Multitudes. Operations on sets.
Displaying sets. Power of set

I welcome you to the first lesson on higher algebra, which appeared... on the eve of the site’s fifth anniversary, after I had already created more than 150 articles on mathematics, and my materials began to be compiled into a completed course. However, I hope that I am not late - after all, many students begin to delve into lectures only for state exams =)

A university vyshmat course is traditionally based on three pillars:

– mathematical analysis (limits, derivatives etc.)

– and finally, the 2015/16 academic year season opens with lessons Algebra for Dummies, Elements of mathematical logic, on which we will analyze the basics of the section, as well as get acquainted with basic mathematical concepts and common notations. I must say that in other articles I do not overuse “squiggles” , however, this is just a style, and, of course, they need to be recognized in any condition =). I inform newly arrived readers that my lessons are practice-oriented, and the following material will be presented in this spirit. For more complete and academic information, please refer to educational literature. Go:

A bunch of. Examples of sets

Set is a fundamental concept not only of mathematics, but of the entire surrounding world. Take any object in your hand right now. Here you have a set consisting of one element.

In a broad sense, set is a collection of objects (elements) that are understood as a single whole(according to certain characteristics, criteria or circumstances). Moreover, these are not only material objects, but also letters, numbers, theorems, thoughts, emotions, etc.

Sets are usually denoted in capital letters (optionally, with subscripts: etc.), and its elements are written in curly braces, for example:

– many letters of the Russian alphabet;
– set of natural numbers;

Well, it's time to get to know each other a little:
– many students in the 1st row

... I'm glad to see your serious and concentrated faces =)

The sets are final(consisting of a finite number of elements), and a set is an example infinite multitudes. In addition, the so-called empty set:

– a set in which there is not a single element.

The example is well known to you - the set in the exam is often empty =)

The membership of an element in a set is written with the symbol, for example:

– the letter “be” belongs to many letters of the Russian alphabet;
- letter "beta" Not belongs to many letters of the Russian alphabet;
– the number 5 belongs to the set of natural numbers;
- but the number 5.5 is no longer there;
– Voldemar does not sit in the front row (and, moreover, does not belong to the multitude or =)).

In abstract and not very algebra, the elements of a set are denoted by small Latin letters and, accordingly, the fact of ownership is drawn up in the following style:

– the element belongs to the set.

The above sets are written direct transfer elements, but this is not the only way. It is convenient to define many sets using some sign (s), which is inherent all its elements. For example:

– the set of all natural numbers less than one hundred.

Remember: a long vertical stick expresses the verbal phrase “which”, “such that”. Quite often a colon is used instead: - let's read the entry more formally: "the set of elements belonging to the set of natural numbers, such that » . Well done!

This set can also be written by direct enumeration:

More examples:
– and if there are quite a lot of students in the 1st row, then such an entry is much more convenient than directly listing them.

– a set of numbers belonging to the segment . Please note that this means multiple valid numbers (more on them later), which are no longer possible to list separated by commas.

It should be noted that the elements of a set do not have to be “homogeneous” or logically interconnected. Take a large bag and start randomly putting various items into it. There is no pattern in this, but, nevertheless, we are talking about a variety of subjects. Figuratively speaking, a set is a separate “package” in which “by the will of fate” a certain collection of objects ended up.

Subsets

Almost everything is clear from the name itself: a set is subset set if every element of the set belongs to the set. In other words, the set is contained in the set:

An icon is called an icon inclusion.

Let's return to the example, in which this is a set of letters of the Russian alphabet. Let us denote by – the set of its vowels. Then:

You can also select a subset of consonant letters and, in general, an arbitrary subset consisting of any number of randomly (or non-randomly) taken Cyrillic letters. In particular, any Cyrillic letter is a subset of the set.

It is convenient to depict relationships between subsets using a conventional geometric diagram called Euler circles.

Let be the set of students in the 1st row, be the set of students in the group, and be the set of university students. Then the inclusion relation can be depicted as follows:

The set of students from another university should be depicted as a circle that does not intersect the outer circle; many students of the country - a circle that contains both of these circles, etc.

We see a typical example of inclusions when considering numerical sets. Let us repeat school material that is important to keep in mind when studying higher mathematics:

Number sets

As you know, historically the first to appear were natural numbers intended for counting material objects (people, chickens, sheep, coins, etc.). This set has already been encountered in the article, the only thing is that we are now slightly modifying its designation. The fact is that numerical sets are usually denoted by bold, stylized or thick letters. I prefer to use bold font:

Sometimes zero is included in the set of natural numbers.

