Designation, recording and representation of numerical sets

A bunch of is a set of any objects that are called elements of this set.

For example: many schoolchildren, many cars, many numbers .

In mathematics, set is considered much more broadly. We will not delve too deeply into this topic, since it relates to higher mathematics and may create learning difficulties at first. We will consider only that part of the topic that we have already dealt with.

Designations

A set is most often denoted by capital letters of the Latin alphabet, and its elements by lowercase letters. In this case, the elements are enclosed in curly braces.

For example, if our friends name is Tom, John and Leo , then we can define a set of friends whose elements will be Tom, John and Leo.

Let's denote many of our friends using a capital Latin letter F(friends), then put an equal sign and list our friends in curly brackets:

F = (Tom, John, Leo)

Example 2. Let's write down the set of divisors of the number 6.

Let us denote this set by any capital Latin letter, for example, by the letter D

then we put an equal sign and list the elements of this set in curly brackets, that is, we list the divisors of the number 6

D = (1, 2, 3, 6)

If some element belongs to a given set, then this membership is indicated using the membership sign ∈. For example, the divisor 2 belongs to the set of divisors of the number 6 (the set D). It is written like this:

Reads like: “2 belongs to the set of divisors of the number 6”

If some element does not belong to a given set, then this non-membership is indicated using a crossed out membership sign ∉. For example, the divisor 5 does not belong to the set D. It is written like this:

Reads like: "5 do not belong set of divisors of the number 6″

In addition, a set can be written by directly listing the elements, without capital letters. This can be convenient if the set consists of a small number of elements. For example, let's define a set of one element. Let this element be our friend Volume:

( Volume )

Let's define a set that consists of one number 2

{ 2 }

Let's define a set that consists of two numbers: 2 and 5

{ 2, 5 }

Set of natural numbers

This is the first set we started working with. Natural numbers are the numbers 1, 2, 3, etc.

Natural numbers appeared due to the need of people to count those other objects. For example, count the number of chickens, cows, horses. Natural numbers arise naturally when counting.

In previous lessons, when we used the word "number", most often it was a natural number that was meant.

There are many in mathematics natural numbers indicated by capital Latin letter N.

For example, let's point out that the number 1 belongs to the set of natural numbers. To do this, we write down the number 1, then using the membership sign ∈ we indicate that the unit belongs to the set N

1 ∈ N

Reads like: “one belongs to the set of natural numbers”

Set of integers

The set of integers includes all positive and , as well as the number 0.

A set of integers is denoted by a capital letter Z .

Let us point out, for example, that the number −5 belongs to the set of integers:

−5 ∈ Z

Let us point out that 10 belongs to the set of integers:

10 ∈ Z

Let us point out that 0 belongs to the set of integers:

In the future, we will call all positive and negative numbers one phrase - whole numbers.

Set of rational numbers

Rational numbers are the same ones common fractions which we are still studying today.

A rational number is a number that can be represented as a fraction, where a- numerator of the fraction, b- denominator.

The numerator and denominator can be any numbers, including integers (with the exception of zero, since you cannot divide by zero).

For example, imagine that instead of a is the number 10, but instead b- number 2

10 divided by 2 equals 5. We see that the number 5 can be represented as a fraction, which means the number 5 is included in the set of rational numbers.

It is easy to see that the number 5 also applies to the set of integers. Therefore, the set of integers is included in the set of rational numbers. This means that the set of rational numbers includes not only ordinary fractions, but also integers of the form −2, −1, 0, 1, 2.

Now let's imagine that instead of a the number is 12, but instead b- number 5.

12 divided by 5 equals 2.4. We see that the decimal fraction 2.4 can be represented as a fraction, which means it is included in the set of rational numbers. From this we conclude that the set of rational numbers includes not only ordinary fractions and integers, but also decimal fractions.

We calculated the fraction and got the answer 2.4. But we could isolate the whole part of this fraction:

When you isolate the whole part of a fraction, you get a mixed number. We see that a mixed number can also be represented as a fraction. This means that the set of rational numbers also includes mixed numbers.

As a result, we come to the conclusion that the set of rational numbers contains:

• whole numbers
• common fractions
• decimals
• mixed numbers

The set of rational numbers is denoted by a capital letter Q.

