Operations on sets and their properties. Sets. Operations on Sets Operations on Sets
Basic concepts of set theory
The concept of a set is a fundamental concept in modern mathematics. We will consider it original and construct set theory intuitively. Let us give a description of this initial concept.
Lots of Is a collection of objects (objects or concepts), which is thought of as a single whole. The objects included in this collection are called elements sets.
You can talk about a lot of first-year students of the mathematics department, about a lot of fish in the ocean, etc. Mathematics is usually interested in a variety of mathematical objects: a set of rational numbers, a set of rectangles, etc.
The sets will be denoted by capital letters of the Latin alphabet, and its elements by small ones.
If is an element of the set M, then they say "belongs M"And write:. If some object is not an element of the set, then they say “does not belong M"And write (sometimes).
There are two main ways to define sets: enumeration 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, the elements of which are the numbers 2, 4, 7 and only they. This method is not always applicable, since, for example, the set of all real numbers cannot be set in this way.
Characteristic property elements of the set M Is such a property that any element possessing this property belongs to M, and any element that does not have this property does not belong to M... The set of elements with a property is denoted as follows:
or 
.
The most common sets have their own special designations. In the future, we will adhere to the following notation:
N= Is the set of all natural numbers;
Z= - the set of all integers;
- the set of all rational numbers;
R- the set of all real (real) numbers, i.e. rational numbers (infinite decimal periodic fractions) and irrational numbers (infinite decimal non-periodic fractions);
- the set of all complex numbers.
Let us give more special examples of specifying sets by specifying a characteristic property.
Example 1.
The set of all natural divisors of 48 can be written as follows: 
(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 as follows:.
Example 3. - an 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 containing only one element can be considered. For example, the set of integer roots of the equation 
... Moreover, it is convenient to speak of 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. Sets and are called equal(denoted by A = B) if these sets consist of the same elements.
Definition 2. If each element of the set belongs to the set, then we call subset sets.
Legend: ("included in"); ("Includes").
It is clear that Ø and the set itself are subsets of the set. Any other subset of the set is called its the right part... If and, then they say that “ A – proper subset"Or that" And it is strictly included in"And write.
The following statement is obvious: multitudes and are equal if and only if and.
This statement is based on a universal method for proving the equality of two sets: to prove that the sets and are equal, it suffices to show that ,a is a subset of the set .
This is the most common method, although not the only one. Later, having become acquainted with operations on sets and their properties, we will indicate another way of proving the equality of two sets - using transformations.
In conclusion, we note that often in one or another mathematical theory one deals with subsets of the same set U which is called universal in this theory. For example, in school algebra and mathematical analysis, the set is universal R real numbers, in geometry - a set of points in space.
Set operations and their properties
On sets, you can perform actions (operations) that resemble addition, multiplication and subtraction.
Definition 1. Consolidation sets and is called a set, denoted by, each element of which belongs to at least one of the sets or.
The operation itself, as a result of which such a set is obtained, is called a union.
Brief record of definition 1:
Definition 2. Crossing sets and is called a set, denoted by, containing all those and only those elements, each of which belongs to and, and.
The operation itself, which results in a set, is called intersection.
Definition 2 in short:
For example, if 
, 
, then 
, .
Sets can be depicted as geometric shapes, which allows you to visually illustrate operations on sets. This method was proposed by Leonard Euler (1707–1783) for the analysis of logical reasoning, was widely used and was further developed in the works of the English mathematician John Venn (1834–1923). Therefore, 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 some set of indices.
Definition . Consolidation a set 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 . Crossing a set 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 when the set of indices is finite, for example, 
, then to denote the union and intersection of a collection of sets in this case, they usually use the notation:
and 
.
For example, if 
, 
, 
, then , .
The concepts of union and intersection of sets are repeatedly encountered in the school mathematics course.
Example 1. Lots 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. Lots of M system solutions
is the intersection of the solution sets for each of the inequalities of this system. The set of solutions to the first equation is the set of points of a straight line, i.e. ... Lots of . The set consists of one element - the intersection points of the lines.
Example 3. The set of solutions to the equation
where 
, is the union of the sets of solutions to each of the equations, i.e.
Definition 3. Difference sets and called the set, denoted by, and consisting of all those and only those elements that belong, but do not belong .– shaded part; ... with operations of union, intersection and complement. The resulting mathematical structure is called algebra of sets or Boolean algebra of sets(including the Irish mathematician and logician George Boole (1816-1864)). Let us denote 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. Therefore, they are called the laws of the algebra of sets.
Mathematical analysis is a 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 value, set, function, infinitesimal function, limit, derivative, integral.
