does not exhaust all points of the segment, which contradicts the initial assumption. The theorem has been proven.

The set equivalent to the set of all points of a segment is called continuum power set.

Since the sets of points of intervals, segments and the entire line are equivalent to each other, they all have the power of a continuum.

To prove that a given set has the cardinality of a continuum, it is enough to indicate a one-to-one correspondence between this set and the set of points on a segment, interval, or the entire line.

Example 1.24.

From Fig. 1.8 it follows that the set of points of the parabola y= x 2 is equivalent to the set of points on the line –¥< x < ¥ и, следовательно, имеет мощность континуума.

You can also set the continuum power using the following theorems on continuum power sets(given without evidence).

Theorem 1. The set of all subsets of a countable set is countable.

Theorem 2. The set of irrational numbers has the power of a continuum.

Theorem 3. Set of all points n- dimensional space for any n has the power of continuum.

Theorem 4. The set of all complex numbers has the cardinality of a continuum.

Theorem 5. The set of all continuous functions defined on the interval [ a, b] has the power of continuum.

So, the cardinalities of infinite sets may differ. The power of the continuum is greater than the power of a countable set. The answer to the question whether there are sets of higher cardinality than the cardinality of the continuum is given by the following theorem (given without proof).

Theorem on sets of higher cardinality. The set of all subsets of a given set has a higher cardinality than the given set.

From this theorem it follows that there are no sets with the largest cardinality.

Test questions for topic 1

1. Let aÎ A. Does it follow from this that ( a} A?

2. In what case A AÇ IN?

3. Name a set that is a subset of any set.

4. Can a set be equivalent to its subset?

5. Which set has more cardinality: the set of natural numbers or the set of points on the segment?

TOPIC 2. RELATIONSHIPS. FUNCTIONS

Relationship. Basic concepts and definitions

Definition 2.1.Ordered pair<x, y> called a collection of two elements x And y, arranged in a certain order.

Two ordered pairs<x, y> and<u, v> are equal to each other if and only if x = u And y= v.

Example 2.1.

<a, b>, <1, 2>, <x, 4> – ordered pairs.

Similarly, we can consider triplets, quadruples, n-ki elements<x 1 , x 2 ,… x n>.

Definition 2.2.Direct(or Cartesian)work two sets A And B is the set of ordered pairs such that the first element of each pair belongs to the set A, and the second – to the set B:

A ´ B = {<a, b>, ç aÎ A And bÏ IN}.

In general, the direct product n sets A 1 ,A 2 ,…A n called a set A 1´ A 2 ´…´ A n, consisting of ordered sets of elements<a 1 , a 2 , …,a n> length n, such that i- th a i belongs to the set A i,a i Î A i.

Example 2.2.

Let A = {1, 2}, IN = {2, 3}.

Then A ´ B = {<1, 2>, <1, 3>,<2, 2>,<2, 3>}.

Example 2.3.

Let A= {x ç0 £ x£ 1) and B= {yç2 £ y£3)

Then A ´ B = {<x, y >, ç0 £ x£1&2£ y£3).

Thus, many A ´ B consists of points lying inside and on the border of a rectangle formed by straight lines x= 0 (y-axis), x= 1,y= 2i y = 3.

The French mathematician and philosopher Descartes was the first to propose a coordinate representation of points on a plane. This is historically the first example of a direct product.

Definition 2.3.Binary(or double)ratio r is called the set of ordered pairs.

If a couple<x, y>belongs r, then it is written as follows:<x, y> Î r or, what is the same, xr y.

Example2.4.

r = {<1, 1>, <1, 2>, <2, 3>}

Similarly we can define n-local relation as a set of ordered n-OK.

Since a binary relation is a set, the methods for specifying a binary relation are the same as the methods for specifying a set (see Section 1.1). A binary relation can be specified by listing ordered pairs or by specifying a general property of ordered pairs.

Example 2.5.