For example, we point out that a fraction belongs to the set of rational numbers. To do this, we write down the fraction itself, then using the membership sign ∈ we indicate that the fraction belongs to the set of rational numbers:

∈ Q

Let us point out that the decimal fraction 4.5 belongs to the set of rational numbers:

4,5 ∈ Q

Let us point out that a mixed number belongs to the set of rational numbers:

∈ Q

The introductory lesson on sets is complete. We'll look at sets much better in the future, but for now what's covered in this lesson will suffice.

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The phrase " number sets" is quite common in mathematics textbooks. There you can often find phrases like this:

“Blah blah blah, where belongs to the set of natural numbers.”

Often, instead of the end of a phrase, you can see something like this. It means the same as the text a little above - a number belongs to the set of natural numbers. Many people quite often do not pay attention to which set this or that variable is defined in. As a result, completely incorrect methods are used when solving a problem or proving a theorem. This happens because the properties of numbers belonging to different sets may differ.

There are not so many numerical sets. Below you can see the definitions of various number sets.

The set of natural numbers includes all integers greater than zero—positive integers.

For example: 1, 3, 20, 3057. The set does not include the number 0.

This number set includes all integers greater and less than zero, and also zero.

For example: -15, 0, 139.

Rational numbers, generally speaking, are a set of fractions that cannot be canceled (if a fraction is canceled, then it will already be an integer, and for this case there is no need to introduce another number set).

An example of numbers included in the rational set: 3/5, 9/7, 1/2.

,

where is a finite sequence of digits of the integer part of a number belonging to the set of real numbers. This sequence is finite, that is, the number of digits in the integer part of a real number is finite.

– an infinite sequence of numbers that are in the fractional part of a real number. It turns out that the fractional part contains an infinite number of numbers.

Such numbers cannot be represented as a fraction. Otherwise, such a number could be classified as a set of rational numbers.

Examples of real numbers:

Let's take a closer look at the meaning of the root of two. The integer part contains only one digit - 1, so we can write:

In the fractional part (after the dot), the numbers 4, 1, 4, 2 and so on appear sequentially. Therefore, for the first four digits we can write:

I dare to hope that now the definition of the set of real numbers has become clearer.

Conclusion

It should be remembered that the same function can exhibit completely different properties depending on which set the variable belongs to. So remember the basics - they will come in handy.

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Mathematical analysis is the branch of mathematics that deals with the study of functions based on the idea of ​​an infinitesimal function.

Basic concepts mathematical analysis are magnitude, set, function, infinite small function, limit, derivative, integral.

Size Anything that can be measured and expressed by number is called.

Many is a collection of some elements united by some common feature. Elements of a set can be numbers, figures, objects, concepts, etc.

Sets are denoted by uppercase letters, and elements of the set are denoted by lowercase letters. Elements of sets are enclosed in curly braces.

If element x belongs to the set X, then write x∈X (∈ - belongs).
If set A is part of set B, then write A ⊂ B (⊂ - contained).

A set can be defined in one of two ways: by enumeration and by using a defining property.

• A=(1,2,3,5,7) - set of numbers
• Х=(x 1 ,x 2 ,...,x n ) - set of some elements x 1 ,x 2 ,...,x n
• N=(1,2,...,n) — set of natural numbers
• Z=(0,±1,±2,...,±n) — set of integers

The set (-∞;+∞) is called number line, and any number is a point on this line. Let a be an arbitrary point on the number line and δ be a positive number. The interval (a-δ; a+δ) is called δ-neighborhood of point a.

A set X is bounded from above (from below) if there is a number c such that for any x ∈ X the inequality x≤с (x≥c) holds. The number c in this case is called top (bottom) edge set X. A set bounded both above and below is called limited. The smallest (largest) of the upper (lower) faces of a set is called exact top (bottom) edge of this multitude.

Basic number sets

If a set does not contain a single element, then it is called empty set and is recorded Ø .

Elements of logical symbolism

Notation ∀x: |x|<2 → x 2 < 4 означает: для каждого x такого, что |x|<2, выполняется неравенство x 2 < 4.

Quantifiers are often used when writing mathematical expressions.

Quantifier is called a logical symbol that characterizes the elements following it in quantitative terms.

• ∀- general quantifier, is used instead of the words “for everyone”, “for anyone”.
• ∃- existence quantifier, is used instead of the words “exists”, “is available”. The symbol combination ∃! is also used, which is read as if there is only one.

Set Operations

Two sets A and B are equal(A=B) if they consist of the same elements.
For example, if A=(1,2,3,4), B=(3,1,4,2) then A=B.