The magnitude everything that can be measured and expressed by a number is called.
Many is called a set of some elements united by some common feature. The elements of a set can be numbers, figures, objects, concepts, etc.
Sets are indicated by uppercase letters, and elements are indicated by multiples in lowercase letters. Set elements are enclosed in curly braces.
If element x belongs to the set X then write x∈NS (∈
- belongs).
If set A is part of set B, then write A ⊂ B (⊂
- contains).
A set can be specified in one of two ways: by enumeration and using a defining property.
For example, the following sets are specified by enumeration:- A = (1,2,3,5,7) - a set of numbers
- X = (x 1, x 2, ..., x n) - the set of some elements x 1, x 2, ..., x n
- N = (1,2, ..., n) - set of natural numbers
- Z = (0, ± 1, ± 2, ..., ± n) - 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 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 above (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) bounds of a set is called exact top (bottom) edge this set.
Basic number sets
| N | (1,2,3, ..., n) The set of all |
| Z | (0, ± 1, ± 2, ± 3, ...) The set whole numbers. The set of integers includes many natural numbers. |
| Q |
Lots of rational numbers. In addition to integers, there are also fractions. A fraction is an expression of the form, where p- an 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, as a fraction with the denominator "one": 2 = 2/1. Thus, any rational number can be written as a decimal fraction - of course or an infinite periodic. |
| R |
Many of all real numbers. Irrational numbers are infinite non-periodic fractions. These include: Together, two sets (rational and irrational numbers) - form a set of real (or real) numbers. |
If the set does not contain any element, then it is called empty set and is recorded Ø .
Elements of logical symbology
Notation ∀x: | x |<2 → x 2 < 4 означает: для каждого x такого, что |x|<2, выполняется неравенство x 2 < 4.
QuantorQuantifiers are often used when writing mathematical expressions.
Quantifier is a logical symbol that characterizes the following elements in quantitative terms.
- ∀- generality quantifier, is used instead of the words "for all", "for any".
- ∃- existential quantifier, is used instead of the words "exists", "is". The combination of characters ∃ !, which is read as there is only one, is also used.
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.
Consolidation (sum) sets A and B is called a set A ∪ B, the elements of which 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 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)
Difference sets A and B is 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 А = (1,2,3,4), B = (3,4,5,6), then А Δ В = (1,2) ∪ (5,6) = (1,2,5 , 6)
Properties of operations on sets
Permutability propertiesA ∪ B = B ∪ A
A ∩ B = B ∩ A
(A ∪ B) ∪ C = A ∪ (B ∪ C)
(A ∩ B) ∩ C = A ∩ (B ∩ C)
Countable and uncountable sets
In order to compare any two sets A and B, a correspondence is established between their elements.
If this correspondence is one-to-one, then the sets are called equivalent or equivalent, A B or B A.
Example 1The set of points of the leg BC and the hypotenuse AC of the triangle ABC are of equal power.
Theories
There are two main approaches to the concept of a set - naive and axiomatic set theory.
Axiomatic set theory
Today, a set is defined as a model that satisfies the ZFC axioms (Zermelo - Fraenkel axioms with the axiom of choice). With this approach, in some mathematical theories there are collections of objects that are not sets. Such collections are called classes (of different orders).
Element of the set
The objects that make up the set are called elements of the set or by points of the set. Sets are most often denoted by large letters of the Latin alphabet, its elements - by small ones. If a is an element of the set A, then write a ∈ A (a belongs to A). If a is not an element of the set A, then write a∉A (and does not belong to A).
Some kinds of sets
- An ordered set is a set on which an ordering relation is specified.
- A set (specifically, an ordered pair). Unlike just a set, it is written inside parentheses: ( x 1, x 2, x 3, ...), and the elements can be repeated.
By hierarchy:
Set of sets Subset Superset
By limitation:
Set operations
Literature
- Stoll R.R. Sets. Logics. Axiomatic theories. - M .: Education, 1968 .-- 232 p.
see also
Wikimedia Foundation. 2010.
See what "Set Element" is in other dictionaries:
element of the set- - [L.G. Sumenko. The English Russian Dictionary of Information Technology. M .: GP TsNIIS, 2003.] element of a set An object of any nature, which together with other similar objects constitutes a set. Often, instead of the term, an element in ... ...
Element of the set- an object of any nature, which together with other similar objects constitutes a set. Often, instead of the term element in this sense, they use "point of a set", "member of a set", etc. ... ...
