And also many elements that... Operations on sets and their properties

People constantly have to deal with various collections of objects, which led to the emergence of the concept of number, and then the concept of set, which is one of the basic simplest mathematical concepts and doesn't give in precise definition. The following remarks are intended to clarify what is a set, but do not purport to define it.

A set is a collection, set, collection of things united according to some characteristic or according to some rule. The concept of set arises through abstraction. Considering any collection of objects as a set, one is abstracted from all connections and relationships between various items, components of sets, but retain their objects personality traits. Thus, the set consisting of five coins and the set consisting of five apples are different sets. On the other hand, a set of five coins arranged in a circle and a set of the same coins placed one on top of the other are the same set.

Let's give some examples of sets. We can talk about the multitude of grains of sand that make up a heap of sand, about the multitude of all the planets of our solar system, about the set of all people in this moment in any house, about the set of all the pages of this book. In mathematics we also constantly meet various sets, for example the set of all roots given equation, a lot of everyone natural numbers, the set of all points on a line, etc.

Mathematical discipline that studies general properties sets, i.e., the properties of sets that do not depend on the nature of their constituent objects, is called set theory. This discipline began to develop rapidly in late XIX and the beginning of the 20th century. Founder scientific theory sets - German mathematician G. Kantor.

Cantor's work on set theory grew out of consideration of questions of convergence trigonometric series. This is a very common phenomenon: very often consideration of specific mathematical problems leads to the construction of very abstract and general theories. The significance of such abstract constructions is determined by the fact that they are associated not only with that specific task, from which they grew, but have applications in a number of other matters. In particular, this is exactly the case with set theory. The ideas and concepts of set theory penetrated literally all branches of mathematics and significantly changed its face. Therefore, it is impossible to get a correct idea of modern mathematics without becoming familiar with the elements of set theory. Especially great importance has set theory for the theory of functions of a real variable.

A set is considered given if with respect to any object it can be said whether it belongs to the set or does not belong. In other words, a set is completely determined by the specification of all objects belonging to it. If the set \(M\) consists of objects \(a,\,b,\,c,\,\ldots\) and only of these objects, then we write

\(M=\(a,\,b,\,c,\,\ldots\)\)

The objects that make up a set are usually called its elements. The fact that the object m is an element of the set \(M\) is written in the form

\(\Large(m\in M)\)

and reads: “\(m\) belongs to \(M\)”, or “\(m\) is an element of \(M\)”. If the object \(m\) does not belong to the set \(M\) , then they write: \(m\notin M\) . Each object can serve as only one element of a given set; in other words, all elements (of the same set are different
from each other.

Elements of the set \(M\) can themselves be sets, however, to avoid contradictions, we have to require that the set \(M\) itself not be one of its own elements: \(M\notin M\) .

A set that does not contain a single element is called empty set. For example, the set of all real roots equations

\(x^2+1=0\)

There is empty set. The empty set will be further denoted by \(\varnothing\) .

If for two sets \(M\) and \(N\) each element \(x\) of the set \(M\) is also an element of the set \(N\) then we say that \(M\) is included in \ (\) that \(M\) is a part of \(N\) , that \(M\) is a subset of \(M\) or that \(M\) is contained in \(N\) ; this is written as

\(M\subseteq N\) or \(N\supseteq M\)

For example, the set \(M=\(1,2\)\) is part of the set \(N=\(1,2,3\)\) .

It is clear that there is always \(M\subseteq M\) . It is convenient to assume that the empty set is part of any set.

Two sets equal, if they consist of the same elements. For example, the set of roots of the equation \(x^2-3x+2=0\) and the set \(M=\(1,2\)\) are equal to each other.

Let's define rules for operating on sets.