1. r = {<1, 2>, <2, 1>, <2, 3>) – the relation is specified by enumerating ordered pairs;

2. r = {<x, y> ç x+ y = 7, x, y– real numbers) – the relation is specified by specifying the property x+ y = 7.

Additionally, a binary relation can be given binary relation matrix. Let A = {a 1 , a 2 , …, a n) is a finite set. Binary relation matrix C is a square matrix of order n, whose elements c ij are defined as follows:

c ij =

Example 2.6.

A= (1, 2, 3, 4). Let's define a binary relation r in the three listed ways.

1. r = {<1, 2>, <1, 3>, <1, 4>, <2, 3>, <2, 4>, <3, 4>) – the relation is specified by enumerating all ordered pairs.

2. r = {<a i, a j> ç a i < a j; a i, a jÎ A) – the relation is specified by indicating the property “less than” on the set A.

3. – the relation is specified by the binary relation matrix C.

Example 2.7.

Let's look at some binary relationships.

1. Relations on the set of natural numbers.

a) the relation £ holds for pairs<1, 2>, <5, 5>, but does not hold for the pair<4, 3>;

b) the relation “have a common divisor other than one” holds for pairs<3, 6>, <7, 42>, <21, 15>, but does not hold for the pair<3, 28>.

2. Relations on the set of points of the real plane.

a) the relation “to be at the same distance from the point (0, 0)” is satisfied for points (3, 4) and (–2, Ö21), but is not satisfied for points (1, 2) and (5, 3);

b) the relation “to be symmetrical about the axis OY" is performed for all points ( x, y) And (- x, –y).

3. Relationships with many people.

a) the attitude of “living in the same city”;

b) the attitude of “studying in the same group”;

c) the “being older” attitude.

Definition 2.4. The domain of definition of a binary relation r is the set D r = (x çthere is y such that xr y).

Definition 2.5. The range of values of a binary relation r is the set R r = (y çexists x such that xr y).

Definition 2.6. The domain of specifying a binary relation r is called the set M r = D r ÈR r .

Using the concept of direct product, we can write:

rÎ D r´ R r

If D r= R r = A, then we say that the binary relation r defined on the set A.

Example 2.8.

Let r = {<1, 3>, <3, 3>, <4, 2>}.

Then D r ={1, 3, 4}, R r = {3, 2}, Mr= {1, 2, 3, 4}.

Operations on relationships

Since relations are sets, all operations on sets are valid for relations.

Example 2.9.

r 1 = {<1, 2>, <2, 3>, <3, 4>}.

r 2 = {<1, 2>, <1, 3>, <2, 4>}.

r 1 È r 2 = {<1, 2>, <1, 3>, <2, 3>, <2, 4>, <3, 4>}.

r 1 Ç r 2 = {<1, 2>}.

r 1 \ r 2 = {<2, 3>, <3, 4>}.

Example 2.10.

Let R– set of real numbers. Let us consider the following relations on this set:

r 1 – "£"; r 2 – " = "; r 3 – " < "; r 4 – "³"; r 5 – " > ".

r 1 = r 2 È r 3 ;

r 2 = r 1 Ç r 4 ;

r 3 = r 1 \ r 2 ;

r 1 = ;

Let us define two more operations on relations.

Definition 2.7. The relationship is called reverse to attitude r(denoted r – 1), if

r – 1 = {<x, y> ç< y, x> Î r}.

Example 2.11.

r = {<1, 2>, <2, 3>, <3, 4>}.

r – 1 = {<2, 1>, <3, 2>, <4, 3>}.

Example 2.12.

r = {<x, y> ç x – y = 2, x, y Î R}.

r – 1 = {<x, y> ç< y, x> Î r} = r – 1 = {<x, y> ç y–x = 2, x, y Î R} = {<x, y> ç– x+ y = 2, x, y Î R}.

Definition 2.8.Composition of two relations r and s called relation

s r= {<x, z> çthere is such a thing y, What<x, y> Î r And< y, z> Î s}.