If we add the same numbers with the opposite sign and zero to the set, we get set of integers:

Innovators and lazy people write down its elements with icons "plus minus":))

It is quite clear that the set of natural numbers is a subset of the set of integers:
– since every element of the set belongs to the set. Thus, any natural number can safely be called an integer.

The name of the set is also “telling”: whole numbers – that means, no fractions.

And, since they are integers, let us immediately remember the important signs of their divisibility by 2, 3, 4, 5 and 10, which will be required in practical calculations almost every day:

An integer is divisible by 2 without a remainder, if it ends in 0, 2, 4, 6 or 8 (i.e. any even digit). For example, numbers:
400, -1502, -24, 66996, 818 – divisible by 2 without a remainder.

And let’s immediately look at the “related” sign: an integer is divisible by 4, if a number made up of its last two digits (in the order they appear) divisible by 4.

400 – divisible by 4 (since 00 (zero) is divisible by 4);
-1502 – not divisible by 4 (since 02 (two) is not divisible by 4);
-24, of course, is divisible by 4;
66996 – divisible by 4 (since 96 is divisible by 4);
818 – not divisible by 4 (since 18 is not divisible by 4).

Conduct a simple substantiation of this fact yourself.

Divisibility by 3 is a little more difficult: an integer is divisible by 3 without a remainder if the sum of the digits included in it divisible by 3.

Let's check whether the number 27901 is divisible by 3. To do this, sum up its digits:
2 + 7 + 9 + 0 + 1 = 19 – not divisible by 3
Conclusion: 27901 is not divisible by 3.

Let's sum up the digits of -825432:
8 + 2 + 5 + 4 + 3 + 2 = 24 – divisible by 3
Conclusion: the number -825432 is divisible by 3

Integer divisible by 5, if it ends with a five or a zero:
775, -2390 – divisible by 5

Integer divisible by 10 if it ends in zero:
798400 – divisible by 10 (and obviously by 100). Well, everyone probably remembers that in order to divide by 10, you just need to remove one zero: 79840

There are also signs of divisibility by 6, 8, 9, 11, etc., but there is practically no practical use from them =)

It should be noted that the listed signs (seemingly so simple) are strictly proven in number theory. This section of algebra is generally quite interesting, but its theorems... are just like a modern Chinese execution =) And that was enough for Voldemar at the last desk... but it’s okay, soon we’ll do life-giving physical exercises =)

The next numerical set is set of rational numbers:
– that is, any rational number can be represented as a fraction with an integer numerator and natural denominator.

Obviously, the set of integers is subset set of rational numbers:

And in fact, any integer can be represented as a rational fraction, for example: etc. Thus, an integer can quite legitimately be called a rational number.

A characteristic “identifying” feature of a rational number is the fact that when dividing the numerator by the denominator, the result is either
– integer,

or
– final decimal,

or
– endless periodic decimal (replay may not start immediately).

Enjoy division and try to do this action as little as possible! In the organizational article Higher mathematics for dummies and in other lessons I have repeatedly repeated, repeat, and will repeat this mantra:

In higher mathematics we strive to perform all operations in ordinary (proper and improper) fractions

Agree that dealing with a fraction is much more convenient than with the decimal number 0.375 (not to mention infinite fractions).

Let's move on. In addition to rational numbers, there are many irrational numbers, each of which can be represented as an infinite NON-PERIODIC decimal fraction. In other words, there is no pattern in the “infinite tails” of irrational numbers:
(“year of birth of Leo Tolstoy” twice)
etc.

There is plenty of information about the famous constants “pi” and “e”, so I will not dwell on them.

The combination of rational and irrational numbers forms set of real numbers:

– icon associations sets.

The geometric interpretation of a set is familiar to you - this is the number line:


Each real number corresponds to a certain point on the number line, and vice versa - each point on the number line necessarily corresponds to a certain real number. Essentially, I have now formulated continuity property real numbers, which, although it seems obvious, is strictly proven in the course of mathematical analysis.

The number line is also denoted by an infinite interval, and the notation or equivalent notation symbolizes the fact that it belongs to the set of real numbers (or simply “x” is a real number).