Union or sum of sets

Let there be sets \(M,N,P,\ldots\) . The union or sum of these sets is the set \(X\) consisting of all elements belonging to at least one of the “summands”

\(X=M+N+P+\ldots\) or \(X=M\cup N\cup P\cup\ldots\)

Moreover, even if the element \(x\) belongs to several terms, then it appears in the sum \(M\) only once. It's clear that

\(M+M=M\cup M=M\)

and if \(M\subseteq N\) , then

\(M+N=M\cup N=N\)

Intersection of many

By crossing or common part sets \(M,N,P,\ldots\) . is called a set \(Y\) consisting of all those elements that simultaneously belong to all sets \(M,N,P,\ldots\) .

It is clear that \(M\cdot M=M\) , and if \(M\subseteq N\) , then \(M\cdot N=M\) .

If the intersection of the sets \(M\) and \(N\) is empty: \(M\cdot N=\varnothing\) , then these sets are said to be do not intersect.

To denote the operation of sum and intersection of sets, the signs \(\textstyle(\sum)\) and \(\textstyle(\prod)\) are also used. Thus,

\(E=\sum E_i\) is the sum of the sets \(E_i\) , and \(F=\prod E_i\) is their intersection.

\(M(N+P)=MN+MP,\)

as well as by law

\(M+NP=(M+N)(M+P).\)

Set difference

The difference of two sets \(M\) and \(N\) is the set \(Z\) of all those elements from \(Z\) that do not belong to \(N\):

\(Z=M-N\) or \(Z=M\setminus N\) .

If \(N\subseteq M\) , then the difference \(Z=M\setminus N=M-N\) is also called the complement to the set \(N\) with respect to \(M\) .

It is not difficult to show that it is always

\(M(N-P)=MN-MP\) and \((M-N)+MN=M.\)

Thus, the rules for operating on sets differ significantly from normal rules arithmetic.

Finite and infinite sets

Sets consisting of a finite number of elements are called finite sets. If the number of elements of a set is unlimited, then such a set is called infinite. For example, the set of all natural numbers is infinite.

Let's consider two sets \(M\) and \(N\) and ask the question whether the number of elements in these sets is the same or not.

If the set \(M\) is finite, then the number of its elements is characterized by some natural number - the number of its elements. In this case, to compare the number of elements of the sets \(M\) and \(N\), it is enough to count the number of elements in \(M\), the number of elements in \(N\) and compare the resulting numbers. It is also natural to assume that if one of the sets \(M\) and \(N\) is finite and the other is infinite, then the infinite set contains more elements than the final one.

However, if both sets \(M\) and \(N\) are infinite, then simply counting the elements yields nothing. Therefore, the following questions immediately arise: is everything infinite sets have the same number of elements, or are there infinite sets with more and less elements? If the second is true, then how can we compare the number of elements in infinite sets? We will now deal with these questions.

One-to-one correspondence between sets

Let again \(M\) and \(N\) be two finite sets. How do you know which of these sets has more elements without counting the number of elements in each set? To do this, we will make pairs by combining one element from \(M\) and one element from \(N\) into a pair. Then, if some element from \(M\) does not have a paired element from \(N\) , then \(M\) has more elements than \(N\) . Let us illustrate this reasoning with an example.

Let there be a certain number of people and a certain number of chairs in the hall. To find out what is more, just ask people to take their seats. If someone is left without a seat, then there are more people, and if, say, everyone is sitting and all the seats are occupied, then there are as many people as there are chairs. The described method of comparing the number of elements in sets has the advantage over direct counting of elements that it can be applied without any special changes not only to finite, but also to infinite sets.

Consider the set of all natural numbers

\(M=\(1,\,2,\,3,\,4,\,\ldots\)\)

and the set of all even numbers

\(N=\(2,\,4,\,6,\,8,\,\ldots\)\)

Which set contains more elements? At first glance it seems that the first. However, we can form pairs from the elements of these sets, as follows.

