Lots of numbers. Laws of actions on various numbers. The set is closed under the operation Relationship between the complements of open and closed sets
Let us now prove some special properties of closed and open sets.
Theorem 1. The sum of a finite or countable number of open sets is an open set. The product of a finite number of open sets is an open set,
Consider the sum of a finite or countable number of open sets:
If , then P belongs to at least one of Let Since is an open set, then some -neighborhood of P also belongs. The same -neighborhood of P also belongs to the sum g, from which it follows that g is an open set. Let us now consider the final product
and let P belong to g. Let us prove, as above, that some -neighborhood of P also belongs to g. Since P belongs to g, then P belongs to everyone. Since - are open sets, then for any there is some -neighborhood of the point belonging to . If the number is taken to be equal to the smallest of which the number is finite, then the -neighborhood of the point P will belong to everyone and, consequently, to g. Note that we cannot claim that the product of a countable number of open sets is an open set.
Theorem 2. The set CF is open and the set CO is closed.
Let's prove the first statement. Let P belong to CF. It is necessary to prove that some neighborhood P belongs to CF. This follows from the fact that if there were points F in any -neighborhood of P, the point P, which does not belong by condition, would be a limit point for F and, due to its closedness, should belong, which leads to a contradiction.
Theorem 3. The product of a finite or countable number of closed sets is a closed set. The sum of a finite number of closed sets is a closed set.
Let us prove, for example, that the set
closed. Moving on to additional sets, we can write
By theorem, sets are open, and, by Theorem 1, the set is also open, and thus the additional set g is closed. Note that the sum of a countable number of closed sets may also turn out to be an open set.
Theorem 4. A set is an open set and a closed set.
It is easy to check the following equalities:
From these, by virtue of the previous theorems, Theorem 4 follows.
We will say that a set g is covered by a system M of certain sets if every point g is included in at least one of the sets of the system M.
Theorem 5 (Borel). If a closed bounded set F is covered by an infinite system a of open sets O, then from this infinite system it is possible to extract a finite number of open sets that also cover F.
We prove this theorem by inverse. Let us assume that no finite number of open sets from the system a covers and we bring this to a contradiction. Since F is a bounded set, then all points of F belong to some finite two-dimensional interval. Let us divide this closed interval into four equal parts, dividing the intervals in half. We will take each of the resulting four intervals to be closed. Those points of F that fall on one of these four closed intervals will, by virtue of Theorem 2, represent a closed set, and at least one of these closed sets cannot be covered by a finite number of open sets from the system a. We take one of the four closed intervals indicated above where this circumstance occurs. We again divide this interval into four equal parts and reason in the same way as above. Thus, we obtain a system of nested intervals of which each next represents a fourth part of the previous one, and the following circumstance holds: the set of points F belonging to any k cannot be covered by a finite number of open sets from the system a. With an infinite increase in k, the intervals will infinitely shrink to a certain point P, which belongs to all intervals. Since for any k they contain an infinite number of points, the point P is a limiting point for and therefore belongs to F, since F is a closed set. Thus, the point P is covered by some open set belonging to the system a. Some -neighborhood of the point P will also belong to the open set O. For sufficiently large values of k, the intervals D will fall inside the above -neighborhood of the point P. Thus, these will be entirely covered by only one open set O of the system a, and this contradicts the fact that the points belonging to for any k cannot be covered by a finite number of open sets belonging to a. Thus the theorem is proven.
Theorem 6. An open set can be represented as the sum of a countable number of half-open intervals in pairs without common points.
Recall that we call a half-open interval in a plane a finite interval defined by inequalities of the form .
Let us draw on the plane a grid of squares with sides parallel to the axes and with a side length equal to one. The set of these squares is a countable set. From these squares, let us choose those squares all of whose points belong to a given open set O. The number of such squares may be finite or countable, or perhaps there will be no such squares at all. We divide each of the remaining squares of the grid into four identical squares and from the newly obtained squares we again select those whose points all belong to O. We again divide each of the remaining squares into four equal parts and select those squares whose all points belong to O, etc. Let us show that every point P of the set O will fall into one of the selected squares, all points of which belong to O. Indeed, let d be the positive distance from P to the boundary of O. When we reach squares whose diagonal is less than , then we can, obviously, assert that point P has already fallen into a square, all the volumes of which belong to O. If the selected squares are considered half-open, then they will not have common points in pairs, and the theorem is proven. The number of selected squares will necessarily be countable, since the finite sum of half-open intervals is obviously not an open set. Denoting by DL those half-open squares that we obtained as a result of the above construction, we can write
A countable set is an infinite set whose elements can be numbered by natural numbers, or it is a set equivalent to the set of natural numbers.