Everything is transparent with embeddings: the set of rational numbers is subset sets of real numbers:
, thus, any rational number can safely be called a real number.

A lot of irrational numbers are also subset real numbers:

At the same time, subsets and do not intersect- that is, not a single irrational number can be represented as a rational fraction.

Are there any other number systems? Exist! This is, for example, complex numbers, which I recommend getting acquainted with literally in the coming days or even hours.

Well, for now we move on to the study of operations on sets, the spirit of which has already materialized at the end of this section:

Actions on sets. Venn diagrams

Venn diagrams (similar to Euler circles) are a schematic representation of actions with sets. Again, I warn you that I will not consider all operations:

1) Intersection AND and is indicated by the icon

The intersection of sets is a set, each element of which belongs to And many, And to many. Roughly speaking, intersection is the common part of sets:

So, for example, for sets:

If sets do not have identical elements, then their intersection is empty. We just came across this example when considering numerical sets:

The sets of rational and irrational numbers can be schematically represented by two disjoint circles.

The intersection operation is also applicable for a larger number of sets; in particular, Wikipedia has a good one an example of the intersection of sets of letters of three alphabets.

2) An association sets are characterized by a logical connective OR and is indicated by the icon

A union of sets is a set, each element of which belongs to the set or to many:

Let's write the union of sets:
– roughly speaking, here you need to list all the elements of the sets and , and the same elements (in this case, the unit is at the intersection of sets) should be specified once.

But the sets, of course, may not intersect, as is the case with rational and irrational numbers:

In this case, you can draw two non-intersecting shaded circles.

The union operation is also applicable for a larger number of sets, for example, if , then:

In this case, the numbers do not have to be arranged in ascending order. (I did this purely for aesthetic reasons). Without further ado, the result can be written like this:

3) By difference And does not belong to the set:

The difference is read as follows: “a without be.” And you can reason in exactly the same way: consider the sets . To write down the difference, you need to “throw away” from the set all the elements that are in the set:

Example with number sets:
– here all natural numbers are excluded from the set of integers, and the entry itself reads like this: “a set of integers without a set of natural numbers.”

Mirrored: difference sets and are called a set, each element of which belongs to the set And does not belong to the set:

For the same sets
– what is in the set is “thrown out” from the set.

But this difference turns out to be empty: . And in fact, if you exclude integers from the set of natural numbers, then, in fact, there will be nothing left :)

In addition, it is sometimes considered symmetrical difference, which unites both “crescents”:
– in other words, this is “everything except the intersection of sets.”

4) Cartesian (direct) product sets and is called a set everyone ordered pairs in which element , and element

Let's write down the Cartesian product of sets:
– it is convenient to enumerate pairs using the following algorithm: “first, we sequentially attach each element of the set to the 1st element of the set, then we append each element of the set to the 2nd element of the set, then we append each element of the set to the 3rd element of the set”:

Mirrored: Cartesian product sets and the set of all is called ordered pairs in which In our example:
– here the recording scheme is similar: first we sequentially add all the elements of the set to “minus one”, then to “de” we add the same elements:

But this is purely for convenience - in both cases, the pairs can be listed in any order - it is important to write down here All possible pairs.

And now the highlight of the program: the Cartesian product is nothing more than the set of points of our native Cartesian coordinate system .

Exercise for self-fixing of material:

Perform operations if:

A bunch of It is convenient to describe it by listing its elements.

And a little thing with intervals of real numbers:

Let me remind you that the square bracket means inclusion numbers into the interval, and the round one - its non-inclusion, that is, “minus one” belongs to the set, and “three” Not belongs to the set. Try to figure out what the Cartesian product of these sets is. If you have any difficulties, follow the drawing ;)

A short solution to the problem at the end of the lesson.

Displaying Sets

Display many into many is rule, according to which each element of the set is associated with an element (or elements) of the set. In the event that the correspondence is made the only one element, then this rule is called clearly defined function or just function.

A function, as many people know, is most often denoted by a letter - it puts in correspondence to each element has a single value belonging to the set.

Well, now I will again disturb many students of the 1st row and offer them 6 topics for essays (many):

Installed (voluntary or forced =)) The rule assigns each student of the set a single topic of the set's essay.

...and you probably couldn’t even imagine that you would play the role of a function argument =) =)

The elements of the set form domain functions (denoted by ), and the elements of the set are range functions (denoted by ).