Table 1

\((\color(blue)\begin(array)(c|c|c|c|c|c) (\color(black)M) &(\color(black)1) &(\color(black) 2) &(\color(black)3) &(\color(black)4) &(\color(black)\cdots)\\\hline (\color(black)N) &(\color(black)2 ) &(\color(black)4) &(\color(black)6) &(\color(black)8) &(\color(black)\cdots) \end(array))\)

No element of \(M\) and no element of \(N\) is left without a pair. True, we could also form pairs like this:

table 2

\((\color(blue)\begin(array)(c|c|c|c|c|c|c) (\color(black)M)&(\color(black)1)&(\color( black)2)&(\color(black)3)&(\color(black)4)&(\color(black)5)&(\color(black)\cdots)\\\hline (\color(black )N)&(\color(black)-)&(\color(black)2)&(\color(black)-)&(\color(black)4)&(\color(black)-)&( \color(black)\cdots) \end(array))\)

Then many elements from \(M\) are left without pairs. On the other hand, we could make pairs like this:

Table 3

\((\color(blue)\begin(array)(c|c|c|c|c|c|c|c|c) (\color(black)M)&(\color(black)-)& (\color(black)1)&(\color(black)-)&(\color(black)2)&(\color(black)-)&(\color(black)3)&(\color(black )-)&(\color(black)\cdots)\\\hline (\color(black)N)&(\color(black)2)&(\color(black)4)&(\color(black) 6)&(\color(black)8)&(\color(black)10)&(\color(black)12)&(\color(black)14)&(\color(black)\cdots) \end (array))\)

Now many elements from \(M\) remain without pairs.

Thus, if the sets \(A\) and \(B\) are infinite, then in various ways pair formations correspond to different results. If there is a way of forming pairs in which each element \(A\) and each element \(B\) has a paired element, then we say that between the sets \(A\) and \(B\) it is possible to establish one-to-one correspondence. For example, between the sets \(M\) and \(N\) considered above, one can establish a one-to-one correspondence, as
this can be seen from the table. 1.

If a one-to-one correspondence can be established between the sets \(A\) and \(B\), then they are said to have same number of elements or are equally powerful. If at any method of pair formation, some elements from \(A\) always remain without pairs, then we say that the set \(A\) contains more elements than \(B\), or that the set \(A\) has greater cardinality than \(B\) .

Thus, we received an answer to one of the questions posed above: how to compare the number of elements in infinite sets. However, this does not bring us any closer to answering another question: do infinite sets exist at all? having different powers? To get an answer to this question, let us examine some of the simplest types of infinite sets.

Countable sets. If it is possible to establish a one-to-one correspondence between the elements of the set \(A\) and the elements of the set of all natural numbers

\(Z=\(1,\,2,\,3,\,\ldots\),\)

then they say that the set \(A\) countably. In other words, the set \(A\) is countable if all its elements can be enumerated using natural numbers, that is, written in the form sequences

\(a_1,~a_2,~\ldots,~a_n,~\ldots\)

Table 1 shows that the set of all even numbers is countable (the top number is now considered to be the number of the corresponding bottom number).

Countable sets are, so to speak, the smallest of infinite sets: every infinite set contains a countable subset.

If two non-empty finite sets do not intersect, then their sum contains more elements than each of the terms. For infinite sets this rule may not hold. Indeed, let \(G\) be the set of all even numbers, \(H\) the set of all odd numbers, and \(Z\) the set of all natural numbers. As Table 4 shows, the sets \(G\) and \(H\) are countable. However, the set \(Z=G+H\) is again countable.

Table 4

\((\color(blue)\begin(array)(c|c|c|c|c|c) (\color(black)G)&(\color(black)2)&(\color(black) 4)&(\color(black)6)&(\color(black)8)&(\color(black)\cdots)\\\hline (\color(black)H)&(\color(black)1 )&(\color(black)3)&(\color(black)5)&(\color(black)7)&(\color(black)\cdots)\\\hline (\color(black)Z) &(\color(black)1)&(\color(black)2)&(\color(black)3)&(\color(black)4)&(\color(black)\cdots) \end(array ))\)

Violation of the rule “the whole is greater than the part” for infinite sets shows that the properties of infinite sets are qualitatively different from the properties of finite sets. The transition from the finite to the infinite is accompanied in full agreement with the well-known position of dialectics - qualitative change properties.

