Relations between sets.
1) sets do not have common elements
2) two sets have common elements
3) one set is a subset of another. The set is called subset set A if every element of set B is an element of set A. It is also said that set B is included in set A
4) two sets are equal. The sets are called equal or matching. If every element of set A is an element of set B and vice versa.
The empty set is a subset of any set.
Union of sets and its properties. Intersection of sets and its properties.
1. a) union of two sets. The union of two sets A and B is the set C, which consists of all those elements that belong to the set A or the set B. The union is defined by shading and is denoted
A B B A B A B
1) A U B=C, 2) 3) AU B=A, 4) AUB=A=B.
b) properties of the operation of union of sets:
commutative property: АУВ=ВUA
Associative property: АU (ВУС)=(АУВ) UC
absorption law: AUA=A; AUØ=A; AUU=U.
2. a) intersection of two sets. The intersection of two sets A and B is a set C containing all the elements that belong to set B at the same time.
A B A B A B
1) A∩B= Ø, 2) 3) A∩B=B 4) A∩B=A=B.
b) intersection properties:
commutative property: A∩B= B∩A
Associative property: A∩(B∩C)=(A∩B)∩C
absorption law: A∩A=A, A∩ Ø= Ø, A∩U=A
Distributive properties linking union and intersection operations.
They can be proved on Euler circles.
1). AU (B∩C)=(AUB)∩(AUC)
2). A∩(BUC)=(A∩B) U (A∩C)
Proof. Denote the left side of the equality by M, and the right side by H. To prove the validity of this equality, we prove that the set M is included in H, and H in M.
Let 1). (arbitrarily chosen element).
The principle of expanding the numerical set. Sets of integers and rational numbers, their properties.
1. The expandable set is a subset of the extended set (natural numbers are a subset of integers) N is the set of natural numbers, Z is the set of integers, Q is the set of rational numbers, R is the set of real numbers.
2. Operation of arithmetic operations in extensible R
set, which is algebraic,
Similarly, in the extended set. If in Q
Expandable set of arithmetic operations Z
are not fulfilled, i.e. operation is not N
algebraic, then in the extended set this
the operation becomes algebraic.
For example: subtraction in the set of natural numbers
non-algebraic operation, and in the set of integers - algebraic. Division in the set of integers is non-algebraic, and in the set of rational numbers it is algebraic.
Set of integers(Z) includes the set of natural numbers, the number 0 and the opposite numbers of natural numbers. The set of integers can be arranged on the number line so that each integer corresponds to one and only one point on the number line. The converse is not true, any point will not always correspond to an integer.
Integers are located on the number line at the same distance from 0. The number 0 is called the neutral element. A number that is at the same distance from a given number to the left of 0 is called the opposite. The sum of two opposite numbers is 0.
Z - is linearly ordered, i.e. for any numbers A and B taken from Z, one of the following relations is true A = B, A<В, А>B. Z is a countable set. A set is called countable if it is equivalent to the set of natural numbers, i.e. it is possible to establish correspondences between the given set and the set N.
Let us show that Z is countable, i.e., to each natural number one-to-one (uniquely) there corresponds an integer. In order to establish such a correspondence, let's put each odd natural number in correspondence with a negative integer. And to each even natural number we assign a positive number. Having established such a correspondence, one can show that it will be one-to-one, which means that the set Z is countable.
Z is discrete. A set is discrete if it is ordered and between any two elements of this set there is a finite number of elements of this set.
The set of rational numbers (Q). The need to measure various quantities led to the consideration of fractional numbers. Fractions first appeared in the DR. Egypt, but were considered only as fractions of 1, i.e. only fractions of the form 1\n were considered. Fractions appeared on a geometric basis when measuring the lengths of segments. Nr. let a segment A be given, in order to measure this segment, another segment E is chosen as a unit of length and fits into the given one. if it turns out that the segment E fits an equal number of times, then the length of the segment A is expressed as a natural number. But it often turned out that the segment E was laid down an unequal number of times. Then it was divided into smaller parts and a segment E 1 was obtained, and this segment was already laid in a given segment A. Then the length of the segment A was measured by a pair of natural numbers. The first number showed how many times segment E fit in segment A. The second number showed how many times segment E 1 fit in the remainder of segment A after measuring segment E. This pair of numbers determined the fraction. An entry of the form m\n is called a fraction, where m and n are natural numbers. Two fractions are called equivalent (equivalent) if the product of the numerator of the first fraction by the denominator of the second is equal to the product of the denominator of the first fraction by the numerator of the second.
