There may be a lot of everything. An infinite number in the multitude. Set of rational numbers

Task 1

Compare the elements of the sets in the first and second rows. Is there an element in the first row that is not in the second row? Is there an element in the second row that is not in the first row?

Solution
There are no elements in the first row that are not in the second row

There are no elements in the second row that are not in the first row

Task 2

Compare the sets in the first and second rows. Which row has an extra element?

Task 3

Is the equation correct? Why?

Solution
A) Right. In these equalities, the elements are the same, only in a different order.

b) Not true. There is a triangle on the left side of the equality, but not on the right side.

V) Right. The left side is not equal to the right side because their elements are different.

Task 4

Let A = (0; 1; 2). Which of the sets B - (2; 0; 1), C = ( 1; 0), D = ( 3; 2; 1; 0) are equal to the set A, and which are not equal to it? Make notes and explain them.

Solution
A = B: These sets have the same elements, written in a different order.

C is not equal to A: Set C lacks element 2, which set A does.

D is not equal to A: Set A is missing element 3, which is the same as set D.

Task 5

D = ( a; ; 5 ). Make up a set A that is equal to a set D, and a set B that is not equal to a set D.

Task 6

a) Compose all sets equal to the set ( O; / \ );
b) Compose all the sets equal to the set (a; b; c).

Solution
a) (O; /\), (/\; O).

b) (a; b; c), (a; c; b), (c; a; b), (b; a; c).

Task 7

How many elements does it contain:

a) many days of the week
b) many desks in the first row;
c) a set of letters of the Russian alphabet;
d) Murka's cat has many tails;
e) Petya has a lot of noses;
f) many horses grazing on the moon?

Solution
a) set of days of the week = 7;

b) set of desks in the first row = 3;

c) the set of letters of the Russian alphabet = 33;

d) the set of tails in Murka's cat = 1;

e) Petya's set of noses = 1;

f) set of horses grazing on the moon = 0.

Task 8

a) Do tropical palms grow in your school garden? How many palm trees are in the school garden?
b) What is the number of six-legged horses, two-year-old children in the class, crocodiles in the Moscow River?
c) Think of some examples of an empty set.

Solution
a) Palm trees do not grow in the school garden. Empty set Ø

b) The empty set. Ø

c) Two-meter flies, wooden gloves.

Task 9

Find the correct notation for the empty set, and cross out the rest:

Task 10

a) How many times is 56 greater than 8?
b) How many times is 8 less than 56?
c) How many more units is 56 than 8?
d) How much is 8 less than 56?

Solution
a) 56 is 7 times more than 8.

b) 8 is less than 56 by 7 times.

c) 56 is more than 8 by 48 units.

d) 8 is less than 56 by 48 units.

Task 11

a) A hat costs a ruble, and a coat costs 9 times more. How much is the coat and hat together?
b) The mass of the watermelon is b kg, and the mass of the pumpkin is 2 kg less. What is the total mass of watermelon and pumpkin?
c) The bucket contains c l of water, and the pot - 7 times less. How much larger is the bucket than the pot?
d) There was (d m of fabric) in the piece. 8 identical dresses were sewn from this fabric, spending n m for each dress. How many meters of fabric are left in the piece

Solution
a) (a * 9) + a

b) (b - 2) + b

c) c - (c: 7)

d) d - (8 * n)

Task 12

Guess who it is?

The page uses tasks and assignments from the book by L. G. Peterson “Mathematics. Grade 3 Part 1." 2008
Link to the author's site:

This topic contains a lot of terminology, so I will add the content of the topic, which will make it easier to navigate the material.

Let's start with what, in fact, is meant by the word "set". On an intuitive level, a set is understood as a certain collection of objects, called set elements. For example, you can talk about a lot of pears on the table, a lot of letters in the word "lot", and so on. Georg Cantor (German mathematician, founder of modern theory sets) wrote that by "a set, I mean in general all that many things that can be thought of as a single whole, i.e. such a collection of certain elements that, by means of one law, can be combined into one whole." For some time, the concept of a set, introduced by Cantor, was assumed to be fairly obvious and did not require additional explanations. It seemed that the appearance of the works of Bolzano, and then of Cantor at the end of the 19th - beginning of the 20th century, would put an end to many questions (for example, finally resolve the aporias of Zeno, solve the problem of infinity, etc.) and become the beginning of a new mathematics. The brilliant German mathematician David Hilbert noted that "No one will expel us from the paradise created by Cantor."

