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Elements of set theory.

"Under many we understand the unification into one whole of certain, completely distinguishable objects of our intuition or our thought” - this is how Georg Cantor, the founder of set theory, described the concept of “set”.

The basic premises of Cantor's set theory boil down to the following:

1° A set can consist of any distinguishable objects.

2° A set is uniquely defined by the set of its constituent objects.

3° Any property defines a set of objects that have this property.

If x is an object, P is a property, P(x) is a designation that x has property P, then (x|P(x)) denotes the entire class of objects that have property P. The objects that make up a class or set are called elements class or set.

The term " a bunch of" is used as a synonym for the concepts set, collection, collection of some elements. Thus, we can talk about:

a) many bees in the hive,

b) a set of points on a segment,

c) the set of vertices of a square or the sets of its sides and diagonals,

d) many students in the audience, etc.

In the above examples, in cases a), c)-d), the corresponding sets consist of a certain finite number of objects, such sets are called final. The set of points on a segment (example b)) cannot be counted, therefore such sets are called endless. A set that does not contain a single element is called empty many.

The simplest form of specifying a set is to list its elements, for example A = (4, 7, 13) (set A consists of three elements - integers 4, 7, 13). Another frequently used form of assignment is indicating the properties of the elements of the set, for example A = (x| x 2 ≤ 4) - the set of numbers x that satisfy the specified condition.

Sets are usually denoted by capital letters A, B, C,..., and their elements by small letters: a, b, c,... The notation a ∈ A (read: a belongs to A) or A ∋ a (read: A contains a) means , that a is an element of the set A. An empty set is denoted by the symbol Ø.

If every element of set B is also an element of set A, set B is called subset set A (designation - B ⊆ A or A ⊇ B).

Each set is its own subset (this is the “widest” subset of the set). The empty set is a subset of any set (this is the "narrowest" subset). Any other subset of set A contains at least one element of set A, but not all of its elements. Such subsets are called true, or proper subsets. For true subsets of a set A, the notation B ⊂ A or A ⊃ B is used. If at the same time B ⊆ A and A ⊆ B, that is, each element of the set B belongs to A, and at the same time, each element of A belongs to B, then A and B , obviously, consist of the same elements and, therefore, coincide. In this case, the set equals sign is used: A = B. (The symbols ∈, ∋, ⊂, ⊃, ⊆, ⊇ are called inclusion symbols).

Geometrically, sets are usually depicted as certain sets of points on a plane. The pictures themselves are called Euler-Venn diagrams (Eulerian circles). That is, Euler-Venn diagrams are geometric representations of sets or geometric representations of relationships between volumes of concepts through intersecting contours (circles or ellipses), proposed by the English logician John Venn (1834 - 1923) at the end of the century before last. In his works on visual graphic representation of logical figures, he relied on a number of graphic systems proposed by Euler (1707 - 1783), I. Lambert (1728 - 1777), Gergonne (1771 -1859), B. Bolzano (1781 -1848).

Here are some of the diagrams. The construction of the diagram consists of drawing a large rectangle representing the universal set U, and inside it - circles (or some other closed figures) representing sets. The shapes must intersect in the most general way required by the problem and must be labeled accordingly. Points lying inside different areas of the diagram can be considered as elements of the corresponding sets. With the diagram constructed, you can shade certain areas to indicate newly formed sets.

Set operations are considered to obtain new sets from existing ones.

Definition. Association sets A and B is a set consisting of all those elements that belong to at least one of the sets A, B (Fig. 1):

Definition. By crossing sets A and B is a set consisting of all those and only those elements that belong simultaneously to both set A and set B (Fig. 2):

Definition. By difference sets A and B is the set of all those and only those elements of A that are not contained in B (Fig. 3):

Definition. Symmetrical difference sets A and B is the set of elements of these sets that belong either only to set A or only to set B (Fig. 4):

Another common designation for the symmetric difference is: A ∆ B, instead of A + B.

Definition. An absolute complement set A is the set of all those elements that do not belong to set A (Fig. 5):

The following equalities are valid: (AUB)= A∩B; (A∩B)= AUB.