The constructed mapping of sets has a very important characteristic: it is one-to-one or bijective(bijection). In this example this means that to each the student is matched one unique topic of the essay, and back - for each The topic of the essay is assigned to one and only one student.

However, one should not think that every mapping is bijective. If you add a 7th student to the 1st row (to the set), then the one-to-one correspondence will disappear - or one of the students will be left without a topic (there will be no display at all), or some topic will go to two students at once. The opposite situation: if a seventh topic is added to the set, then the one-to-one mapping will also be lost - one of the topics will remain unclaimed.

Dear students in the 1st row, do not be upset - the remaining 20 people after the classes will go to clean the university territory from autumn foliage. The caretaker will give out twenty goliks, after which a one-to-one correspondence will be established between the main part of the group and the brooms..., and Voldemar will also have time to run to the store =)). The area of ​​definition corresponds to his own unique“y”, and vice versa - for any value of “y” we can unambiguously restore “x”. So it is a bijective function.

! Just in case, I will eliminate any possible misunderstanding: my constant reservation about the scope of definition is not accidental! A function may not be defined for all “X”s, and, moreover, it may be one-to-one in this case as well. Typical example:

But for the quadratic function nothing similar is observed, firstly:
– that is, different values ​​of “x” were displayed in same the meaning of "yay"; and secondly: if someone calculated the value of the function and told us that , then it is not clear whether this “y” was obtained at or at ? Needless to say, there is not even a hint of mutual unambiguity here.

Task 2: view graphs of basic elementary functions and write down the bijective functions on a piece of paper. Checklist at the end of this lesson.

Power of set

Intuition suggests that the term characterizes the size of a set, namely the number of its elements. And our intuition does not deceive us!

The cardinality of an empty set is zero.

The cardinality of the set is six.

The power of the set of letters of the Russian alphabet is thirty-three.

And in general - the power of any final of a set is equal to the number of elements of a given set.

...perhaps not everyone fully understands what it is final set – if you start counting the elements of this set, sooner or later the counting will end. As they say, the Chinese will eventually run out.

Of course, sets can be compared in terms of cardinality and their equality in this sense is called equal power. Equivalence is determined as follows:

Two sets are of equal cardinality if a one-to-one correspondence can be established between them.

The set of students is equivalent to the set of essay topics, the set of letters of the Russian alphabet is equivalent to any set of 33 elements, etc. Notice what exactly anyone set of 33 elements - in this case, only their number matters. The letters of the Russian alphabet can be compared not only with many numbers
1, 2, 3, …, 32, 33, but generally with a herd of 33 cows.

The situation with infinite sets is much more interesting. Infinities are different too! ...green and red The smallest infinite sets are counting multitudes. Quite simply, the elements of such a set can be numbered. The reference example is a set of natural numbers . Yes - it is infinite, but each of its elements, in PRINCIPLE, has a number.

There are a lot of examples. In particular, the set of all even natural numbers is countable. How to prove this? You need to establish its one-to-one correspondence with the set of natural numbers or simply number the elements:

A one-to-one correspondence is established, therefore, the sets are equal and the set is countable. Paradoxically, from the point of view of power, there are as many even natural numbers as there are natural numbers!

The set of integers is also countable. Its elements can be numbered, for example, like this:

Moreover, the set of rational numbers is also countable . Since the numerator is an integer (and they, as just shown, can be numbered), and the denominator is a natural number, then sooner or later we will “get” to any rational fraction and assign a number to it.

But the set of real numbers is already uncountable, i.e. its elements cannot be numbered. This fact, although obvious, is strictly proven in set theory. The cardinality of the set of real numbers is also called continuum, and compared to countable sets this is a “more infinite” set.

Since there is a one-to-one correspondence between the set and the number line (see above), then the set of points on the number line is also uncountable. And what’s more, there are the same number of points on both the kilometer and millimeter segments! Classic example:


By rotating the beam counterclockwise until it aligns with the beam, we will establish a one-to-one correspondence between the points of the blue segments. Thus, there are as many points on the segment as there are on the segment and !

This paradox is apparently connected with the riddle of infinity... but now we will not bother ourselves with the problems of the universe, because the next step is

Task 2 One-to-one functions in lesson illustrations

2. In how many ways can the coach determine which of the 12 athletes ready to participate in the 4x100 m relay will run in the first, second, third and fourth stages?