However, the appearance of paradoxes (Russell, Burali-Forti) put an end to "Cantor's paradise". One of the formulations of Russell's paradox, known as the "barber's paradox" is as follows: in a certain village, the barber shaves those and only those residents of the village who do not shave themselves. Who, then, shaves the barber himself? Let's say he shaves himself. Those. he belongs to those inhabitants of the village who shave themselves - and in fact, according to the condition of these inhabitants, barbers have no right to shave. Therefore, the assumption that the barber shaves himself leads to a contradiction. Let's try it differently: let the barber not shave himself. If he does not shave himself, then according to his condition, a barber is obliged to shave - again a contradiction! Attempts were made to resolve the contradictions of the set theory proposed by Cantor. The Cantorian set theory itself was called "naive" by mathematicians. The goal of many mathematical works was to build such a system of axioms in which such paradoxes would be impossible. But the task was not so simple. On this moment, as far as I know, there is no unified axiomatics of set theory. The system of axioms of Zermelo-Fraenkel (ZFC) is considered to be the most common, in which the so-called "axiom of choice" stands apart. There are variations of this system: for example, the author of the B-method, Jean-Raymond Abrial, proposed a typed set theory, on the basis of which he created a formal method for developing programs.

Set notation. Membership of an element in a set. Empty set.

Sets are usually written in curly braces. For example, the set of all vowels of the Russian alphabet will be written as follows:

$$$a, e, e, u, o, y, s, u, u, i $ $$

And the set of all integers greater than 8 but less than 15 will be:

$$\{9,10,11,12,13,14 \} $$

A set may not contain any element at all. In this case it is called empty set and denoted as $\varnothing$.

Most often in the mathematical literature, sets are denoted by capital letters Latin alphabet. For example:

$$A=$0, 5, 6, -9 $,\; B=$\Delta, +, -5, 0$.$$

There are also well-established designations of certain sets. For example, the set of natural numbers is usually denoted by the letter $N$; the set of integers - with the letter $Z$; the set of rational numbers - with the letter $Q$; set of all real numbers- the letter $R$. There are other well-established designations, but we will refer to them as necessary.

A set that contains a finite number of elements is called finite set. If a set contains an infinite number of elements, it is called endless.

For example, the above set $A=$0, 5, 6, -9 $$ is a finite set because it contains 4 elements (i.e. finite number elements). The set of natural numbers $N$ is infinite. Generally speaking, we cannot always immediately say with certainty whether a certain set is infinite or not. For example, let $F$ be the set of prime numbers.

What is a prime number: show/hide

The prime numbers are called integers large 1s that are only divisible by 1 or itself. For example, 2, 3, 5, 7 and so on. For comparison: the number 12 is not a prime number, since it is divisible not only by 12 and 1, but also by other numbers (for example, by 3). The number 12 is composite.

The question arises: is the set $F$ infinite or not? Is there a largest prime number? To answer this question, it took a whole theorem, proved by Euclid, that the set of prime numbers is infinite.

Under the power of the set for finite sets, the number of elements of a given set is understood. The cardinality of $A$ is denoted as $|A|$.

For example, since the finite set $A=$0, 5, 6, -9 $$ contains 4 elements, the cardinality of the set $A$ is 4, i.e. $|A|=4$.

If we know that some object $a$ belongs to the set $A$, then we write it like this: $a\in A$. For example, for the set $A$ above, we can write that $5\in A$, $-9\in A$. If the object $a$ does not belong to the set $A$, then it is denoted as follows: $a\notin A$. For example, $19\notin A$. By the way, the elements of sets can be other sets, for example:

$$ M=$-9,1,0, \( a, g$, \varnothing \) $$

The elements of the set $M$ are the numbers -9, 1, 0, as well as the set $ $ a,\; g$$ and the empty set $\varnothing$. In general, to simplify perception, the set can be represented as a portfolio. An empty set is an empty portfolio. This analogy will come in handy a bit later.

Subset. Universal set. Set equality. Bulean.