In conclusion, we present another formula for calculating the number of elements in the union of three sets (for the general case of their relative arrangement shown in the figure):

m(AUBUC)=m(A)+m(B)+m(C)-m(A∩B)-m(B∩C)-m(A∩C)+m(A∩B∩C)

Example 1. Write down the set of all natural divisors of the number 15 and the number of its elements.

Example 2. Given sets A=(2,3,5,8,13,15), B=(1,3,4,8,16), C=(12,13,15,16), D= (0,1 ,20).

Find AUB, CUD, B∩C, A∩D, A\C, C\A, B\D, AUBUC, A∩B∩C, BUD∩C, A∩C\D.

AUB= (1,2,3,4,5,8,13,15,16)

CUD= (0,1,12,13,15,16,20)

AUBUC= (1,2,3,4,5,8,12,13,15,16)

BU(D∩C)= (1,3,4,8,16)

(A∩C)\D= (13.15)

Example 3. The school has 1400 students. Of these, 1,250 can ski and 952 can skate. 60 students do not know how to ski or skate. How many students can skate and ski?

A∩B is the set of students who cannot ski or skate.

By condition m(A∩B)=60, we also use the equality (AUB)= A∩B, then m((AUB))=60.

So m(AUB)=m( U)-m((AUB))=1400-60=1340.

By condition m(A)=1250, m(B)=952, we obtain m(A∩B)=m(A)+m(B)-m(AUB)=1250+952-1340=862

Example 4. A group of 25 students passed the examination session with the following results: 2 people received only “excellent”; 3 people received excellent, good and satisfactory grades; 4 people only “good”; 3 people received good and satisfactory grades. The number of students who passed the session only as “excellent”, “good” is equal to the number of students who passed the session only as “satisfactory”. There are no students who received only excellent and satisfactory grades. 22 students received satisfactory or good grades. How many students did not appear for the exams? How many students passed the session only “satisfactorily”?x. Then from the condition, we get

We find the number of students who did not appear for the exam as follows:

Answer: 6 students received only “satisfactory” marks, 1 student did not appear for the exams.

Example 5.

Equality of sets.

Sets A And IN are considered equal if they consist from the same elements.

Equality of sets is denoted as follows: A = B.

If the sets are not equal, then write A ¹ B.

Writing the equality of two sets A = B is equivalent to writing AÌ IN, or INÌ A.

For example, the set of solutions to the equation x 2 - 5x+ 6 = 0 contains the same elements (the numbers 2 and 3) as the set of prime numbers less than five. These two sets are equal. (A prime number is a natural number that is divisible without a remainder only by 1 and itself; 1 is not a prime number.)

Intersection (multiplication) of sets.

A bunch of D, consisting of all elements belonging to and set A and set B, is called the intersection of sets A And IN and is designated D = A IN.

Let's consider two sets: X= (0, 1, 3, 5) and Y= (1, 2, 3, 4). The numbers 1 and 3 and only they belong to both sets simultaneously X And Y. The set (1, 3) made up of them contains all common sets X And Y elements. Thus, the set (1, 3) is the intersection of the considered sets X And Y:

{1, 3} = {0, 1, 3, 5} {1, 2, 3, 4}.

For the segment [-1; 1] and interval ]0; 3[ intersection, i.e., a set consisting of common elements, is the interval ]0; 1] (Fig. 1).

Rice. 1. Intersecting the segment [-1; 1] and interval ]0; 3[ is the interval ]0; 1]

The intersection of a set of rectangles and a set of rhombuses is a set of squares.

The intersection of a set of eighth-grade students of a given school and a set of members of a chemistry club of the same school is the set of eighth-grade students who are members of a chemistry club.

The intersection of sets (and other operations - see below) is well illustrated by visually depicting sets on a plane. Euler suggested using circles for this. Image of intersection (in gray) of sets A And IN using Euler circles is shown in Fig. 2.

Rice. 3. Euler-Venn diagram of intersection (highlighted in gray) of sets A And IN, which are subsets of a certain universe, depicted as a rectangle

If the sets A And IN do not have common elements, then they say that these sets do not intersect or that their intersection is the empty set, and write A IN = Æ.

For example, the intersection of the set of even numbers and the set of odd numbers is empty.

The intersection of the numerical intervals ]-1 is also empty; 0] and -1; 0] and )