Let's prove that set of all rational numbers countably. To do this, let’s arrange all the rational numbers in the following table:

Table 5

\(\)

Here the first line contains all natural numbers in ascending order, the second line contains 0 and integers negative numbers in descending order, in the third line - positive irreducible fractions with denominator 2 in ascending order, in the fourth row - negative irreducible fractions with denominator 2 in descending order, etc. It is clear that each rational number appears once and only once in this table. Let's renumber now
all the numbers in this table are in the order indicated by the arrows. Then all rational numbers will be placed in the order of one sequence:

Number of the seat occupied
rational number 1 2 3 4 5 6 7 8 9. . .
Rational number 1. 2, O, 3, - 1, 4 -2 _

This establishes a one-to-one correspondence between all rational numbers and all natural numbers. Therefore, the set of all rational numbers is countable.

Continuum power sets

If it is possible to establish a one-to-one correspondence between the elements of the set \(M\) and the points of the segment \(0\leqslant x\leqslant1\), then the set \(M\) is said to have continuum power. In particular, according to this definition, the set of points of the segment \(0\leqslant x\leqslant1\) itself has the cardinality of a continuum.

From Fig. 1 it is clear that the set of points of any segment \(AB\) has the cardinality of a continuum. Here, a one-to-one correspondence is established geometrically, through design.

It is easy to show that sets of points of any interval \(x\in\) and the entire number line \(x\in[-\infty,+\infty]\) have the cardinality of continuum.

Much more interesting is this fact: the set of points of the square \(0\leqslant x\leqslant1,\) \(0\leqslant y\leqslant1\) has the cardinality of a continuum. Thus, roughly speaking, there are “as many” points in a square as there are in a segment.

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From a huge variety of all kinds sets special interest represent the so-called number sets , that is, sets whose elements are numbers. It is clear that to work comfortably with them you need to be able to write them down. We will begin this article with the notation and principles of writing numerical sets. Next, let’s look at how numerical sets are depicted on a coordinate line.

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Writing numerical sets

Let's start with accepted notations. As you know, capital letters are used to denote sets. Latin alphabet. Number sets like special case sets are also denoted. For example, we can talk about number sets A, H, W, etc. Of particular importance are the sets of natural, integer, rational, real, complex numbers etc., their own designations were adopted for them:

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

From here it is clear that you should not denote a set consisting, for example, of two numbers 5 and −7 as Q, this designation will be misleading, since the letter Q usually denotes the set of all rational numbers. To denote the specified numerical set, it is better to use some other “neutral” letter, for example, A.

Since we are talking about notation, let us also recall here about the notation of an empty set, that is, a set that does not contain elements. It is denoted by the sign ∅.

Let us also recall the designation of whether an element belongs or does not belong to a set. To do this, use the signs ∈ - belongs and ∉ - does not belong. For example, the notation 5∈N means that the number 5 belongs to the set of natural numbers, and 5,7∉Z – decimal 5,7 does not belong to the set of integers.

And let us also recall the notation adopted for including one set into another. It is clear that all elements of the set N are included in the set Z, thus the number set N is included in Z, this is denoted as N⊂Z. You can also use the notation Z⊃N, which means that the set of all integers Z includes the set N. The relations not included and not included are indicated by ⊄ and , respectively. Non-strict inclusion signs of the form ⊆ and ⊇ are also used, meaning included or coincides and includes or coincides, respectively.

We've talked about notation, let's move on to the description of numerical sets. In this case, we will only touch on the main cases that are most often used in practice.

Let's start with numerical sets containing a finite and small number of elements. It is convenient to describe numerical sets consisting of a finite number of elements by listing all their elements. All number elements are written separated by commas and enclosed in , which is consistent with the general rules for describing sets. For example, a set consisting of three numbers 0, −0.25 and 4/7 can be described as (0, −0.25, 4/7).