Sometimes sets of equal cardinality to any subset of the set of natural numbers are called countable, that is, all finite sets are also considered countable.
A countable set is the “smallest” infinite set, that is, in any infinite set there is a countable subset.
Properties:
1. Any subset of a countable set is at most countable.
2. The union of a finite or countable number of countable sets is countable.
3. The direct product of a finite number of countable sets is countable.
4. The set of all finite subsets of a countable set is countable.
5. The set of all subsets of a countable set is continuous and, in particular, is not countable.
Examples of countable sets:
Prime numbers Natural numbers, Integers, Rational numbers, Algebraic numbers, Period ring, Computable numbers, Arithmetic numbers.
Theory of real numbers.
(Real = real - reminder for us guys.)
The set R contains rational and irrational numbers.
Real numbers that are not rational are called irrational numbers
Theorem: There is no rational number whose square is equal to the number 2
Rational numbers: ½, 1/3, 0.5, 0.333.
Irrational numbers: root of 2=1.4142356…, π=3.1415926…
The set R of real numbers has the following properties:
1. It is ordered: for any two different numbers a and b one of two relations holds a or a>b
2. The set R is dense: between two different numbers a and b contains an infinite number of real numbers X, i.e. numbers satisfying the inequality a There's also a 3rd property, but it's huge, sorry Bounded sets. Properties of upper and lower boundaries. Limited set- a set that in a certain sense has a finite size. bounded above if there is a number such that all elements do not exceed: The set of real numbers is called bounded below, if there is a number , such that all elements are at least: A set bounded above and below is called limited. A set that is not bounded is called unlimited. As follows from the definition, a set is unbounded if and only if it not limited from above or not limited below. Number sequence. Consistency limit. Lemma about two policemen. Number sequence is a sequence of elements of number space. Let be either the set of real numbers or the set of complex numbers. Then the sequence of elements of the set is called numerical sequence. Example. A function is an infinite sequence of rational numbers. The elements of this sequence, starting from the first, have the form . Sequence limit- this is an object to which the members of the sequence approach as the number increases. In particular, for number sequences, a limit is a number in any neighborhood of which all terms of the sequence starting from a certain point lie. The theorem about two policemen... If the function is such that for everyone in some neighborhood of the point , and the functions and have the same limit at , then there is a limit of the function at equal to the same value, that is Let two sets X and Y be given, whether they coincide or not. Definition.
The set of ordered pairs of elements, the first of which belongs to X and the second to Y, is called Cartesian product of sets and is designated . Example.
Let If Example.
Let Example.
Let If , then. Elements from Let Definition.
Binary algebraic operation ( internal law of composition) on Thus, any ordered pair Typically, binary operations are denoted by the symbols Definition.
A bunch of Example.
Consider the set of non-negative integers Comment.
As follows from the definition, specifying an algebraic operation * on Let Definition.
By external law
compositions on a set those. the law by which any element Example.
Matrix multiplication distributive regarding the internal law of composition * in The external law of composition is called distributive relative to the internal law of composition * in Y, if Example.
Matrix multiplication Binary algebraic operation on a set Comment.
The properties of commutativity and associativity are independent. Example.
Consider the set of integers. Operation on those. the operation is commutative, but not associative. Example.
Consider the set
Definition.
Element Lemma.
If Proof
.
Let Definition.
Element Example.