The set $A$ is called subset set $B$ if all elements of the set $A$ are also elements of the set $B$. Notation: $A\subseteq B$.

For example, consider the sets $K=$ -9,5$$ and $T=$8,-9,0,5,p, -11$$. Each element of the set $K$ (ie -9 and 5) is also an element of the set $T$. Therefore, the set $K$ is a subset of the set $T$, i.e. $K\subseteq T$.

Since all elements of any set $A$ belong to the set $A$ itself, then the set $A$ is a subset of the set $A$ itself. The empty set $\varnothing$ is a subset of any set. Those. for an arbitrary set $A$ the following is true:

$$A\subseteq A; \; \varnothing\subseteq A.$$

Let's introduce one more definition - a universal set.

Universal set(universe) $U$ has the property that all other sets considered in this problem are its subsets.

In other words, the universe contains elements of all sets that are considered within the framework of a certain task. For example, consider the following problem: a survey of students of a certain academic group is being conducted. Each student is asked to indicate the mobile operators of the Russian Federation whose SIM cards he uses. The data of this survey can be presented in the form of sets. For example, if student Vasily uses SIM cards from MTS and Life, then you can write the following:

$$ Vasilij=$MTC, Life $ $$

Similar sets can be made for each student. The universe in this model will be a set that lists all operators in Russia. In principle, as a universe, one can also take a set that lists all operators in the CIS, as well as the set of all mobile operators in the world. And this will not be a contradiction, because any operator in Russia is included in the set of operators in both the CIS and the whole world. Thus, the universe is defined only within the framework of a certain specific task, and it is often possible to consider several universal sets.

The sets $A$ and $B$ are called equal if they consist of the same elements. In other words, if every element of the set $A$ is also an element of the set $B$, and every element of the set $B$ is also an element of the set $A$, then $A=B$.

The definition of set equality can also be written in another way: if $A\subseteq B$ and $B\subseteq A$, then $A=B$.

Consider a pair of sets: the first will be $$\Delta, k $$, and the second will be $$k, \Delta$$. Each element of the first set (ie $\Delta$ and $k$) is also an element of the second set. Each element of the second set (ie $k$ and $\Delta$) is also an element of the second set. Output: $$\Delta, k $=$k, \Delta$$. As you can see, the order in which elements are written in the set does not matter.

Consider a couple more sets: $X=$k, \Delta, k, k,k $$ and $Y=$\Delta, k $$. Each element of the set $X$ is also an element of the set $Y$; every element of the set $Y$ is also an element of the set $X$. Hence $$k, \Delta, k, k, k $=$\Delta, k $$. Taking into account such equalities in set theory, it is customary not to repeat the same elements twice in the notation. For example, the set of digits of the number 1111111555559999 would be $$1,5,9$$. There are, of course, exceptions: the so-called multisets. In the notation of multisets, elements can be repeated, but in classical set theory, repetitions of elements are not allowed.

Using the concept of set equality, subsets can be classified.

If $A\subseteq B$, and $A\neq B$, then the set $A$ is called own (strict) subset sets $B$. It is also said that the set $A$ is strictly included in the set $B$. Write it like this: $A \subset B$.

If some subset of the set $A$ coincides with the set $A$ itself, then this subset is called improper. In other words, the set $A$ is an improper subset of the set $A$ itself.

For example, for the sets $K=$ -9,5$$ and $T=$8,-9,0,5,p, -11$$ considered above, we have: $K\subseteq T$, with this $K\neq T$. Therefore, the set $K$ is a proper subset of the set $T$, which is written as $K\subset T$. One can also say this: the set $K$ is strictly included in the set $T$. The notation $K\subset T$ is more specific than $K\subseteq T$. The point is that by writing $K\subset T$ we guarantee that $K\neq T$. While the notation $K\subseteq T$ does not exclude the case of equality $K=T$.

Note on terminology: show/hide

Generally speaking, there is some confusion in terminology. The above definition of improper sets is accepted in the American and Russian literature. However, in another part of Russian literature there is a slightly different interpretation of the concept of improper sets.

If $A\subseteq B$, and $A\neq B$ and $A\neq \varnothing$, then the set $A$ is called a proper (strict) subset of the set $B$. It is also said that the set $A$ is strictly included in the set $B$. Write it like this: $A \subset B$. The sets $B$ and $\varnothing$ are called improper subsets of the set $B$.