Example 2.13.

r = {<x, y> ç y = sinx}.

s= {<x, y> ç y = Ö x}.

s r= {<x, z> çthere is such a thing y, What<x, y> Î r And< y, z> Î s} = {<x, z> çthere is such a thing y, What y = sinx And z= Ö y} = {<x, z> ç z= Ö sinx}.

The definition of the composition of two relations corresponds to the definition of a complex function:

y = f(x), z= g(y) Þ z= g(f(x)).

Example 2.14.

r = {<1, 1>, <1, 2>, <1, 3>, <3, 1>}.

s = {<1, 2>, <1, 3>, <2, 2>, <3, 2>, <3, 3>}.

Finding process s r in accordance with the definition of composition, it is convenient to depict it in a table in which all possible values are enumerated x, y, z. for each pair<x, y> Î r we need to consider all possible pairs< y, z> Î s(Table 2.1).

Table 2.1

<x, y> Î r	< y, z> Î s	<x, z> Î s r
<1, 1> <1, 1> <1, 2> <1, 3> <1, 3> <3, 1> <3, 1>	<1, 2> <1, 3> <2, 2> <3, 2> <3, 3> <1, 2> <1, 3>	<1, 2> <1, 3> <1, 2> <1, 2> <1, 3> <3, 2> <3, 3>

Note that the first, third and fourth, as well as the second and fifth rows of the last column of the table contain identical pairs. Therefore we get:

s r= {<1, 2>, <1, 3>, <3, 2>, <3, 3>}.

Properties of relationships

Definition 2.9. Attitude r called reflective on a set X, if for any xÎ X running xr x.

From the definition it follows that every element<x,x > Î r.

Example 2.15.

a) Let X– finite set, X= (1, 2, 3) and r = {<1, 1>, <1, 2>, <2, 2>, <3, 1>, <3, 3>). Attitude r reflectively. If X is a finite set, then the main diagonal of the reflexive relation matrix contains only ones. For our example

b) Let X r relation of equality. This attitude is reflexive, because every number is equal to itself.

c) Let X- a lot of people and r"live in the same city" attitude. This attitude is reflexive, because everyone lives in the same city with themselves.

Definition 2.10. Attitude r called symmetrical on a set X, if for any x, yÎ X from xry should yr x.

It's obvious that r symmetrical if and only if r = r – 1 .

Example 2.16.

a) Let X– finite set, X= (1, 2, 3) and r = {<1, 1>, <1, 2>, <1, 3>, <2, 1>, <3, 1>, <3, 3>). Attitude r symmetrically. If X is a finite set, then the symmetric relation matrix is symmetric with respect to the main diagonal. For our example

b) Let X– set of real numbers and r relation of equality. This relationship is symmetrical, because If x equals y, then y equals x.

c) Let X– many students and r"study in the same group" attitude. This relationship is symmetrical, because If x studies in the same group as y, then y studies in the same group as x.

Definition 2.11. Attitude r called transitive on a set X, if for any x, y,zÎ X from xry And yr z should xr z.

Simultaneous fulfillment of conditions xry, yr z, xr z means that the pair<x,z> belongs to the composition r r. Therefore for transitivity r it is necessary and sufficient for the set r r was a subset r, i.e. r rÍ r.

Example 2.17.

a) Let X– finite set, X= (1, 2, 3) and r = {<1, 1>, <1, 2>, <2, 3>, <1, 3>). Attitude r transitive, because along with pairs<x,y>and<y,z>have a couple<x,z>. For example, along with pairs<1, 2>, And<2, 3>there is a pair<1, 3>.

b) Let X– set of real numbers and r ratio £ (less than or equal to). This relation is transitive, because If x£ y And y£ z, That x£ z.

c) Let X- a lot of people and r"being older" attitude. This relation is transitive, because If x older y And y older z, That x older z.

Definition 2.12. Attitude r called equivalence relation on a set X, if it is reflexive, symmetric and transitive on the set X.

Example 2.18.

a) Let X– finite set, X= (1, 2, 3) and r = {<1, 1>, <2, 2>, <3, 3>). Attitude r is an equivalence relation.

b) Let X– set of real numbers and r relation of equality. This is an equivalence relation.

c) Let X– many students and r"study in the same group" attitude. This is an equivalence relation.