SET, in mathematics, a collection of certain objects. These objects are called set members. The number of elements can be infinite or finite, or even zero (the number of elements in an empty set is denoted by 0). Each… … Scientific and technical encyclopedic dictionary
element- A generalized term, which, depending on the relevant conditions, can be understood as a surface, line, point. Notes 1. An element can be a surface (part of a surface, a plane of symmetry of several surfaces), a line (a profile ... Technical translator's guide
Part of something. One of the possible etymologies of this word is the name of a number of consonants in Latin letters L, M, N (el em en). Element (philosophy) Element is a mandatory accessory of the flag, banner and standard. Element of the set Elementary ... ... Wikipedia
Element- the primary (for this research, model) component of a complex whole. See Set Element, System Element ... Economics and Mathematics Dictionary
The set is one of the key objects of mathematics, in particular, set theory. “By a set we mean the unification into one whole of certain, completely distinguishable objects of our intuition or our thought” (G. Kantor). It is not in full ... ... Wikipedia
element- 02.01.14 element (character sign or symbol): A single stroke or space in a barcode symbol or a single polygonal or circular cell in a matrix symbol, forming a symbol sign in ... ... Dictionary-reference book of terms of normative and technical documentation
A; m. [from lat. elementum element, original substance] 1. Part of which l .; component. Decompose the whole into elements. What are the elements of culture? The nature of e. production. The constituent elements of which l. // Typical movement, one ... ... encyclopedic Dictionary
The concept of a set refers to the axiomatic concepts of mathematics.
Definition... A set is a set, group, collection of elements that have some common property or feature for all of them.
Designation: A, B.
Definition... Two sets A and B are equal if and only if they consist of the same elements. A = B.
The notation a ∈ A (a ∉ A) means that a is (is not) an element of the set A.
Definition... A set containing no elements is called empty and denoted by ∅.
Usually, in specific cases, the elements of all the sets under consideration are taken from one, sufficiently wide set U, which is called universal set.
Cardinality of the set is denoted as | M | ...
Comment : for finite sets, cardinality is the number of elements.
Definition... If | A | = | B | , then the sets are called equal.
To illustrate operations on sets, the following are often used Euler - Venn diagrams... The construction of the diagram consists in the image of a large rectangle representing the universal set U, and inside it - circles representing the sets.
The following operations are defined on sets:
Union А∪В: = (х / х∈А∨х∈В)
Intersection А∩В: = (х / х∈А & х∈В)
Difference А \ В: = (х / х∈А & х∈В)
Complement A U \ A: = (x / x U & x ∉ A)
Task 1.1. Given: a) A, B⊆Z, A = (1; 3; 4; 5; 9), B = (2; 4; 5; 10). b) A, B⊆R, A = [-3; 3), B = (2; 10].
Solution.
a) A∩B = (4; 5), A∪B = (1; 2; 3; 4; 5; 9; 10), A \ B = (1; 3; 9), B \ A = (2 ; 10), B = Z \ B;
b) A∩B = (2; 3), A∪B = [-3; 10], A \ B = [-3,2], B \ A =, BZ \ B = (-∞, 2] ∪ (10, + ∞).
1) Given: a) A, B ⊆ Z, A = (1; 2; 5; 7; 9; 11), B = (1; 4; 6; 7).
b) A, B ⊆ R, A = [-3; 7), B = [-4; 4].
Find: A∩B, A∪B, A \ B, B \ A, B.
2) Given: a) A, B ⊆ Z, A = (3; 6; 7; 10), B = (2; 3; 10; 12).
b) A, B ⊆ R, A =.
Find: A∩B, A∪B, A \ B, B \ A, B.
3) Given: a) A, B ⊆ Z, A = (1; 2; 5; 7; 9; 11), B = (1; 4; 6; 7).
b) A, B ⊆ R, A =.
4) Given: a) A, B ⊆ Z, A = (0; 4; 6; 7), B = (-3; 3; 7).
b) A, B ⊆ R, A = [-15; 0), B = [-2; 1].
Find: A∩B, A∪B, A \ B, B \ A, A.
5) Given: a) A, B ⊆ Z, A = (0; 9), B = (-6; 0; 3; 9).
b) A, B ⊆ R, A = [-10; 5), B = [-1; 6].
Find: A ∩ B, A ∪ B, A \ B, B \ A, B.
6) Given: a) A, B ⊆ Z, A = (0; 6; 9), B = (-6; 0; 3; 7).
b) A, B ⊆ R, A = [-8; 3), B =.
Find: A ∩ B, A ∪ B, A \ B, B \ A, B.
7) Given: a) A, B ⊆ Z, A = (-1; 0; 2; 10), B = (-1; 2; 9; 10).
b) A, B ⊆ R, A = [-10; 9), B = [-5; 15].