3. In a circular diagram, the circle is divided into 5 sectors. The sectors are painted with different colors taken from a set containing 10 colors. in how many ways can this be done?

4. find the value of the expression

c)(7!*5!)/(8!*4!)

TO EVERYONE WHO DECIDED, thank you)))

No. 1. 1. Give the concept of a complex number. Name three forms of representing complex numbers (1 point).

2. Given complex numbers: z1=-4i and z2=-5+i. Indicate their form of representation, find the real and imaginary parts of the indicated numbers (1 point).
3. Find their sum, difference and product (1 point).
4. Write down the numbers that are complex conjugates of the data (1 point).
No. 2. 1. How is a complex number represented on the complex plane (1 point)?
2. Given a complex number. Draw it on the complex plane. (1 point).
3. Write down the formula for calculating the modulus of a complex number and calculate (2 points).
No. 3. 1. Define a matrix, name the types of matrices (1 point).
2. Name linear operations on matrices (1 point).
3. Find a linear combination of two matrices if, (2 points).
No. 4. 1. What is the determinant of a square matrix? Write down the formula for calculating the 2nd order determinant (1 point).
2. Calculate the second-order determinant: (1 point).
3. Formulate a property that can be used to calculate the 2nd order determinant? (1 point)
4. Calculate the determinant using its properties (1 point).
No. 5. 1. In what cases is the determinant of a square matrix equal to zero (1 point)?
2. Formulate Sarrus’ rule (draw a diagram) (1 point).
3. Calculate the 3rd order determinant (by any of the methods) (2 points).
No. 6. 1. Which matrix is ​​called the inverse of a given matrix (1 point)?
2. For which matrix can the inverse be constructed? Determine whether there is a matrix inverse of the matrix. (2 points).
3. Write down the formula for calculating the elements of the inverse matrix (1 point).
No. 7. 1. Define the rank of a matrix. Name the methods for finding the rank of a matrix. What is the rank of the matrix? (2 points).
2. Determine between which values ​​the rank of matrix A lies: A= . Calculate some minor of the 2nd order (2 points).
No. 8. 1. Give an example of a system of linear algebraic equations (1 point).
2. What is called a solution to a system? (1 point).
3. What system is called joint (incompatible), definite (indefinite)? Formulate a criterion for system compatibility (1 point).
4. The extended matrix of the system is given. Write down the system corresponding to this matrix. Using the Kronecker-Capelli criterion, draw a conclusion about the compatibility or incompatibility of this system. (1 point).
No. 9. 1. Write a system of linear algebraic equations in matrix form. Write down a formula for finding unknowns using the inverse matrix. (1 point).
2. In what case can a system of linear algebraic equations be solved using the matrix method? (1 point).
3. Write the system in matrix form and determine whether it can be solved using the inverse matrix? How many solutions does this system have? (2 points).
No. 10. 1. Which system is called square? (1 point).
2. State Cramer’s theorem and write Cramer’s formulas. (1 point).
3. Using Cramer’s formulas, solve the system. (2 points).

help me please! as much as you can! urgent need!

1.What is called a quadratic trinomial
2.What is a discriminant
3Which equation is called a quadratic equation?
4. What equations are called equivalent?
5. Which equation is called an incomplete quadratic equation?
6. How many roots can an incomplete quadratic equation have?
7. How many roots does a quadratic equation have if the discriminant:
a) positive; b) equal to zero; c) negative?
8. What formula can be used to find the roots of a quadratic equation if its discriminant is non-negative?
9. Which equation is called a reduced quadratic equation?
10. What formula can be used to find the roots of the reduced square
equation if its discriminant is non-negative?
11. Formulate:
a) Vieta’s theorem; b) the theorem converse to Vieta’s theorem.
12. Which equation is called rational with unknown x? What is the root of an equation with unknown x? What does it mean to solve an equation? What equations are called equivalent?
13. Which equation is called a biquadratic equation? How do you solve a biquadratic equation? How many roots can a biquadratic equation have?
opinion?
14. Give an example of a splitting equation and explain how to solve it. What does it mean “an equation splits into two equations”?
15. How can you solve an equation, one part of which is zero,
and the other is an algebraic fraction?
16. What is the rule for solving rational equations? What
what can happen if you deviate from this rule?

Using a simple example, let us recall what is called a subset, what subsets there are (proper and improper), the formula for finding the number of all subsets, as well as a calculator that gives the set of all subsets.