Sometimes, when the number of elements of a numerical set is quite large, but the elements obey a certain pattern, an ellipsis is used for description. For example, the set of all odd numbers from 3 to 99 inclusive can be written as (3, 5, 7, ..., 99).

So we smoothly approached the description of numerical sets, the number of elements of which is infinite. Sometimes they can be described using all the same ellipses. For example, let’s describe the set of all natural numbers: N=(1, 2. 3, …) .

They also use the description of numerical sets by indicating the properties of its elements. In this case, the notation (x| properties) is used. For example, the notation (n| 8·n+3, n∈N) specifies the set of natural numbers that, when divided by 8, leave a remainder of 3. This same set can be described as (11,19, 27, ...).

In special cases, numerical sets with an infinite number of elements are the known sets N, Z, R, etc. or numerical intervals. Basically, numerical sets are represented as Union their constituent individual numerical intervals and numerical sets with a finite number of elements (which we talked about just above).

Let's show an example. Let the number set consist of the numbers −10, −9, −8.56, 0, all the numbers of the segment [−5, −1,3] and the numbers of the open number line (7, +∞). Due to the definition of a union of sets, the specified numerical set can be written as {−10, −9, −8,56}∪[−5, −1,3]∪{0}∪(7, +∞) . This notation actually means a set containing all the elements of the sets (−10, −9, −8.56, 0), [−5, −1.3] and (7, +∞).

Similarly, by combining different number intervals and sets of individual numbers, any number set (consisting of real numbers) can be described. Here it becomes clear why such types of numerical intervals as interval, half-interval, segment, open number ray and a numerical ray: all of them, coupled with notations for sets of individual numbers, make it possible to describe any numerical sets through their union.

Please note that when writing a number set, its constituent numbers and numerical intervals are ordered in ascending order. This is not a necessary but desirable condition, since an ordered numerical set is easier to imagine and depict on a coordinate line. Also note that such records do not use numerical intervals with common elements, since such records can be replaced by combining numerical intervals without common elements. For example, the union of numerical sets with common elements [−10, 0] and (−5, 3) is the half-interval [−10, 3) . The same applies to the union of numerical intervals with the same boundary numbers, for example, the union (3, 5]∪(5, 7] is a set (3, 7] , we will dwell on this separately when we learn to find the intersection and union of numerical sets

Representation of number sets on a coordinate line

In practice, it is convenient to use geometric images of numerical sets - their images on. For example, when solving inequalities, in which it is necessary to take into account ODZ, it is necessary to depict numerical sets in order to find their intersection and/or union. So it will be useful to have a good understanding of all the nuances of depicting numerical sets on a coordinate line.

It is known that there is a one-to-one correspondence between the points of the coordinate line and the real numbers, which means that the coordinate line itself is a geometric model of the set of all real numbers R. Thus, to depict the set of all real numbers, you need to draw a coordinate line with shading along its entire length:

And often they don’t even indicate the origin and the unit segment:

Now let's talk about the image of numerical sets that represent some final number individual numbers. For example, let's depict the number set (−2, −0.5, 1.2) . The geometric image of this set, consisting of three numbers −2, −0.5 and 1.2, will be three points of the coordinate line with the corresponding coordinates:

Note that usually for practical purposes there is no need to carry out the drawing exactly. Often a schematic drawing is sufficient, which implies that it is not necessary to maintain the scale, and it is only important to maintain mutual arrangement points relative to each other: any point with a smaller coordinate must be to the left of a point with a larger coordinate. The previous drawing will schematically look like this:

Separately, from all kinds of numerical sets, numerical intervals (intervals, half-intervals, rays, etc.) are distinguished, which represent their geometric images; we examined them in detail in the section. We won't repeat ourselves here.