In other words, the empty set in this interpretation is excluded from proper subsets and goes into the category of improper ones. The choice of terminology is a matter of taste.

The set of all subsets of some set $A$ is called boolean or degree sets $A$. Boolean is denoted as $P(A)$ or $2^A$.

Let the set $A$ contain $n$ elements. The Boolean of $A$ contains $2^n$ elements, i.e.

$$ \left| P(A) \right|=2^(n),\;\; n=|A|. $$

Let's consider a couple of examples using the concepts introduced above.

Example #1

From the proposed list, select those statements that are true. Justify your answer.

$$-3,5, 9 $\subseteq $-3, 9, 8, 5, 4, 6 $ $;
$$-3,5, 9$\subset $-3, 9, 8, 5, 4, 6$ $;
$$-3.5, 9$\in $-3, 9, 8, 5, 4, 6$ $;
$\varnothing \subseteq \varnothing$;
$\varnothing=$\varnothing$$;
$\varnothing \in \varnothing$;
$A=$9, -5, 8 \(7, 6 $ \);\; |A|=5$.

We are given two sets: $$-3,5, 9 $$ and $$-3, 9, 8, 5, 4, 6 $$. Each element of the first set is also an element of the second set. Therefore, the first set is a subset of the second, i.e. $$-3,5, 9 $\subseteq $-3, 9, 8, 5, 4, 6 $$. The first point is correct.
In the first paragraph, we found out that $$-3,5, 9 $\subseteq $-3, 9, 8, 5, 4, 6 $$. In this case, these sets are not equal to each other, i.e. $$-3,5, 9 $\neq $-3, 9, 8, 5, 4, 6 $$. Hence, the set $$-3,5, 9 $$ is a proper (strict in other terminology) subset of the set $$-3, 9, 8, 5, 4, 6 $$. This fact is written as $$-3,5, 9$\subset $-3, 9, 8, 5, 4, 6$$. So the second point is true.
The set $$-3,5, 9 $$ is not an element of the set $$-3, 9, 8, 5, 4, 6 $$. The third point is false. For comparison: the statement $$-3,5, 9 $\in $9, 8, 5, 4, \(-3,5,9$, 6 \)$ is true.
The empty set is a subset of any set. Therefore the statement $\varnothing \subseteq \varnothing$ is true.
The assertion is false. The set $\varnothing$ does not contain any elements, but the set $$\varnothing $$ contains one element, so $\varnothing=$\varnothing $$ is not true. To make this clearer, you can refer to the analogy that I described above. The set is a portfolio. The empty set $\varnothing$ is an empty portfolio. The set $$\varnothing $$ is a portfolio containing an empty portfolio. Naturally, an empty portfolio and a non-empty portfolio with something inside are different portfolios :)
The empty set contains no elements. None. So the statement $\varnothing \in \varnothing$ is false. By comparison, the statement $\varnothing\in$\varnothing $$ is true.
The set $A$ contains 4 elements, namely: 9, -5, 8 and $$7, 6 $$. Therefore, the cardinality of $A$ is 4, i.e., $|A|=4$. Therefore, the statement that $|A|=5$ is false.

Answer: The statements in paragraphs #1, #2, #4 are true.

Example #2

Write the boolean of the set $A=$-5,10,9$$.

The set $A$ contains 3 elements. In other words: the cardinality of $A$ is 3, $|A|=3$. Therefore, the set $A$ has $2^3=8$ subsets, i.e. the boolean of set $A$ will consist of eight elements. We list all subsets of the set $A$. Let me remind you that the empty set $\varnothing$ is a subset of any set. So the subsets are:

$$ \varnothing, $-5 $, $ 10$, $ 9$, $-5,10 $, $-5, 9 $, $-10, 9 $ , $-5, 10, 9 $ $$

Let me remind you that the subset $$-5, 10, 9 $$ is improper, since it coincides with the set $A$. All other subsets are their own. All subsets written above are elements of the Boolean set $A$. So:

$$ P(A)=\left$\varnothing, \(-5 $, $ 10$, $ 9$, $-5,10 $, $-5, 9 $ , $-10, 9 $, $-5, 10, 9 $ \right\) $$

Boolean found, it remains only to write down the answer.