Let r X.

Definition 2.13. Let r– equivalence relation on the set X And xÎ X. Equivalence class, generated by the element x, is called a subset of the set X, consisting of those elements yÎ X, for which xry. Equivalence class generated by element x, denoted by [ x].

Thus, [ x] = {yÎ X|xry}.

The equivalence classes form partition sets X, i.e., a system of non-empty pairwise disjoint subsets of it, the union of which coincides with the entire set X.

Example 2.19.

a) The equality relation on the set of integers generates the following equivalence classes: for any element x from this set [ x] = {x), i.e. each equivalence class consists of one element.

b) The equivalence class generated by the pair<x, y> is determined by the relation:

[<x, y>] = .

Each equivalence class generated by a pair<x, y>, defines one rational number.

c) For the relation of belonging to one student group, the equivalence class is the set of students of the same group.

Definition 2.14. Attitude r called antisymmetric on a set X, if for any x, yÎ X from xry And yr x should x = y.

From the definition of antisymmetry it follows that whenever a pair<x,y> belongs at the same time r And r – 1 , the equality must be satisfied x = y. In other words, r Ç r – 1 consists only of pairs of the form<x,x >.

Example 2.20.

a) Let X– finite set, X= (1, 2, 3) and r = {<1, 1>, <1, 2>, <1, 3>, <2, 2>, <2, 3>, <3, 3>). Attitude r antisymmetric.

Attitude s= {<1, 1>, <1, 2>, <1, 3>, <2, 1>, <2, 3>, <3, 3>) is non-antisymmetric. For example,<1, 2> Î s, And<2, 1> Î s, but 1¹2.

b) Let X– set of real numbers and r ratio £ (less than or equal to). This relationship is antisymmetric, because If x £ y, And y £ x, That x = y.

Definition 2.15. Attitude r called partial order relation(or just a partial order) on the set X, if it is reflexive, antisymmetric and transitive on the set X. Many X in this case it is called partially ordered and the indicated relation is often denoted by the symbol £, if this does not lead to misunderstandings.

The inverse of the partial order relation will obviously be a partial order relation.

Example 2.21.

a) Let X– finite set, X= (1, 2, 3) and r = {<1, 1>, <1, 2>, <1, 3>, <2, 2>, <2, 3>, <3, 3>). Attitude r

b) Attitude AÍ IN on the set of subsets of some set U there is a partial order relation.

c) The divisibility relation on the set of natural numbers is a partial order relation.

Functions. Basic concepts and definitions

In mathematical analysis, the following definition of a function is accepted.

Variable y called a function of a variable x, if according to some rule or law each value x corresponds to one specific value y = f(x). Variable change area x is called the domain of definition of a function, and the domain of change of a variable y– range of function values. If one value x corresponds to several (and even infinitely many values) y), then the function is called multivalued. However, in the course on the analysis of functions of real variables, multi-valued functions are avoided and single-valued functions are considered.

Let's consider another definition of function in terms of relationships.

Definition 2.16. Function is any binary relation that does not contain two pairs with equal first components and different second ones.

This property of a relationship is called unambiguity or functionality.

Example 2.22.

A) (<1, 2>, <3, 4>, <4, 4>, <5, 6>) – function.

b) (<x, y>: x, y Î R, y = x 2) – function.

V) (<1, 2>, <1, 4>, <4, 4>, <5, 6>) is a relation, but not a function.

Definition 2.17. If f– function, then D f – domain of definition, A Rf – range functions f.

Example 2.23.

For example 2.22 a) D f – {1, 3, 4, 5}; Rf – {2, 4, 6}.

For example 2.22 b) D f = Rf = (–¥, ¥).

Each element x D f function matches the only one element y Rf. This is denoted by the well-known notation y = f(x). Element x called function argument or element preimage y with function f, and the element y function value f on x or element image x at f.