Find: A∩B, A∪B, A \ B, B \ A, B.
8) Given: a) A, B ⊆ Z, A = (1; 2; 9; 37), B = (-1; 1; 9; 11; 15).
b) A, B ⊆ R, A = [-8; 1), B = [-5; 7].
Find: A ∩ B, A ∪ B, A \ B, B \ A, B.
9) Given: a) A, B ⊆ Z, A = (-1; 0; 9; 17), B = (-1; 1; 9; 10; 25).
b) A, B ⊆ R, A = [-4; 9), B = [-5; 7].
Find: A∩B, A∪B, A \ B, B \ A, B.
10) Given: a) A, B⊆Z, A = (1; 7; 9; 17), B = (-2; 1; 9; 10; 25).
b) A, B⊆R, A =.
Find: A ∩ B, A ∪ B, A \ B, B \ A, A.
Task 1.1. Using the Euler-Venn diagrams, prove the identity:
A \ (B \ C) = (A \ B) ∪ (A ∩ C).
Solution.
Let's build Venn diagrams.
The left side of the equality is shown in Figure a), the right one - in Figure b). From the diagrams, the equality of the left and right sides of this relation is obvious.
Tasks for independent solution
Using Euler-Venn diagrams, prove the identities:
1) A \ (B ∪ C) = (A \ B) ∩ (A \ C);
2) A ∪ (B \ C) = (A ∩ B) \ C;
3) A ∪ (B \ C) = (A ∩ B) \ (A ∩ C);
4) (A \ B) \ C = (A \ B) \ (B \ C);
5) (A \ B) \ C = (A \ B) ∪ (A∩C);
6) A∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C);
7) (A ∩ B) \ (A ∩ C) = (A ∩ B) \ C;
8) A∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C);
9) (A ∪ B) \ C = (A \ C) ∪ (B \ C)
10) A∪ (A ∩ B) = A ∪ B
Task 1.3. In literature class, the teacher decided to find out which of the 40 students in the class had read books A, B, C. The results of the survey were as follows: book A was read by 25 students; Book B was read by 22 students; Book C was read by 22 students; books A or B were read by 33 students; books A or C were read by 32 students; books B or C were read by 31 students; all books were read by 10 students. Determine: 1) How many students have read only Book A?
2) How many students read only Book B?
3) How many students have read only Book C?
4) How many students have read only one book?
5) How many students have read at least one book?
6) How many students have not read a single book?
Solution.
Let U be the set of students in the class. Then
| U | = 40, | A | = 25, | B | = 22, | C | = 22, | A ∪ B | = 33, | A ∪ C | = 32, | B ∪ C | = 31, | A ∩ B ∩ C | = 10
Let's try to illustrate the problem.
We divide the set of students who have read at least one book into seven subsets k 1, k 2, k 3, k 4, k 5, k 6, k 7, where
k 1 - the set of students who have read only book A;
k 3 - the set of students who have read only book B;
k 7 - the set of students who have read only book C;
k 2 - the set of students who have read books A and B and have not read book C;
k 4 - the set of students who have read books A and C and have not read book B;
k 6 - the set of students who have read books B and C and have not read book A;
k 5 is the set of students who have read books A, B and C.
Let's calculate the cardinality of each of these subsets.
| k 2 | = | A ∩ B | - | A ∩ B ∩ C |; | k 4 | = | A ∩ C | - | A ∩ B ∩ C |;
| k 6 | = | B ∩ C | - | A ∩ B ∩ C |; | k 5 | = | A ∩ B ∩ C |.
Then | k 1 | = | A | - | k 2 | - | k 4 | - | k 5 |, | k 3 | = | B | - | k 2 | - | k 6 | - | k 5 |, | k 7 | = | C | - | k 6 | - | k | - | k 5 |.
Find | A ∩ B |, | A ∩ C |, | B ∩ C |.
| A ∩ B | = | A | + | B | - | A ∩ B | = 25 + 22 - 33 = 14,
| A ∩ C | = | A | + | C | - | A ∩ C | = 25 + 22 - 32 = 15,
| B ∩ C | = | B | + | C | - | B ∩ C | = 22 + 22 - 31 = 13.
Then k 1 = 25-4-5-10 = 6; k 3 = 22-4-3-10 = 5; k 7 = 22-5-3-10 = 4;
| A ∪ B ∪ C | = | A ∪ B | + | C | - | (A ∪ B) ∪ C | ...
It is clear from the figure that | C | - | (A ∪ B) ∪ C | = | k 7 | = 4, then | A ∪ B ∪ C | = 33 + 4 = 37 - the number of students who have read at least one book.