By union (sum) sets A and B is a set A ∪ B whose elements belong to at least one of these sets.
For example, if A=(1,2,4), B=(3,4,5,6), then A ∪ B = (1,2,3,4,5,6)

By intersection (product) sets A and B is called a set A ∩ B, the elements of which belong to both the set A and the set B.
For example, if A=(1,2,4), B=(3,4,5,2), then A ∩ B = (2,4)

By difference The sets A and B are called the set AB, the elements of which belong to the set A, but do not belong to the set B.
For example, if A=(1,2,3,4), B=(3,4,5), then AB = (1,2)

Symmetrical difference sets A and B is called the set A Δ B, which is the union of the differences of the sets AB and BA, that is, A Δ B = (AB) ∪ (BA).
For example, if A=(1,2,3,4), B=(3,4,5,6), then A Δ B = (1,2) ∪ (5,6) = (1,2,5 ,6)

Properties of set operations

A ∪ B = B ∪ A
A ∩ B = B ∩ A

(A ∪ B) ∪ C = A ∪ (B ∪ C)
(A ∩ B) ∩ C = A ∩ (B ∩ C)

Countable and uncountable sets

In order to compare any two sets A and B, a correspondence is established between their elements.

If this correspondence is one-to-one, then the sets are called equivalent or equally powerful, A B or B A.

The set of points on the leg BC and the hypotenuse AC of triangle ABC are of equal power.

Of the large number of diverse sets, numerical sets are especially interesting and important, i.e. those sets whose elements are numbers. Obviously, to work with numerical sets you need to have the skill of writing them down, as well as depicting them on a coordinate line.

Writing numerical sets

The generally accepted designation for any set is capital Latin letters. Number sets are no exception. For example, we can talk about number sets B, F or S, etc. However, there is also a generally accepted marking of numerical sets depending on the elements included in it:

N – set of all natural numbers; Z – set of integers; Q – set of rational numbers; J – set of irrational numbers; R – set of real numbers; C is the set of complex numbers.

It becomes clear that designating, for example, a set consisting of two numbers: - 3, 8 with the letter J can be misleading, since this letter marks a set of irrational numbers. Therefore, to designate the set - 3, 8, it would be more appropriate to use some kind of neutral letter: A or B, for example.

Let us also recall the following notation:

• ∅ – an empty set or a set that has no constituent elements;
• ∈ or ∉ is a sign of whether an element belongs or does not belong to a set. For example, the notation 5 ∈ N means that the number 5 is part of the set of all natural numbers. The notation - 7, 1 ∈ Z reflects the fact that the number - 7, 1 is not an element of the set Z, because Z – set of integers;
• signs that a set belongs to a set:
  ⊂ or ⊃ - “included” or “includes” signs, respectively. For example, the notation A ⊂ Z means that all elements of the set A are included in the set Z, i.e. the number set A is included in the set Z. Or vice versa, the notation Z ⊃ A will clarify that the set of all integers Z includes the set A.
  ⊆ or ⊇ are signs of the so-called non-strict inclusion. Mean "included or matches" and "includes or matches" respectively.

Let us now consider the scheme for describing numerical sets using the example of the main standard cases most often used in practice.

We will first consider numerical sets containing a finite and small number of elements. It is convenient to describe such a set by simply listing all its elements. Elements in the form of numbers are written, separated by a comma, and enclosed in curly braces (which corresponds to the general rules for describing sets). For example, we write the set of numbers 8, - 17, 0, 15 as (8, - 17, 0, 15).

It happens that the number of elements of a set is quite large, but they all obey a certain pattern: then an ellipsis is used in the description of the set. For example, we write the set of all even numbers from 2 to 88 as: (2, 4, 6, 8, …, 88).

Now let's talk about describing numerical sets in which the number of elements is infinite. Sometimes they are described using the same ellipsis. For example, we write the set of all natural numbers as follows: N = (1, 2, 3, ...).

It is also possible to write a numerical set with an infinite number of elements by specifying the properties of its elements. The notation (x | properties) is used. For example, (n | 8 n + 3, n ∈ N) defines the set of natural numbers that, when divided by 8, leave a remainder of 3. This same set can be written as: (11, 19, 27, …).

In special cases, numerical sets with an infinite number of elements are the well-known sets N, Z, R, etc., or numerical intervals. But basically, numerical sets are a union of their constituent numerical intervals and numerical sets with a finite number of elements (we talked about them at the very beginning of the article).