Answer: $P(A)=\left$\varnothing, \(-5 $, $ 10$, $ 9$, $-5,10 $, $-5, 9 $ , $-10, 9 $, $-5, 10, 9 $ \right\)$.

Methods for specifying sets.

First way is a simple enumeration of the elements of a set. Naturally, this method is suitable only for finite sets. For example, using this method, the set first three natural numbers will be written like this:

$$ \{1,2,3\} $$

Often in the literature you can find designations of this nature: $T=$0,2,4,6,8, 10, \ldots $$. Here, the set is not defined by enumeration of elements, as it seems at first glance. It is impossible to enumerate all the even non-negative numbers that make up the set $T$, because there are infinitely many of these numbers. An entry like $T=$0,2,4,6,8, 10, \ldots $$ is allowed only when it does not cause discrepancies.

Second way- set the set using the so-called characteristic condition (characteristic predicate) $P(x)$. In this case, the set is written in the following form:

$$$x| P(x)$$$

The notation $$x| P(x)$$ reads as follows: "the set of all elements $x$ for which the statement $P(x)$ is true". What exactly the phrase "characteristic condition" means is easier to explain with an example. Consider this statement:

$$P(x)="x\; - \;natural\; number,\; last\; digit\; of which\;is\;7"$$

Substitute the number 27 instead of $x$ in this statement. We get:

$$P(27)="27\; - \;natural\; number,\; last\; digit\; of which\;is\; 7"$$

This is a true statement, since 27 is indeed a natural number whose last digit is 7. Let's substitute the number $\frac(2)(5)$ into this statement:

$$P\left(\frac(2)(5)\right)="\frac(2)(5)\; - \;natural\; number,\; last\; digit\; of which \;is\; ; 7"$$

This statement is false, since $\frac(2)(5)$ is not a natural number. So, for some objects $x$ the statement $P(x)$ may be false, for some it is true (and for some it is not defined at all). We are only interested in those objects for which $P(x)$ is true. It is these objects that form the set specified using the characteristic condition $P(x)$ (see example No. 3).

Third way- set the set using the so-called generating procedure. The generating procedure describes how to get the elements of a set from already known elements or some other objects (see example No. 4).

Example #3

Write the set $A=\(x| x\in Z \wedge x^2< 10\}$ перечислением элементов.

The set $A$ is defined with the help of the characteristic condition. The characteristic condition in this case is expressed by the notation "$x\in Z \wedge x^2< 10$" (знак "$\wedge$" означает "и"). Расшифровывается эта запись так: "$x$ - целое число, и $x^2 < 10$". Иными словами, в множество $A$ должны входить лишь целые числа, квадрат которых меньше 10. Таких чисел всего 7, т.е.

$$ A=$0,-1,1,-2,2,-3,3$ $$

The set $A$ is now defined by enumeration of elements.

Answer: $A=$0,-1,1,-2,2,-3,3$$.

Example #4

Describe the elements of the set $M$, which is given by such a generating procedure:

$3\in M$;
If the element is $x\in M$, then $3x\in M$.
The set $M$ is a subset of any set $A$ that satisfies conditions #1 and #2.

Let's leave condition #3 alone for now and see what elements are included in the set $M$. The number 3 is included there according to the first paragraph. Since $3\in M$, then, according to point #2, we have: $3\cdot 3\in M$, i.e. $9\in M$. Since $9\in M$, then according to point #2 we get: $3\cdot 9\in M$, i.e. $27\in M$. Since $27\in M$, then by the same point #2 we have: $81\in M$. In short, the constructed set 3, 9, 27, 81 and so on is natural degrees number 3.

$$3^1=1; \; 3^2=9; \; 3^3=27; \; 3^4=81;\; \ldots$$

So, it seems that the required set is given. And it looks like this: $$3,9,27,81,\ldots $$. However, do conditions #1 and #2 really define only this set?