Consider the set of integers The set of natural numbers consists of the numbers 1, 2, 3, 4, ..., used for counting objects. The set of all natural numbers is usually denoted by the letter N
: N
= {1, 2, 3, 4, ..., n, ...}
. 1. For any natural numbers a And b equality is true a + b = b + a
. This property is called the commutative law of addition. 2. For any natural numbers a, b, c
equality is true (a + b) + c = a + (b + c)
. This property is called the combined (associative) law of addition. Laws of multiplication of natural numbers 3. For any natural numbers a And b equality is true ab = ba. This property is called the commutative law of multiplication. 4. For any natural numbers a, b, c
equality is true (ab)c = a(bc)
. This property is called the combined (associative) law of multiplication. 5. For any values a, b, c
equality is true (a + b)c = ac + bc
. This property is called the distributive law of multiplication (relative to addition). 6. For any values a equality is true a*1 = a. This property is called the law of multiplication by one. The result of adding or multiplying two natural numbers is always a natural number. Or, to put it another way, these operations can be performed while remaining in the set of natural numbers. This cannot be said regarding subtraction and division: for example, from the number 3 it is impossible, remaining in the set of natural numbers, to subtract the number 7; The number 15 cannot be divided by 4 completely. Divisibility of a sum. If each term is divisible by a number, then the sum is divisible by that number. Divisibility of a product. If in a product at least one of the factors is divisible by a certain number, then the product is also divisible by this number. These conditions, both for the sum and for the product, are sufficient but not necessary. For example, the product 12*18 is divisible by 36, although neither 12 nor 18 is divisible by 36. Test for divisibility by 2. In order for a natural number to be divisible by 2, it is necessary and sufficient that its last digit be even. Test for divisibility by 5. In order for a natural number to be divisible by 5, it is necessary and sufficient that its last digit be either 0 or 5. Test for divisibility by 10. In order for a natural number to be divisible by 10, it is necessary and sufficient that the units digit be 0. Test for divisibility by 4. In order for a natural number containing at least three digits to be divisible by 4, it is necessary and sufficient that the last digits be 00, 04, 08 or the two-digit number formed by the last two digits of this number is divisible by 4. Test for divisibility by 2 (by 9). In order for a natural number to be divisible by 3 (by 9), it is necessary and sufficient that the sum of its digits is divisible by 3 (by 9). Consider a number line with the origin at the point O. The coordinate of the number zero on it will be a point O. Numbers located on the number line in a given direction are called positive numbers. Let a point be given on the number line A with coordinate 3. It corresponds to the positive number 3. Now let us plot the unit segment from the point three times O, in the direction opposite to the given one. Then we get the point A", symmetrical to the point A relative to the origin O. Point coordinate A" there will be a number - 3. This number is the opposite of the number 3. Numbers located on the number line in the direction opposite to the given one are called negative numbers. Numbers opposite to natural numbers form a set of numbers N"
: N"
= {- 1, - 2, - 3, - 4, ...}
. If we combine the sets N
, N"
and singleton set {0}
, then we get a set Z
all integers: Z
= {0} ∪ N
∪ N"
. For integers, all the above laws of addition and multiplication are true, which are true for natural numbers. In addition, the following subtraction laws are added: a - b = a + (- b)
; a + (- a) = 0
. To make the operation of dividing integers by any number not equal to zero feasible, fractions are introduced: Where a And b- integers and b not equal to zero. If we add the set of all positive and negative fractions to the set of integers, we get the set of rational numbers Q
: Moreover, each integer is also a rational number, since, for example, the number 5 can be represented in the form , where the numerator and denominator are integers. This is important when performing operations on rational numbers, one of which can be an integer. The main property of a fraction. If the numerator and denominator of a given fraction are multiplied or divided by the same natural number, you get a fraction equal to the given one: This property is used when reducing fractions. Adding fractions. The addition of ordinary fractions is defined as follows: That is, to add fractions with different denominators, the fractions are reduced to a common denominator. In practice, when adding (subtracting) fractions with different denominators, the fractions are reduced to the lowest common denominator. For example, like this: To add fractions with the same numerators, simply add the numerators and leave the denominator the same. Multiplying fractions. Multiplication of ordinary fractions is defined as follows: That is, to multiply a fraction by a fraction, you need to multiply the numerator of the first fraction by the numerator of the second fraction and write the product in the numerator of the new fraction, and multiply the denominator of the first fraction by the denominator of the second fraction and write the product in the denominator of the new fraction. Dividing fractions. Division of ordinary fractions is defined as follows: That is, to divide a fraction by a fraction, you need to multiply the numerator of the first fraction by the denominator of the second fraction and write the product in the numerator of the new fraction, and multiply the denominator of the first fraction by the numerator of the second fraction and write the product in the denominator of the new fraction. Raising a fraction to a power with a natural exponent. This operation is defined as follows: That is, to raise a fraction to a power, the numerator is raised to that power and the denominator is raised to that power. Theorem. Any rational number can be represented as a finite or infinite periodic fraction. For example, A sequentially repeating group of digits after the decimal point in the decimal notation of a number is called a period, and a finite or infinite decimal fraction that has such a period in its notation is called periodic. In this case, any finite decimal fraction is considered an infinite periodic fraction with a zero in the period, for example: The result of addition, subtraction, multiplication and division (except division by zero) of two rational numbers is also a rational number. On the number line, which we considered in connection with the set of integers, there may be points that do not have coordinates in the form of a rational number. Thus, there is no rational number whose square is 2. Therefore, the number is not a rational number. There are also no rational numbers whose squares are 5, 7, 9. Therefore, the numbers , , are irrational. The number is also irrational. No irrational number can be represented as a periodic fraction. They are represented as non-periodic fractions. The union of the sets of rational and irrational numbers is the set of real numbers R