So, from all relations, functions stand out in that each element from the domain of definition has the only one image.

Definition 2.18. If D f = X And Rf = Y, then they say that the function f determined on X and takes its values on Y, A f called mapping the set X to Y(X ® Y).

Definition 2.19. Functions f And g are equal if their domain is the same set D, and for anyone x Î D equality is true f(x) = g(x).

This definition does not contradict the definition of equality of functions as equality of sets (after all, we defined a function as a relation, i.e., a set): sets f And g are equal if and only if they consist of the same elements.

Definition 2.20. Function (display) f called surjective or just surjection, if for any element y Y there is an element x Î X, such that y = f(x).

So every function f is a surjective mapping (surjection) D f® Rf.

If f is a surjection, and X And Y are finite sets, then ³ .

Definition 2.21. Function (display) f called injective or just injection or one-to-one, if from f(a) = f(b) should a = b.

Definition 2.22. Function (display) f called bijective or just bijection, if it is both injective and surjective.

If f is a bijection, and X And Y are finite sets, then = .

Definition 2.23. If the range of the function D f consists of one element, then f called constant function.

Example 2.24.

A) f(x) = x 2 is a mapping from the set of real numbers to the set of non-negative real numbers. Because f(–a) = f(a), And a ¹ – a, then this function is not an injection.

b) For everyone x R= (– , ) function f(x) = 5 – constant function. It displays many R to set (5). This function is surjective, but not injective.

V) f(x) = 2x+ 1 is an injection and a bijection, because out of 2 x 1 +1 = 2x 2 +1 follows x 1 = x 2 .

Definition 2.24. Function that implements the display X 1´ X 2 ´...´ Xn ® Y called n-local function.

Example 2.25.

a) Addition, subtraction, multiplication and division are two-place functions on a set R real numbers, i.e. functions like R 2® R.

b) f(x, y) = is a two-place function that implements the mapping R ´ ( R \ )® R. This function is not an injection, because f(1, 2) = f(2, 4).

c) The lottery winnings table specifies a two-place function that establishes a correspondence between pairs of N 2 (N– a set of natural numbers) and a set of winnings.

Since functions are binary relations, it is possible to find inverse functions and apply the composition operation. The composition of any two functions is a function, but not for every function f attitude f–1 is a function.

Example 2.26.

A) f = {<1, 2>, <2, 3>, <3, 4>, <4, 2>) – function.

Attitude f –1 = {<2, 1>, <3, 2>, <4, 3>, <2, 4>) is not a function.

b) g = {<1, a>, <2, b>, <3, c>, <4, D>) is a function.

g -1 = {<a, 1>, <b, 2>, <c, 3>, <D, 4>) is also a function.

c) Find the composition of functions f from example a) and g-1 from example b). We have g -1f = {<a, 2>, <b, 3>, <c, 4>, <d, 2>}.

fg-1 = Æ.

Note that ( g -1f)(a) = f(g -1 (a)) = f(1) = 2; (g -1f)(c) = f(g -1 (c)) = f(3) = 4.

An elementary function in mathematical analysis is any function f, which is a composition of a finite number of arithmetic functions, as well as the following functions:

1) Fractional-rational functions, i.e. functions of the form

a 0 + a 1 x + ... + a n x n

b 0 + b 1 x + ... + b m x m.

2) Power function f(x) = x m, Where m– any constant real number.

3) Exponential function f(x) = e x.

4) logarithmic function f(x) = log a x, a >0, a 1.

5) Trigonometric functions sin, cos, tg, ctg, sec, csc.

6) Hyperbolic functions sh, ch, th, cth.

7) Inverse trigonometric functions arcsin, arccos etc.

For example, the function log 2 (x 3 +sincos 3x) is elementary, because it is a composition of functions cosx, sinx, x 3 , x 1 + x 2 , logx, x 2 .

An expression describing the composition of functions is called a formula.

For a multiplace function, the following important result is valid, obtained by A. N. Kolmogorov and V. I. Arnold in 1957 and which is a solution to Hilbert’s 13th problem:

Theorem. Any continuous function n variables can be represented as a composition of continuous functions of two variables.