Properties of the set of rational numbers. 1). Q is linearly ordered, i.e. for any rational numbers A and B, one of the relations A=B, A>B, A<В. Рациональное число , если a*d>b*c . Let us prove that Q is linearly ordered and the relation is a relation of strict order.
Let's prove antisymmetry. From what, from what fraction. T.K. in the set of natural numbers, the relation "greater than" is antisymmetric, we can write .
Let's prove transitivity more relationship.
If , then
Since the product (bc)n=(cn)b and the relation "greater than" in the set of natural numbers is transitive → (ad)n>(dm)b | reduce by d
Since the properties of antisymmetry and transitivity are satisfied, the relation "greater than" is a relation of strict order.
2). Any rational number can be associated with a single point on the number line. The converse is not true.
3). Q is an everywhere dense set. A numerical set is called everywhere dense if it is linearly ordered and between any two of its elements there is an infinite number of elements of the given set. To prove this, we choose two rational numbers k 1 , k 2 on the real line. prove. That there are infinitely many rational numbers between them. We use the operation of finding the arithmetic mean
To 1 to 4 to 3 to 5 to 2
The number k is rational, since the operations of addition and division by 2 are defined. The process of finding the arithmetic mean is always feasible and infinite, i.e. between k and k there are infinitely many rational numbers.
4). Q is a countable set, since it is equivalent to the set of natural numbers.
3 . The difference between sets, the complement of one set to another. Difference and addition properties. set difference A and B is a set C whose elements belong to set A, but do not belong to set B. If set B is a subset of set A, then the difference between sets A and B is called addition set B to set A.
A B \ - difference A B
A \u003d (a 1, a 2, a 3 ... a k) n (A) \u003d k
B=(b 1 , b 2 , b 3 ,…b t ) n(B)=t
Let us prove that n(AUB)=k+t
AUB=(a 1 , a 2 , a 3 ,…a k , b k+1 , b k+2 ,…b k+t )
A∩B=Ø n(AUB)=k+t
n(AUB)=n(A)+n(B).
2. If the sets intersect. The number of elements of the union of two finite intersecting sets is equal to the difference between the sum of the number of these sets and the number of intersection of these sets. Proof.
A=(a 1 , a 2 , a 3 ,…a s, a s+1, a s+2…… a s+t ) n(A)=s+t
B=(a 1 , a 2 , a 3 , …a s , b s+1 , b s + 2 , b s + 3 ,…s+k ) n(B)=s+k
A∩B=(a 1 , a 2 , a 3 ,…a s ) n(A∩B)=s
AUB=(a 1 , a 2 ,…a s …a s+t , b s+1 , b s + 2 , b s + 3 …b s + k )
n(AUB=s+t+k=s+t+k+s-s=(s+t)+(s+k)-s, then
n(AUB)=n(A)+n(B)-n(A∩B);
3. The number of elements in the complement of a finite set A to a finite set B is equal to the difference in the number of these sets. Proof.
B=(b 1 , b 2 , b 3 …b k )
A \u003d (b 1, b 2, b 3, ... ... b m) m (B\A)=(b m+1 , b m+2 ,…b k ) n(B\A)=k-m Þ From preschool age, the child operates with natural numbers, either counting objects, or counting many fingers on his hands. The main concept in introducing the concept of the set of natural numbers N is the attitude , which is determined by the following Peano axioms. Axiom 1. in multitude N there is an element that does not immediately follow any element of this set, which is called the unit and is denoted by the symbol 1. Axiom 2. For each element P sets N, there is only one element ( n+1) immediately following P. Axiom 3. For each element P from N there is at most one element ( n-1), which is immediately followed by P. Axiom 4. Any subset R sets N coincides with N, if the following properties hold for it: 1) 1 is contained in R; 2) from what P contained in R, it follows that and ( n+1) contained in R. Based on Peano's axioms, we formulate the definition of the set of natural numbers. Definition.
A bunch of N, whose elements satisfy axioms 1-4, i.e. are in relation "directly follow", is called set of natural numbers and its elements are natural numbers. An extension of the set of natural numbers N is set of integers Z, which is the union of the natural numbers, the number zero, and the numbers opposite to the natural numbers. An extension of the set of integers is set of rational numbers Q, which is the union of integers and fractional numbers. The set of all numbers representable as an irreducible fraction m/n, Where m can be any integer (not excluding zero), i.e. mÎ Z, A n- natural number, i.e. nÎ N, make up the set of rational numbers .