. DEFINITION 5. Let X be a metric space, ММ Х, аОХ. A point a is called a limit point of M if in any neighborhood of a there are points of the set M\(a). The latter means that in any neighborhood of a there are points of the set M different from a. Notes. 1. A limit point may or may not belong to the set. For example, 0 and 1 are limit points of the set (0,2), but the first does not belong to it, and the second does. 2. A point of a set M may not be its limit point. In this case, it is called an isolated point M. For example, 1 is an isolated point of the set (-1,0)È(1). 3. If the limit point a does not belong to the set M, then there is a sequence of points x n ОM converging to a in this metric space. To prove it, it is enough to take open balls at this point of radii 1/n and select from each ball a point belonging to M. The converse is also true, if for a there is such a sequence, then the point is a limit point. DEFINITION 6. The closure of a set M is the union of M with the set of its limit points. Designation Note that the closure of a ball does not have to coincide with a closed ball of the same radius. For example, in a discrete space, the closure of the ball B(a,1) is equal to the ball itself (consists of one point a) while the closed ball (a,1) coincides with the entire space. Let us describe some properties of the closure of sets. 1. MÌ. This follows directly from the definition of a closure. 2. If M М N, then М 4. 5. The closure of an empty set is empty. This agreement does not follow from the general definition, but is natural. DEFINITION 7. A set M М X is called closed if = M. A set M М X is called open if the set X\M is closed. A set M М X is said to be everywhere dense in X if = X. DEFINITION 8. A point a is called an interior point of the set M if B(a,r)МM for some positive r, i.e., the interior point is included in the set together with some neighborhood. A point a is called an exterior point of the set M if the ball B(a,r)МХ/M for some positive r, i.e., the interior point is not included in the set along with some neighborhood. Points that are neither interior nor exterior points of the set M are called boundary points. Thus, boundary points are characterized by the fact that in each of their neighborhoods there are points both included and not included in M. PROPOSITION 4. In order for a set to be open, it is necessary and sufficient that all its points are interior. Examples of closed sets on a line are , )
,
, Then
.
,
, Then 
.
, where R is the set of all real numbers. Then 
is the set of all Cartesian coordinates of points in the plane.
is a certain family of sets, then the Cartesian product of these sets is the set of all ordered strings of length n:
are row vectors of length n.Algebraic structures with one binary operation
1 Binary algebraic operations
– an arbitrary finite or infinite set.
is an arbitrary but fixed mapping of a Cartesian square 
V 
, i.e.
(1)
(2)
. The fact that 
, is written symbolically in the form 
.
etc. As before, the operation 
means “addition”, and the operation “” means “multiplication”. They differ in the form of notation and, possibly, in axioms, which will be clear from the context. Expression 
we will call it a product, and 
– the sum of elements 
And 
.
is called closed under the operation if for any .
. As binary operations on 
we will consider ordinary addition operations 
and multiplication. Then the sets 
,
will be closed with respect to these operations.
, is equivalent to the closedness of the set 
regarding this operation. If it turns out that a lot 
is not closed under a given operation *, then in this case they say that the operation * is not algebraic. For example, the operation of subtraction on a set of natural numbers is not algebraic.
And 
two sets.
called mapping
,
(3)
and any element 
element is matched 
. The fact that 
, denoted by the symbol 
or 
.
per number 
is an external composition law on the set 
. Multiplying numbers in 
can be considered both as an internal law of composition and as an external one.
, If
per number 
distributive both with respect to the addition of matrices and with respect to the addition of numbers, because,.Properties of binary operations
called:
will be determined in accordance with the rule 
. Let's choose numbers 
and perform the operation on these numbers:
– square matrices of dimension 
with real coefficients. As a binary operation * on 
We will consider matrix multiplication operations. Let 
, Then 
, however 
, i.e. the operation of multiplication on a set of square matrices is associative, but not commutative.
called single or neutral regarding the operation in question on 
, If
– unit element of the set 
, closed under the operation *, then it is unique.
– unit element of the set 
, closed under the operation *. Let's assume that in 
there is one more unit element 
, Then 
, because 
is a single element, and 
, because 
– single element. Hence, 
– the only unit element of the set 
.
called reverse or symmetrical to element 
, If
with addition operation 
. Element 
, then the symmetric element 
there will be an element 
. Really,.Laws of addition of natural numbers
Signs of divisibility of natural numbers
Set of integers
Set of rational numbers
.Laws of arithmetic operations on rational numbers
.Periodic decimals
.Set of real numbers
. Indeed, if a О , a ПМ, then in any neighborhood of a there are points of the set M. They are also points of N. Therefore aО . For points from M this is clear by definition.
.
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