And it remains only to dwell on the image of numerical sets, which are a union of several numerical intervals and sets consisting of individual numbers. There is nothing tricky here: according to the meaning of the union in these cases, on the coordinate line it is necessary to depict all the components of the set of a given numerical set. As an example, let's show an image of a number set (−∞, −15)∪{−10}∪[−3,1)∪ (log 2 5, 5)∪(17, +∞) :

And let us dwell on fairly common cases when the depicted numerical set represents the entire set of real numbers, with the exception of one or several points. Such sets are often specified by conditions like x≠5 or x≠−1, x≠2, x≠3.7, etc. In these cases, geometrically they represent the entire coordinate line, with the exception of corresponding points. In other words, these points need to be “plucked out” from the coordinate line. They are depicted as circles with an empty center. For clarity, let us depict a numerical set corresponding to the conditions (this set essentially exists):

Summarize. Ideally information previous paragraphs should form the same view on the recording and representation of numerical sets as the view on individual numerical intervals: the recording of a numerical set should immediately give its image on the coordinate line, and from the image on the coordinate line we should be ready to easily describe the corresponding numerical set through a union individual intervals and sets consisting of individual numbers.

Bibliography.

Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
Mordkovich A. G. Algebra. 9th grade. At 2 p.m. Part 1. Textbook for students educational institutions/ A. G. Mordkovich, P. V. Semenov. - 13th ed., erased. - M.: Mnemosyne, 2011. - 222 p.: ill. ISBN 978-5-346-01752-3.

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.

Lesson content

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 any element belongs given set, then this affiliation is indicated using the affiliation 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.

In mathematics, the set of natural numbers is denoted by capital letters. 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 isolating the whole part in a fraction, it turns out 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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In mathematics, the concept of set is one of the main, fundamental ones, but there is no single definition of set. One of the most well-established definitions of a set is the following: a set is any collection of definite and distinct objects that can be thought of as a single whole. The creator of set theory, the German mathematician Georg Cantor (1845-1918), said this: “A set is many things that we think of as a whole.”

Sets as a data type have proven to be very convenient for programming complex life situations, since they can be used to accurately model real-world objects and compactly display complex logical relationships. Sets are used in the Pascal programming language, and we will look at one example of a solution below. In addition, based on set theory, the concept of relational databases was created, and based on operations on sets - relational algebra and its operations- used in database query languages, in particular SQL.

Example 0 (Pascal). There is a selection of products sold in several stores in the city. Determine: what products are available in all stores in the city; full range of products in the city.

Solution. We define a basic data type Food (products), it can take values corresponding to the names of products (for example, hleb). We declare a set type, it defines all subsets made up of combinations of values basic type, that is, Food (products). And we form subsets: stores “Solnyshko”, “Veterok”, “Ogonyok”, as well as derived subsets: MinFood (products that are available in all stores), MaxFood (a full range of products in the city). Next, we prescribe operations to obtain derived subsets. The MinFood subset is obtained as a result of the intersection of the Solnyshko, Veterok and Ogonyok subsets and includes those and only those elements of these subsets that are included in each of these subsets (in Pascal, the operation of the intersection of sets is denoted by an asterisk: A * B * C, the mathematical designation for the intersection of sets is given below ). The MaxFood subset is obtained by combining the same subsets and includes elements that are included in all subsets (in Pascal, the operation of combining sets is denoted by the plus sign: A + B + C, the mathematical designation for combining sets is given below).

Code PASCAL

Program Shops; type Food=(hleb, moloko, myaso, syr, sol, sugar, maslo, ryba); Shop = set of Food; var Solnyshko, Veterok, Ogonyok, MinFood, MaxFood: Shop; Begin Solnyshko:=; Veterok:=; Ogonyok:=; ... MinFood:=Solnyshko * Veterok * Ogonyok; MaxFood:=Solnyshko + Veterok + Ogonyok; End.

What types of sets are there?