Any rational number can be written as an infinite decimal periodic fraction, and vice versa, any infinite decimal periodic fraction is a rational number. There are numbers that cannot be represented as an irreducible fraction, i.e. do not belong to the set of rational numbers. Such numbers are set of irrational numbers I, they can be represented as an infinite decimal non-periodic fraction. For example, the length of the diagonal of a square with side 1 must be expressed by some positive number r2=1 2 +1 2 (according to the Pythagorean theorem), i.e. such that r2=2. Number r cannot be an integer, 1 2 = 1, 2 2 = 4, etc. Number r cannot be fractional: if r = m/n- irreducible fraction, where n¹1, then r 2 =m 2 /n 2 will also be an irreducible fraction, where n 2 ¹ 1; which means that m 2 / n 2 is not an integer, and therefore cannot be equal to 2. Therefore, the length of the diagonal of the square is expressed by an irrational number, it is denoted. Similarly, there is no rational number whose square is 5, 7, 10. The corresponding irrational numbers are denoted , , . The opposite numbers are also irrational, they are denoted - ,- ,- . The set of irrational numbers is infinite. For example, the number p, which expresses the ratio of the circumference of a circle to its diameter, cannot be represented as an ordinary fraction - it is an irrational number. A set whose elements are rational and irrational numbers is called set of real numbers and is marked with the letter R. Each real number corresponds to a single point on the coordinate line. Each point of the coordinate line corresponds to a single real number. The set of real numbers is also called number line. We have considered the process of expanding the concept of number from natural to real, which was associated with the needs of practice and with the needs of mathematics itself. The need to perform division led from natural numbers to the concept of fractional positive numbers; then the operation of subtraction led to the concepts of negative numbers and zero; further, the need to extract roots from positive numbers - to the concept of an irrational number. The set on which all these operations are feasible is the set of real numbers, but not all operations are feasible on this set. For example, there is no way to extract the square root of a negative number or solve the quadratic equation x 2 + x + 1 = 0. This means that there is a need to expand the set of real numbers. Let's enter a number i, such that i 2= - 1. This number will allow you to extract roots from negative numbers. So, the extension of the set of real numbers is set of complex numbers, which is denoted by the letter WITH. In detail, with the set of complex numbers, we will get acquainted later. We will use the notation: N- set of natural numbers; Z- set of integers; Q- the set of rational numbers, R- set of real numbers WITH is the set of complex numbers. Positive rational numbers. Least common multiple and greatest common divisor. divisibility signs. Set-theoretic meaning of the difference. Set-theoretic meaning of the sum. QUESTIONS FOR THE COLLOQUIUM 1. From the history of the concept of a natural number. 2. Ordinal and quantitative natural numbers. Check. 3. Set-theoretic meaning of a quantitative natural number and zero. 4. The set-theoretic meaning of the relation “less than”, “equal to” 6. Laws of addition. 8. Relationships "more by" and "less by". 9. Rules for subtracting a number from a sum and a sum from a number. 10. From the history of the emergence and development of ways to write natural numbers and zero. 11. The concept of the number system. 12. Positional and non-positional number systems. 13. Recording and names of numbers in the decimal number system. 14. Addition in decimal number system. 15. Multiplication in decimal number system 16. Ordering of the set of natural numbers. 17. Subtraction in decimal number system. 18. Division in decimal number system. 19. The set of non-negative integers. 20. Divisibility relation and its properties. 23. Prime numbers. Ways to find the greatest common divisor and least common multiple of numbers. 24. The concept of a fraction. 27. Write positive rational numbers as decimal fractions. 28. Real numbers. MODULE 4. GEOMETRIC FIGURES AND MEASUREMENTS It is known that numbers arose from the need for counting and measuring, but if natural numbers are sufficient for counting, then other numbers are needed to measure quantities. However, as a result of measuring quantities, we will consider only natural numbers. Having defined the meaning of a natural number as a measure of magnitude, we will find out what is the meaning of arithmetic operations on such numbers. This knowledge is necessary for a primary school teacher not only to justify the choice of actions when solving problems with quantities, but also to understand another approach to the interpretation of a natural number that exists in elementary mathematics. We will consider a natural number in connection with measurements of positive scalar quantities - lengths, areas, masses, time, etc., therefore, before talking about the relationship between quantities and natural numbers, let us recall some facts related to magnitude and measurement, especially since the concept of magnitude , along with the number, is basic in the initial course of mathematics. In recent years, there has been a tendency to include a significant amount of geometric material in the initial course of mathematics. But in order for the teacher to be able to acquaint students with various geometric shapes (both planes and spaces), to teach them how to correctly depict geometric shapes, he needs appropriate mathematical training. Of course, knowledge about the history of the emergence and development of geometry is needed, since the student, in the process of developing geometric ideas, goes through, in a collapsed form, the main stages in the creation of geometric science. The teacher should be familiar with the leading ideas of the geometry course, know the basic properties of geometric shapes, and be able to construct them. The material in this module will help the teacher to master this material. It, taking into account the training received by students in the school course of mathematics, presents the geometric material necessary for teaching elementary geometry to younger students. The student must be able to: Illustrate with examples from mathematics textbooks for elementary school to the definition of a natural number and operations on numbers, as a result of measuring quantities; Solve elementary construction problems with the help of a compass and a ruler in the amount determined by the content of training; Solve simple problems to prove and calculate the numerical values of geometric shapes; Draw a prism, a rectangular parallelepiped, a pyramid, a cylinder, a cone, a ball on a plane using design rules. The first extension of the concept of number, which students learn after becoming familiar with natural numbers, is the addition of zero. It happens in elementary school. First, "O" is a sign to indicate the absence of a number. Why can't you divide by zero? To divide means to find such X
, What: x-0 = a.