Let's look at an example. Suppose the components of a certain numerical set are the numbers - 15, - 8, - 7, 34, 0, as well as all the numbers of the segment [- 6, - 1, 2] and the numbers of the open number line (6, + ∞). In accordance with the definition of a union of sets, we write the given numerical set as: ( - 15 , - 8 , - 7 , 34 ) ∪ [ - 6 , - 1 , 2 ] ∪ ( 0 ) ∪ (6 , + ∞) . Such a notation actually means a set that includes all the elements of the sets (- 15, - 8, - 7, 34, 0), [- 6, - 1, 2] and (6, + ∞).

In the same way, by combining various numerical intervals and sets of individual numbers, it is possible to give a description of any numerical set consisting of real numbers. Based on the above, it becomes clear why various types of numerical intervals are introduced, such as interval, half-interval, segment, open numerical ray and numerical ray. All these types of intervals, together with the designations of sets of individual numbers, make it possible to describe any numerical set through their combination.

It is also necessary to pay attention to the fact that individual numbers and numerical intervals when writing a set can be ordered in ascending order. In general, this is not a mandatory requirement, but such ordering allows you to represent a numerical set more simply, and also correctly display it on the coordinate line. It is also worth clarifying that such records do not use numerical intervals with common elements, since these records can be replaced by combining numerical intervals, excluding common elements. For example, the union of numerical sets with common elements [- 15, 0] and (- 6, 4) will be the half-interval [- 15, 4). The same applies to the union of numerical intervals with the same boundary numbers. For example, the union (4, 7] ∪ (7, 9] is the set (4, 9]. This point will be discussed in detail in the topic of finding the intersection and union of numerical sets.

In practical examples, it is convenient to use the geometric interpretation of numerical sets - their image on a coordinate line. For example, this method will help in solving inequalities in which it is necessary to take into account ODZ - when you need to display numerical sets in order to determine their union and/or intersection.

We know that there is a one-to-one correspondence between the points of the coordinate line and the real numbers: the entire coordinate line is a geometric model of the set of all real numbers R. Therefore, to depict the set of all real numbers, we draw a coordinate line and apply shading along its entire length:

Often the origin and the unit segment are not indicated:

Consider an image of number sets consisting of a finite number of individual numbers. For example, let's display a number set (- 2, - 0, 5, 1, 2). The geometric model of a given set will be three points of the coordinate line with the corresponding coordinates:

In most cases, it is possible not to maintain the absolute accuracy of the drawing: a schematic image without respect to scale, but maintaining the relative position of the points relative to each other, is quite sufficient, i.e. any point with a larger coordinate must be to the right of a point with a smaller one. With that said, an existing drawing might look like this:

Separately from the possible numerical sets, numerical intervals are distinguished: intervals, half-intervals, rays, etc.)

Now let's consider the principle of depicting numerical sets, which are the union of several numerical intervals and sets consisting of individual numbers. There is no difficulty in this: according to the definition of a union, it is necessary to display on the coordinate line all the components of the set of a given numerical set. For example, let's create an illustration of the number set (- ∞ , - 15) ∪ ( - 10 ) ∪ [ - 3 , 1) ∪ ( log 2 5 , 5 ) ∪ (17 , + ∞) .

It is also quite common for the number set to be drawn to include the entire set of real numbers except one or more points. Such sets are often specified by conditions like x ≠ 5 or x ≠ - 1, etc. In such cases, the sets in their geometric model are the entire coordinate line with the exception of given points. It is generally accepted to say that these points need to be “plucked out” from the coordinate line. The punctured point is depicted as a circle with an empty center. To support what has been said with a practical example, let us display on the coordinate line a set with the given condition x ≠ - 2 and x ≠ 3:

The information provided in this article is intended to help you gain the skill of seeing the recording and representation of numerical sets as easily as individual numerical intervals. Ideally, the written numerical set should be immediately represented in the form of a geometric image on the coordinate line. And vice versa: from the image, a corresponding numerical set should be easily formed through the union of numerical intervals and sets that are separate numbers.

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State educational institution

secondary vocational education

Tula region

"Aleksinsky Mechanical Engineering College"

Numerical

sets

Designed by

teacher

mathematicians

Khristoforova M.Yu.

Number - basic concept , used for characteristics, comparisons, and their parts. Written signs to denote numbers are , and mathematical .