Example 1. Given a set A = (a, c, p, o). Write down all subsets
of this set.
Solution:

Own subsets:(a) , (c) , (p) , (o) , (a, c) , (a, p) , (a, o), (c, p) , (c, o ) ∈, (p, o), (a, c, p) , (a, c, o), (c, p, o).

Not own:(a, c, p, o), Ø.

Total: 16 subsets.

Explanation. A set A is a subset of B if every element of A is also contained in B.

The empty set ∅ is a subset of any set and is called improper;
. any set is a subset of itself, also called improper;
. Any n-element set has exactly 2 n subsets.

The last statement is formula for finding the number of all subsets without listing each one.

Derivation of the formula: Let's say we have a set of n-elements. When composing subsets, the first element may or may not belong to the subset, i.e. we can choose the first element in two ways, similarly for all other elements (total n-elements), we can choose each in two ways, and according to the multiplication rule we get: 2∙2∙2∙ ...∙2=2 n

For mathematicians, we will formulate a theorem and give a rigorous proof.

Theorem. The number of subsets of a finite set consisting of n elements is 2 n.

Proof. A set consisting of one element a has two (i.e. 2 1) subsets: ∅ and (a). A set consisting of two elements a and b has four (i.e. 2 2) subsets: ∅, (a), (b), (a; b).
A set consisting of three elements a, b, c has eight (i.e. 2 3) subsets:
∅, (a), (b), (b; a), (c), (c; a),(c; b), (c; b; a).
It can be assumed that adding a new element doubles the number of subsets.
Let us complete the proof using the method of mathematical induction. The essence of this method is that if a statement (property) is true for some initial natural number n 0 and if, from the assumption that it is true for an arbitrary natural number n = k ≥ n 0, one can prove its validity for the number k + 1, then this property true for all natural numbers.

1. For n = 1 (induction base) (and even for n = 2, 3) the theorem is proven.

2. Let us assume that the theorem has been proven for n = k, i.e. the number of subsets of a set consisting of k elements is equal to 2k.

3. Let us prove that the number of subsets of the set B consisting of n = k + 1 elements is equal to 2 k+1.
We choose some element b of the set B. Consider the set A = B \ (b). It contains k elements. All subsets of set A are subsets of set B that do not contain element b and, by assumption, there are 2 k of them. There are the same number of subsets of the set B containing element b, i.e. 2k
things.

Therefore, all subsets of set B: 2 k + 2 k = 2 ⋅ 2 k = 2 k+1 pieces.
The theorem has been proven.

In example 1, the set A = (a, c, p, o) consists of four elements, n=4, therefore, the number of all subsets is 2 4 =16.

If you need to write down all the subsets, or write a program to write the set of all subsets, then there is an algorithm for solving it: represent the possible combinations in the form of binary numbers. Let's explain with an example.

Example 2. There is a set (a b c), the following numbers are put into correspondence:
000 = (0) (empty set)
001 = (c)
010 = (b)
011 = (b c)
100 = (a)
101 = (a c)
110 = (a b)
111 = (a b c)

Set of all subsets calculator.

The calculator already contains the elements of the set A = (a, c, p, o), just click the Submit button. If you need a solution to your problem, then type the elements of the set in Latin, separated by commas, as shown in the example.

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

The basic concepts of mathematical analysis are quantity, set, function, infinitesimal 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.

For example, the following sets are specified by enumeration:

• 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

N	(1,2,3,...,n) Set of all
Z	(0, ±1, ±2, ±3,...) Set integers. 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- integer, q- natural. Decimal fractions 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 the denominator “one”: 2 = 2/1. Thus, any rational number can be written as a decimal fraction - finite or infinitely periodic.
R	Plenty of everyone 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 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.

Quantifier

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

Commutability properties

A ∪ B = B ∪ A
A ∩ B = B ∩ A

Matching property

(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.

Example 1

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

Recall that “set” is an undefined concept in mathematics. Georg Cantor (1845 – 1918), a German mathematician whose work underlies modern set theory, said that “a set is many things conceived as one.”

Sets are usually denoted in capital Latin letters, elements of the set - in small letters. The words "belongs" and "does not belong" are indicated by the symbols:
And
:
– element belongs to the set ,
– element does not belong to the set .

The elements of the set can be any objects - numbers, vectors, points, matrices, etc. In particular, the elements of a set can be sets.