Methods for specifying functions

1. The simplest way to specify functions is through tables (Table 2.2):

Table 2.2

However, functions defined on finite sets can be defined in this way.

If a function defined on an infinite set (segment, interval) is given at a finite number of points, for example, in the form of trigonometric tables, tables of special functions, etc., then interpolation rules are used to calculate the values of functions at intermediate points.

2. A function can be specified as a formula that describes the function as a composition of other functions. The formula specifies the sequence for calculating the function.

Example 2.28.

f(x) = sin(x + Ö x) is a composition of the following functions:

g(y) = Ö y; h(u, v) = u+ v; w(z) = sinz.

3. The function can be specified as recursive procedure. The recursive procedure specifies a function defined on the set of natural numbers, i.e. f(n), n= 1, 2,... as follows: a) set the value f(1) (or f(0)); b) value f(n+ 1) determined through composition f(n) and other known functions. The simplest example of a recursive procedure is the calculation n!: a) 0! = 1; b) ( n + 1)! = n!(n+ 1). Many numerical methods procedures are recursive procedures.

4. There are possible methods for specifying a function that do not contain a method for calculating the function, but only describe it. For example:

f M(x) =

Function f M(x) – characteristic function of the set M.

So, according to the meaning of our definition, set the function f– means to set the display X ® Y, i.e. define a set X´ Y, so the question comes down to specifying a certain set. However, it is possible to define the concept of a function without using the language of set theory, namely: a function is considered given if a computational procedure is given that, given the value of the argument, finds the corresponding value of the function. A function defined this way is called computable.

Example 2.29.

Determination procedure Fibonacci numbers, is given by the relation

Fn= Fn- 1 + Fn- 2 (n³ 2) (2.1)

with initial values F 0 = 1, F 1 = 1.

Formula (2.1) together with the initial values determines the following series of Fibonacci numbers:

n	0 1 2 3 4 5 6 7 8 9 10 11 …
Fn	1 1 2 3 5 8 13 21 34 55 89 144 …

The computational procedure for determining the value of a function from a given argument value is nothing more than algorithm.

Test questions for topic 2

1. Indicate ways to define a binary relation.

2. The main diagonal of the matrix of which relation contains only ones?

3. For what relationship? r the condition is always met r = r – 1 ?

4. For what attitude r the condition is always met r rÍ r.

5. Introduce equivalence relations and partial order on the set of all lines in the plane.

6. Specify ways to specify functions.

7. Which of the following statements is true?

a) Every binary relation is a function.

b) Every function is a binary relation.

Topic 3. GRAPHS

Euler's first work on graph theory appeared in 1736. In the beginning, this theory was associated with mathematical puzzles and games. However, subsequently graph theory began to be used in topology, algebra, and number theory. Nowadays, graph theory is used in a wide variety of areas of science, technology and practical activity. It is used in the design of electrical networks, transportation planning, and the construction of molecular circuits. Graph theory is also used in economics, psychology, sociology, and biology.

Style: . A set with cardinality continuum is called continual many.

Also term continuum can denote the set of real numbers itself, or even any continuum set.

Properties

Examples

Examples of sets with cardinality continuum:

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See what “Continuum (set theory)” is in other dictionaries:

A theory in which sets (classes) of elements of arbitrary nature are studied. Created primarily by the works of Cantor (as well as R. Dedekind and K. Weierstrass), T. m. by the end of the 19th century. became the basis for constructing the mathematical systems that had developed by that time... ... Philosophical Encyclopedia

Set theory is a branch of mathematics that studies the general properties of sets. Set theory underlies most mathematical disciplines; she had a profound influence on the understanding of the subject itself... ... Wikipedia

SET THEORY- a branch of mathematics that studies the general properties of sets. A set is any combination into one whole of some specific and distinct objects of our perception or thought. In technical mathematics, the general properties of various operations are studied... ... Encyclopedic Dictionary of Psychology and Pedagogy