Consider the set of all natural numbers, i.e. $N$. The number 3 is natural, so $3\in N$. Conclusion: the set $N$ satisfies item #1. Further, for any natural number $x$ the set $N$ also contains the number $3x$. For example, 5 and 15, 7 and 21, 13 and 39 and so on. Hence, the set $N$ satisfies condition #2. And by the way, not only the set $N$ satisfies conditions #1 and #2. For example, the set of all odd natural numbers $N_1=$1,3,5,7,9,11, \ldots$$ also fits the conditions of items #1 and #2. How can we indicate that we need exactly the set $$3,9,27,81,\ldots $$?

A bunch of is a collection of objects considered as a whole. The concept of a set is taken as basic, that is, not reducible to other concepts. The objects that make up a given set are called its elements. Basic relationship between element a and the set containing it A denoted as ( a is an element of the set A; or a belongs A, or A contains a). If a is not an element of the set A, then write ( a not included in A, A does not contain a). A set can be specified by specifying all of its elements, in which case curly braces are used. So ( a, b, c) denotes the set of three elements. A similar notation is also used in the case of infinite sets, with unwritten elements being replaced by ellipsis. So, the set of natural numbers is denoted by (1, 2, 3, ...), and the set of even numbers (2, 4, 6, ...), and the ellipsis in the first case means all natural numbers, and in the second - only even.

Two sets A And B called equal, if they consist of the same elements, i.e. A belongs B and vice versa, each element B belongs A. Then write A = B. Thus, a set is uniquely determined by its elements and does not depend on the order in which these elements are written. For example, a set of three elements a, b, c allows six types of recording:

{a, b, c} = {a, c, b} = {b, a, c} = {b, c, a} = {c, a, b} = {c, b, a}.

For reasons of formal convenience, the so-called "empty set" is also introduced, namely, a set that does not contain a single element. It is denoted, sometimes by the symbol 0 (coincidence with the designation of the number zero does not lead to confusion, since the meaning of the symbol is clear every time).

If each element of the set A included in the set B, That A called a subset B, A B called a superset A. Write ( A included in B or A contained in B, B contains A). Obviously, if and , then A = B. The empty set is, by definition, considered to be a subset of any set.

If each element of the set A included in B, but many B contains at least one element that is not in A, i.e. if and , then A called own subset B, A B - own superset A. In this case, write . For example, the notation and mean the same thing, namely, that the set A not empty.

Note also that it is necessary to distinguish the element a and set ( a) containing a as the only element. Such a difference is dictated not only by the fact that the element and the set play different roles (the relation is not symmetrical), but also by the need to avoid contradiction. So, let A = {a, b) contains two elements. Consider the set ( A) containing as its only element the set A. Then A contains two elements, while ( A) is only one element, and therefore the identification of these two sets is impossible. Therefore, it is recommended to use notation, and not to use notation.

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

Basic concepts mathematical analysis are quantity, set, function, infinitesimal function, limit, derivative, integral.

Value everything that can be measured and expressed by a number is called.

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

Sets are denoted by capital letters, and elements of a set by lowercase letters. Set elements 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 (⊂ - is contained).

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

For example, the enumeration defines the following sets:

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

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

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

Basic Numeric Sets

N	(1,2,3,...,n) The set of all
Z	(0, ±1, ±2, ±3,...) Set whole numbers. The set of integers includes the set of natural numbers.
Q	A bunch of rational numbers. In addition to whole numbers, there are also fractions. A fraction is an expression of the form , where p is an integer, q- natural. Decimals can also be written as . For example: 0.25 = 25/100 = 1/4. Integers can also be written as . For example, in the form of a fraction with a denominator of "one": 2 = 2/1. Thus any rational number can be written decimal— finitely or infinitely periodic.
R	Many of all real numbers. Irrational numbers are infinite non-periodic fractions. These include: Together, two sets (rational and irrational numbers) form the set of real (or real) numbers.

If a set contains no elements, then it is called empty set and recorded Ø .

Elements of logical symbolism

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

quantifier

When writing mathematical expressions, quantifiers are often used.

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

∀- general quantifier, is used instead of the words "for all", "for anyone".
∃- existential quantifier, is used instead of the words "exists", "has". The symbol combination ∃! is also used, which is read as there is only one.

Operations on sets

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.

Union (sum) sets A and B is called the 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)

Intersection (product) sets A and B is called the set A ∩ B, whose elements 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)

difference sets A and B is called a 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)

Symmetric 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)