The objects that make up the sets - the objects of our intuition or intellect - can be of a very different nature. In the example in the first paragraph, we analyzed sets that included a set of products. Sets can consist, for example, of all the letters of the Russian alphabet. In mathematics, sets of numbers are studied, for example, consisting of all:

Natural numbers 0, 1, 2, 3, 4, ...

Prime numbers

Even integers

and so on. (the main numerical sets are discussed in this material).

The objects that make up a set are called its elements. We can say that a set is a “bag of elements.” It is very important: there are no identical elements in a set.

Sets can be finite and infinite. A finite set is a set for which there is a natural number that is the number of its elements. For example, the set of the first five non-negative odd integers is a finite set. A set that is not finite is called infinite. For example, the set of all natural numbers is an infinite set.

If M- a lot, and a- its element, then they write: a∈M, which means " a belongs to the set M".

From the first (zero) example in Pascal with products that are available in certain stores:

hleb∈VETEROK ,

which means: the element "hleb" belongs to many products that are available in the "VETEROK" store.

There are two main ways to define sets: enumeration and description.

A set can be defined by listing all its elements, for example:

VETEROK = {hleb, syr, butter} ,

A = {7 , 14 , 28 } .

An enumeration can only define a finite set. Although you can do this with a description. But infinite sets can only be defined by description.

The following method is used to describe sets. Let p(x) - some statement that describes the properties of a variable x, the range of which is the set M. Then through M = {x | p(x)} denotes the set consisting of all those and only those elements for which the statement p(x) is true. This expression reads like this: "Many M, consisting of all such x, What p(x) ".

For example, record

M = {x | x² - 3 x + 2 = 0}

Example 6. According to a survey of 100 market buyers who bought citrus fruits, oranges were bought by 29 buyers, lemons - 30 buyers, tangerines - 9, only tangerines - 1, oranges and lemons - 10, lemons and tangerines - 4, all three types of fruit - 3 buyers. How many customers have not purchased any of the citrus fruits listed here? How many customers bought only lemons?

Operation of Cartesian product of sets

To determine one more important operation over sets - Cartesian product of sets Let's introduce the concept of an ordered set of lengths n.

The length of the set is the number n its component. A set composed of elements taken in exactly this order is denoted . Wherein i i () set component is .

Now a strict definition will follow, which may not be immediately clear, but after this definition there will be a picture from which it will become clear how to obtain the Cartesian product of sets.

Cartesian (direct) product of sets is called a set denoted by and consisting of all those and only those sets of length n, i-th component of which belongs .

For example, if , , ,

Basic concepts of set theory

The concept of set is a fundamental concept modern mathematics. We will consider it initial and build set theory intuitively. Let us give a description of this initial concept.

A bunch of– is a collection of objects (subjects or concepts), which is thought of as a single whole. The objects included in this collection are called elements multitudes.

We can talk about the many first-year mathematics students, the many fish in the ocean, etc. Mathematicians are usually interested in a set of mathematical objects: the set of rational numbers, the set of rectangles, etc.

We will denote sets in capital letters Latin alphabet, and its elements are small.

If is an element of the set M then they say “belongs” M" and write: . If some object is not an element of a set, then it is said “does not belong” M” and write (sometimes).

There are two main ways to define sets: transfer its elements and indication characteristic property its elements. The first of these methods is used mainly for finite sets. When listing the elements of the set under consideration, its elements are surrounded by curly braces. For example, denotes a set whose elements are the numbers 2, 4, 7 and only them. This method is not always applicable, since, for example, the set of all real numbers cannot be specified in this way.

Characteristic property elements of the set M is a property such that every element possessing this property belongs to M, and any element that does not have this property does not belong M. The set of elements with the property is denoted as follows:

or .

The most frequently occurring sets have their own special designations. In what follows we will adhere to the following notation:

N= – the set of all natural numbers;

Z= – set of all integers;

– the set of all rational numbers;

R– the set of all real (real) numbers, i.e. rational numbers (infinite decimals periodic fractions) and irrational numbers (infinite decimal non-periodic fractions);

– the set of all complex numbers.