For numerical sets, the following notations are generally accepted:

– the set of natural numbers (positive integers);

– an extended set of natural numbers (the number zero is added to the natural numbers);

– the set of all integers, which includes positive and negative integers, as well as zero.

– the set of rational numbers. A rational number is a number that can be written as a fraction
- whole numbers). Since any whole number can be written as a fraction, (for example,
), and not in a unique way, all integers are rational.

– the set of real numbers, which includes all rational numbers, as well as irrational numbers. (For example, numbers are irrational).

Each branch of mathematics uses its own sets. When starting to solve a problem, we first determine the set of objects that will be considered in it. For example, in problems of mathematical analysis, all kinds of numbers, their sequences, functions, etc. are studied. The set that includes all the objects considered in the problem is called universal set (for this task).

The universal set is usually denoted by the letter . The universal set is a maximal set in the sense that all objects are its elements, i.e. the statement
within the task is always true. The minimal set is empty set – , which does not contain any elements.

Set set - this means to indicate a method that allows relative to any element universal set definitely install, belongs many or does not belong. In other words, it is a rule to determine which of two statements
or
, which is true and which is false.

Sets can be defined in various ways. Let's look at some of them.

1. List of set elements. In this way, you can define finite or countable sets. A set is finite or countable if its elements can be numbered, for example, a 1 ,a 2 ,… etc. If there is an element with the highest number, then the set is finite, but if all natural numbers are used as numbers, then the set is an infinite countable set.

1). – a set containing 6 elements (finite set).

2). is an infinite countable set.

3). - a set containing 5 elements, two of which are
And
, are themselves sets.

2. Characteristic property. A characteristic property of a set is a property that every element of the set has, but that no object that does not belong to the set has.

1). - a set of equilateral triangles.

2). – the set of real numbers greater than or equal to zero and less than one.

3).
– the set of all irreducible fractions whose numerator is one less than the denominator.

3. Characteristic function.

Definition 1.1. Characteristic function of the set call the function
, defined on the universal set and taking the value one on those elements of the set which belong , and the value is null on elements that do not belong :

,

From the definition of the characteristic function two obvious statements follow:

1.
,
;

2.
,
.

Let us consider as an example the universal set =
and its two subsets: – the set of numbers less than 7, and – a set of even numbers. Characteristic functions of sets And look like

,
.

Let's write down the characteristic functions And to the table:

( )

A convenient illustration of sets are Euler-Venn diagrams, in which the universal set is depicted as a rectangle, and its subsets as circles or ellipses (Fig. 1.1( a-c)).

As can be seen from Fig. 1.1.( A), selection in the universal set U one set - many A, splits the rectangle into two disjoint regions in which the characteristic function takes different values: =1 inside the ellipse and =0 outside the ellipse. Adding another set - a set B, (Fig. 1.1 ( b)), again divides each of the existing two areas into two sub-areas. Formed
disjoint

areas, each of which corresponds to a certain pair of values ​​of characteristic functions ( ,). For example, pair (01) corresponds to the area in which =0,=1. This region includes those elements of the universal set U, which do not belong to the set A, but belong to the set B.

Adding a third set - a set C, (Fig. 1.1 ( V)), again divides each of the existing four areas into two sub-regions. Formed
non-overlapping areas. Each of them corresponds to a certain triple of values ​​of characteristic functions ( ,,). These triplets can be thought of as area numbers written in binary. For example, No. 101 2 =5 10, i.e. the area in which the elements of sets are located A And C, but there are no elements of the set B, – this is area No. 5. Thus, each of the eight areas has its own binary number, which carries information about whether the elements of this area belong or not to the sets A, B And C.

Adding a fourth, fifth, etc. sets, we obtain 2 4 , 2 5 ,…, 2 n areas, each of which has its own well-defined binary number, composed of the values ​​of the characteristic functions of the sets. We emphasize that the sequence of zeros and ones in any of the numbers is arranged in a certain, pre-agreed order. Only under the condition of ordering, the binary number of the area carries information about the membership or non-belonging of the elements of this area to each of the sets.

Note. Recall that a sequence of n real numbers in linear algebra is considered as an n-dimensional arithmetic vector with coordinates
. The binary number of an area can also be called a binary vector whose coordinates take values ​​in the set
:. The number of distinct n-dimensional binary vectors is 2n.