The concept of number arose in ancient times from the practical needs of people and developed in the process of human development. The scope of human activity expanded and, accordingly, the need for quantitative description and research increased. At first, the concept of number was determined by the needs of counting and measurement that arose in human practical activity, becoming more and more complex. Later, number becomes the basic concept of mathematics, and the needs of this science determine the further development of this concept.

Sets whose elements are numbers are called numerical.

Examples of number sets are:

N=(1; 2; 3; ...; n; ... ) - set of natural numbers;

Zo=(0; 1; 2; ...; n; ... ) - set of non-negative integers;

Z=(0; ±1; ±2; ...; ±n; ...) - set of integers;

Q=(m/n: m Z,n N) is the set of rational numbers.

R-set of real numbers.

There is a relationship between these sets

N Zo Z Q R.

Numbers of the formN = (1, 2, 3, ....) are callednatural . Natural numbers appeared in connection with the need to count objects.

Any , greater than one, can be represented as a product of powers of prime numbers, and in a unique way, up to the order of the factors. For example, 121968=2 4 ·3 2 ·7·11 2

Ifm, n, k - natural numbers, then whenm - n = k they say thatm - minuend, n - subtrahend, k - difference; atm: n = k they say thatm - dividend, n - divisor, k - quotient, numberm also calledmultiples numbersn, and the numbern - divisor numbersm, If the numberm- multiple of a numbern, then there is a natural numberk, such thatm = kn.

From numbers using arithmetic signs and parentheses, they are composednumeric expressions. If you perform the indicated actions in numerical expression, observing the accepted order, you will get a number calledthe value of the expression .

The order of arithmetic operations: the actions in brackets are performed first; Inside any parentheses, multiplication and division are performed first, and then addition and subtraction.

If a natural numberm not divisible by a natural numbern, those. there is no such thingnatural number k, Whatm = kn, then they considerdivision with remainder: m = np + r, Wherem - dividend, n - divisor (m>n), p - quotient, r - remainder .

If a number has only two divisors (the number itself and one), then it is calledsimple : if a number has more than two divisors, then it is calledcomposite.

Any composite natural number can befactorize , and only one way. When factoring numbers into prime factors, usesigns of divisibility .

a Andb can be foundgreatest common divisor. It is designatedD(a,b). If the numbersa Andb are such thatD(a,b) = 1, then the numbersa Andb are calledmutually simple.

For any given natural numbersa Andb can be foundleast common multiple. It is designatedK(a,b). Any common multiple of numbersa Andb divided byK(a,b).

If the numbersa Andb relatively prime , i.e.D(a,b) = 1, ThatK(a,b) = ab .

Numbers of the form:Z = (... -3, -2, -1, 0, 1, 2, 3, ....) are called integers , those. Integers are the natural numbers, the opposite of the natural numbers, and the number 0.

The natural numbers 1, 2, 3, 4, 5.... are also called positive integers. The numbers -1, -2, -3, -4, -5, ..., the opposite of the natural numbers, are called negative integers.

Significant numbers a number is all its digits except the leading zeros.

A sequentially repeating group of digits after the decimal point in the decimal notation of a number is calledperiod, and an infinite decimal fraction having such a period in its notation is calledperiodic . If the period begins immediately after the decimal point, then the fraction is calledpure periodic ; if there are other decimal places between the decimal point and the period, then the fraction is calledmixed periodic .

Numbers that are not integers or fractions are calledirrational .

Each irrational number is represented as a non-periodic infinite decimal fraction.

The set of all finite and infinite decimal fractions is calledmany real numbers : rational and irrational.

The set R of real numbers has the following properties.

1. It is ordered: for any two different numbers α and b, one of two relations holds: a

2. The set R is dense: between any two distinct numbers a and b there is an infinite set of real numbers x, i.e. numbers satisfying the inequality a<х

So, if a

(a 2a< A+bA+b<2b  2 A A<(a+b)/2

Real numbers can be represented as points on a number line. To define a number line, you need to mark a point on the line, which will correspond to the number 0 - the origin, and then select a unit segment and indicate the positive direction.

Each point on the coordinate line corresponds to a number, which is defined as the length of the segment from the origin to the point in question, with a unit segment taken as the unit of measurement. This number is the coordinate of the point. If a point is taken to the right of the origin, then its coordinate is positive, and if to the left, it is negative. For example, points O and A have coordinates 0 and 2, respectively, which can be written as follows: 0(0), A(2).