Since there are 40 students in the class, 3 students have not read a single book.
Answer:- 6 students only read book A.
- 5 students only read Book B.
- 4 students read only book C.
- 15 students read only one book.
- 37 students have read at least one book from A, B, C.
- 3 students have not read a single book.
Tasks for independent solution
1) During the week, films A, B, C were shown in the cinema. Each of the 40 students saw either all 3 films or one of the three. Movie A saw 13 schoolchildren. Movie B saw 16 schoolchildren. Movie C saw 19 schoolchildren. How many schoolchildren have seen only one film?
2) The international conference was attended by 120 people. Of these, 60 speak Russian, 48 - English, 32 - German, 21 - Russian and English, 19 - English and German, 15 - Russian and German, and 10 people speak all three languages. How many conference participants do not speak any of these languages?
3) A school team of 20 people participates in sports competitions, each of whom has a sports category in one or more of the three sports: athletics, swimming and gymnastics. It is known that 12 of them have categories in athletics, 10 in gymnastics and 5 in swimming. Determine the number of schoolchildren from this team who have categories in all sports, if 2 people have categories in athletics and swimming, 4 people in track and field and gymnastics, and 2 people in swimming and gymnastics.
4) A survey of 100 students gave the following results about the number of students studying various foreign languages: Spanish - 28; German - 30; French - 42; Spanish and German - 8; Spanish and French - 10; German and French - 5; all three languages - 3. How many students are learning German if and only if they are learning French? 5) A survey of 100 students revealed the following data on the number of students studying various foreign languages: only German - 18; German, but not Spanish - 23; German and French - 8; German - 26; French - 48; French and Spanish - 8; no language - 24. How many students are studying German and Spanish?
6) In the report on the survey of 100 students, it was reported that the number of students studying different languages is as follows: all three languages - 5; German and Spanish - 10; French and Spanish - 8; German and French - 20; Spanish - 30; German - 23; French - 50. The inspector who submitted this report was dismissed. Why?
7) The international conference was attended by 100 people. Of these, 42 speak French, 28 - English, 30 - German, 10 - French and English, 8 - English and German, 5 - French and German, and 3 people speak all three languages. How many conference participants do not speak any of these languages?
8) 1st year students studying computer science at the university can attend additional disciplines. This year, 25 of them chose to study accounting, 27 chose business, and 12 chose tourism. In addition, there were 20 students taking a course in accounting and business, 5 were studying accounting and tourism, and 3 were studying tourism and business. It is known that none of the students dared to attend 3 additional courses at once. How many students have attended at least 1 additional course?
9) 40 students took part in the Mathematics Olympiad for applicants. They were asked to solve one problem in algebra, one in geometry and one in trigonometry. The problem in algebra was solved by 20 people, in geometry - 18, in trigonometry - 18 people. The problems in algebra and geometry were solved by 7 people, in algebra and trigonometry - 8 people, in geometry and trigonometry - 9 people. No problem was solved by 3 people. How many students have solved only two problems?
10) There are 40 students in the class. Of these, 19 people have triplets in the Russian language, 17 people in mathematics and 22 people in physics. 4 students have triplets in only one Russian language, 4 - only in mathematics and 11 - only in physics. 5 students have triples in Russian, mathematics and physics. 7 people have triples in mathematics and physics. How many students have Cs in two of the three subjects?
Lots of is a collection of objects considered as one whole. The concept of a set is taken as basic, i.e., not reducible to other concepts. The objects that make up a given set are called its elements. Basic relationship between element a and containing the set A denoted as ( a is an element of the set A; or a belongs A, or A contains a). If a not a member of the set A, then they write ( a not included in A, A does not contain a). A set can be specified by specifying all of its elements, and in this case, curly braces are used. So ( a, b, c) denotes a set of three elements. A similar notation is used in the case of infinite sets, and the unwritten elements are 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 are called equal if they consist of the same elements, i.e. A belongs B and, conversely, each element B belongs A... Then they write A = B... Thus, the set is uniquely determined by its elements and does not depend on the writing order of these elements. 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 is included in many B, then A called a subset B, a B called a superset A... They write ( A is included in B or A contained in B, B contains A). Obviously, if and, then A = B... An empty set is by definition considered a subset of any set.
If each element of the set A is included in B but a lot B contains at least one element not included in A, i.e., if and, then A called its 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 is not empty.
Note also that it is necessary to distinguish between the element a and the set ( a) containing a as the only item. This difference is dictated not only by the fact that the element and the set play a different role (the relationship 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 the recording, and not use the recording.