We will give more special examples specifying sets by specifying a characteristic property.

Example 1. Plenty of everyone natural divisors The number 48 can be written like this: (notation is used only for integers, and means that it is divisible by).

Example 2. The set of all positive rational numbers less than 7 is written in the following way: .

Example 3. – interval of real numbers with ends 1 and 5; – a segment of real numbers with ends 2 and 7.

The word "many" suggests that it contains many elements. But it is not always the case. In mathematics, sets can be considered that contain only one element. For example, the set of integer roots of the equation . Moreover, it is convenient to talk about a set that does not contain a single element. Such a set is called empty and is denoted by Ø. For example, the set of real roots of the equation is empty.

Definition 1. The sets are called equal(denoted A=B), if these sets consist of the same elements.

Definition 2. If every element of a set belongs to the set, then it is called subset sets.

Designations: (“included in”); (“includes”).

It is clear that Ø and the set itself are subsets of the set . Any other subset of a set is called its the right part . If and , then they say that “ A – own subset"or what" A is strictly included in" and write.

The following statement is obvious: sets And are equal if and only if and .

Based on this statement universal method proof of equality of two sets: to prove that the sets And are equal, it is enough to show that ,A is a subset of the set .

This is the most commonly used method, although not the only one. Later, having become acquainted with operations on sets and their properties, we will indicate another way to prove the equality of two sets - using transformations.

In conclusion, we note that often in one way or another mathematical theory deal with subsets of the same set U which is called universal in this theory. For example, in school algebra and mathematical analysis set is universal R real numbers, in geometry - a set of points in space.

Operations on sets and their properties

You can perform actions (operations) on sets that resemble addition, multiplication, and subtraction.

Definition 1. Association sets and a set is called, denoted by , each element of which belongs to at least one of the sets or .

The operation itself, which results in such a set, is called a union.

Definition 1 summary:

Definition 2. By crossing sets and is called a set, denoted by , containing all those and only those elements, each of which belongs to both , and .

The operation itself, which results in a set, is called intersection.

A short summary of Definition 2:

For example, if , , That , .

Sets can be represented as geometric shapes, which allows you to clearly illustrate operations on sets. This method was proposed by Leonhard Euler (1707–1783) for the analysis of logical reasoning, was widely used and received further development in the writings of the English mathematician John Venn (1834–1923). That's why such drawings are called Euler-Venn diagrams.

The operations of union and intersection of sets can be illustrated by Euler–Venn diagrams as follows:

– shaded part; – shaded part.

You can define the union and intersection of any collection of sets , where is a certain set of indices.

Definition . Association collection of sets is a set consisting of all those and only those elements, each of which belongs to at least one of the sets.

Definition . By crossing collection of sets is a set consisting of all those and only those elements, each of which belongs to any of the sets.

In the case where the set of indices is finite, for example, , then to denote the union and intersection of a collection of sets in this case, the following notation is usually used:

And .

For example, if , , , That , .

The concepts of union and intersection of sets are encountered repeatedly in school course mathematics.

Example 1. A bunch of M solutions to the system of inequalities

is the intersection of the sets of solutions to each of the inequalities of this system: .

Example 2. A bunch of M system solutions

is the intersection of the sets of solutions to each of the inequalities of this system. The set of solutions to the first equation is the set of points on the line, i.e. . A bunch of . A set consists of one element - the point of intersection of lines.

Example 3. Set of solutions to the equation

Where , is the union of sets of solutions to each of the equations , , i.e.

Definition 3. By difference sets and is a set, denoted by , and consisting of all those and only those elements that belong to but do not belong to .– shaded part; . with the operations of union, intersection and addition. Received mathematical structure called set algebra or Boolean set algebra(after the Irish mathematician and logician George Boole (1816–1864)). We will denote by the set of all subsets of an arbitrary set and call it Boolean sets.

The equalities listed below are valid for any subsets A, B, C universal set U. That's why they are called laws of set algebra.