Direction in mathematics. logic, which deals with the study of fragments of meaningful set theory using mathematical methods. logic. Usually, for this purpose, fragments of set theory are formalized in the form of a formal axiomatic theory. theories. In a narrower sense... ... Mathematical Encyclopedia

Formulation of set theory (See Set theory) in the form of a formal (axiomatic) system (See Axiomatic method). The main incentive for the construction of A. t. m. was the discovery of G. Cantor in the “naive” theory of sets.… … Great Soviet Encyclopedia

Set theory is a branch of mathematics that studies the general properties of sets. Set theory underlies most mathematical disciplines; it had a profound influence on the understanding of the subject of mathematics itself. Contents 1 Theory ... ... Wikipedia

From lat. continuum continuous, continuous. Continuum (in physics) In mathematics: Continuum (set theory) is a set equal to the set of real numbers R, or the class of all such sets. Continuum (topology) connected... ... Wikipedia

Mathematician, theory that studies the problem of infinity by precise means. Subject of M. t. properties of sets (collections, classes, ensembles), ch. arr. endless. Basic content classic M. t. was developed in Germany. mathematician G.... ... Philosophical Encyclopedia

- (from Latin continuum continuous), term used? mathematics, science and philosophy. In mathematics, quantity is understood as infinite sets that are quantitatively equivalent to the set of real ones. numbers. Power, or cardinal number... Philosophical Encyclopedia

R of all real numbers, 2) the set of all points of the interval (0, 1); 3) the set of all irrational numbers from this interval, 4) the set of all points in the space R n, where n is natural; 5) the set of all transcendental numbers; 6) the set of all continuous functions of a real variable quantum mechanics cannot be represented as a countable sum of smaller cardinal numbers. For any cardinal number a such that

In particular,

Continuum hypothesis states that K. m. is the first uncountable cardinal number, i.e.

Lit.: Kuratovsky K., Mostovsky A., Set theory, trans. from English, M., 1970.

B. A. Efimov.

Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.

See what "CONTINUUM POWER" is in other dictionaries:

The cardinality of a set, the cardinal number of a set (lat. cardinalis ← cardo main circumstance, core, core) is a characteristic of sets (including infinite ones), generalizing the concept of the number (number) of elements of a finite ... ... Wikipedia

The task consists of proving or disproving by means of set theory (See Set theory) the following statement, called the continuum hypothesis (KH): the power of the Continuum is the first power, exceeding the power... ...

The cardinal number of a set A is a property of this set that is inherent in any set B equivalent to A. Moreover, the two sets are called. equivalent (or equally powerful) if it is possible to establish a one-to-one relationship between them... ... Mathematical Encyclopedia

Philosophy categories that characterize both the structure of matter and the process of its development. Discontinuity means “granularity”, discreteness of the spatio-temporal structure and state of matter, its constituent elements, types and forms... ... Philosophical Encyclopedia

- (Gödel) Kurt (1906 1978) mathematician and logician, member of the US National Academy of Sciences and the American Philosophical Society, author of the fundamental discovery of the limitations of the axiomatic method and fundamental works in such directions... ...

Mathematician and logician, member of the US National Academy of Sciences and the American Philosophical Society, author of the fundamental discovery of the limitations of the axiomatic method and fundamental works in such areas of mathematical logic as theory... ... History of Philosophy: Encyclopedia

The cardinality of a set or the cardinal number of a set is a generalization of the concept of quantity (the number of elements of a set), which makes sense for all sets, including infinite ones. There are large, there are smaller infinite sets, among them... ... Wikipedia

Philosophy a category characterizing the inexhaustibility of matter and movement, the diversity of phenomena and objects of the material world, forms and trends of its development. Recognizing the objective existence of B. in nature, dialectic. materialism rejects... Philosophical Encyclopedia

The doctrine of the general properties of sets, mainly infinite. The concept of a set, or collection, is one of the simplest mathematical concepts; it is not defined, but can be explained with examples. Yes, you can... ... Great Soviet Encyclopedia