Two cases are possible: 1) A *
X:
dg-0 *
0. This is not possible; 2) a =
0, so we need to find hg. x-0 =
0. Such X
arbitrarily, which contradicts the requirement that each arithmetic operation be unique: There are textbooks where the basic laws of action are considered fair without the necessary justifications. In the course of mathematics in grades 5-6, the construction of a set of rational numbers takes place. It should be noted that the sequence of set extensions is not unique. Possible options: The elementary concept of a fractional number is already given in elementary school as a few fractions of a unit. In the basic school, fractions and actions on them are usually introduced by the method of expedient problems, invented by S. I. Shokhor-Trotsky, for example, when considering the following problem. Students can multiply 15 by 4 by 5, now they need to find From 15. Students can divide by 3 by finding how much one share of 3 costs kilograms, and multiply by 2 to determine how much two such shares are worth. Since it is reasonable to solve the same problem with the same arithmetic operation, they come to the conclusion that these two consecutive operations are equivalent to multiplying 15 by -. When introducing fractional numbers, it is desirable to take into account the experience of students, rely on it. With fractions, students meet in music. The most common fractions in it: two quarters, three quarters, translating into mathematical language: two fourths, three fourths. The top number indicates the number of beats per measure: two or three. The lower number indicates the duration of this beat. In our case, this is a quarter. March, polkas sound in two-quarter time. Waltz in three-quarter time. These memories will help students connect new knowledge with their experience, which is a necessary condition for achieving understanding. When studying the actions of the second stage, it is recommended to arrange various cases of multiplication by a proper fraction in order of increasing difficulty: 1) multiplication by an integer; 2) multiplying an integer by a mixed number; 3) multiplying a fraction by a mixed number; 4) multiplication by a proper fraction; 5) multiplication by a fraction in which the numerator is equal to the denominator. To show that a number, when divided by a proper fraction, increases lichivaetsya, you can consider the following situation: 6: -. Six circles were cut into four parts, of course, there were more parts than circles. To introduce complex cases, a problem is proposed to calculate the area of a rectangle. There are pros and cons to any sequence of learning fractions. If decimal fractions are introduced before ordinary fractions, then it is positive that: The negative is that for ordinary fractions all the theory of fractions must be built anew, since it is impossible to draw general conclusions from a particular case. If ordinary fractions are entered before decimals, then it should be borne in mind that: There are different ways to enter negative numbers. Thus, a problem situation close to the child's experience can be used to provide motivation. Robin Hood, fleeing his pursuers, swam up the river A km, but, being in front of a ford, he was forced to swim down the river and swam b km. Where did he end up from the beginning of his journey (at what distance from the entrance to the river)? Writing out an expression to find the unknown: x = a - b y it is necessary to consider all possible correlations between aik 1) a > k, 2) a = b; 3) but not feasible. Also negative numbers can be entered: These techniques can also be used as one of the aspects of motivation. Another aspect is the inability to perform any action, as in the task above. By introducing comparison and operations on rational numbers and properties of operations, we have obtained a number field. Its further expansion can no longer be dictated by non-performance. The expansion of the concept of number was caused by geometric considerations, namely: the absence of a one-to-one correspondence between the set of rational numbers and the set of points on the number line. For geometry, it is necessary that each point of the number line has an abscissa, i.e. so that each segment with a given unit of measure corresponds to a number that could be taken as its length. The impossibility of extracting a root from a positive number, finding the logarithm of any positive number for any positive base also leads to the need for this extension. This goal is achieved after the field of rational numbers (by adding a system of irrational numbers to it) is expanded to the set of real numbers.
