Numbers – types, concepts and operations, natural and other types of numbers.
Number is a fundamental concept in mathematics that serves to determine quantitative characteristics, numbering, comparison of objects and their parts. Various arithmetic operations are applicable to numbers: addition, subtraction, multiplication, division, exponentiation and others.
The numbers involved in the operation are called operands. Depending on the action performed, they receive different names. In general, the operation scheme can be represented as follows:<операнд1> <знак операции> <операнд2> = <результат>.
In a division operation, the first operand is called the dividend (this is the name of the number that is being divided). The second (by which they divide) is the divisor, and the result is the quotient (it shows how many times the dividend is greater than the divisor).
Types of numbers
Various numbers can be involved in the division operation. The result of division can be integer or fractional. In mathematics there are the following types of numbers:
- Natural numbers are numbers used in counting. Among them, a subset stands out prime numbers, having only two divisors: one and itself. All others except 1 are called composite and have more than two divisors (examples of prime numbers: 2, 5, 7, 11, 13, 17, 19, etc.);
- Integers are a set consisting of negative, positive numbers and zero. When dividing one integer by another, the quotient can be an integer or a real (fractional). Among them we can distinguish a subset of perfect numbers - equal to the amount all of its divisors (including 1), except itself. The ancient Greeks knew only four perfect numbers. Sequence of perfect numbers: 6, 28, 496, 8128, 33550336... So far, not a single odd perfect number is known;
- Rational - representable as a fraction a/b, where a is the numerator and b is the denominator (the quotient of such numbers is usually not calculated);
- Real (real) – containing an integer and a fractional part. The set includes rational and irrational numbers (representable as a non-periodic infinite decimal fraction). The quotient of such numbers is usually a real value.
There are several features associated with performing the arithmetic operation - division. Understanding them is important to obtain the correct result:
- You cannot divide by zero (in mathematics this operation makes no sense);
- Integer division is an operation as a result of which only the integer part is calculated (the fractional part is discarded);
- Calculating the remainder of an integer division allows you to obtain as a result the integer remaining after the operation is completed (for example, when dividing 17 by 2, the integer part is 8, the remainder is 1).
The concept of a real number: real number- (real number), any non-negative or negative number or zero. Real numbers are used to express measurements of each physical quantity.
Real, or real number arose from the need to measure geometric and physical quantities peace. In addition, for performing root extraction operations, calculating logarithms, solving algebraic equations, etc.
Natural numbers were formed with the development of counting, and rational numbers with the need to manage parts of a whole, then real numbers (real) are used to measure continuous quantities. Thus, the expansion of the stock of numbers that are considered led to the set of real numbers, which, in addition to rational numbers, consists of other elements called irrational numbers.
Set of real numbers(denoted R) are sets of rational and irrational numbers collected together.
Real numbers divided byrational And irrational.
The set of real numbers is denoted and often called real or number line. Real numbers consist of simple objects: whole And rational numbers.
A number that can be written as a ratio, wherem is an integer, and n- natural number, isrational number.
Any rational number can easily be represented as a finite fraction or an infinite periodic decimal fraction.
Example,
Infinite decimal, is a decimal fraction that has an infinite number of digits after the decimal point.
Numbers that cannot be represented in the form are irrational numbers.
Example:
Any irrational number can easily be represented as an infinite non-periodic decimal fraction.
Example,
Rational and irrational numbers create set of real numbers. All real numbers correspond to one point on the coordinate line, which is called number line.
For numerical sets the following notation is used:
- N- a bunch of natural numbers;
- Z- set of integers;
- Q- set of rational numbers;
- R- set of real numbers.
Theory of infinite decimal fractions.
A real number is defined as infinite decimal, i.e.:
±a 0 ,a 1 a 2 …a n …
where ± is one of the symbols + or −, a number sign,
a 0 is a positive integer,
a 1 ,a 2 ,…a n ,… is a sequence of decimal places, i.e. elements of a numerical set {0,1,…9}.
An infinite decimal fraction can be explained as a number that lies between rational points on the number line like:
±a 0 ,a 1 a 2 …a n And ±(a 0 ,a 1 a 2 …a n +10 −n) for all n=0,1,2,…
Comparison of real numbers as infinite decimal fractions occurs place-wise. For example, suppose we are given 2 positive numbers:
α =+a 0 ,a 1 a 2 …a n …
β =+b 0 ,b 1 b 2 …b n …
If a 0 0, That α<β ; If a 0 >b 0 That α>β . When a 0 =b 0 Let's move on to the comparison of the next category. Etc. When α≠β , which means that after a finite number of steps the first digit will be encountered n, such that a n ≠b n. If a n n, That α<β ; If a n >b n That α>β .
But it is tedious to pay attention to the fact that the number a 0 ,a 1 a 2 …a n (9)=a 0 ,a 1 a 2 …a n +10 −n . Therefore, if the record of one of the numbers being compared, starting from a certain digit, is a periodic decimal fraction with 9 in the period, then it must be replaced with an equivalent record with a zero in the period.
Arithmetic operations with infinite numbers decimals it is a continuous continuation of the corresponding operations with rational numbers. For example, the sum of real numbers α And β is a real number α+β , which satisfies the following conditions:
∀ a′,a′′,b′,b′′∈ Q(a′⩽ α ⩽ a′′)∧ (b′⩽ β ⩽ b′′)⇒ (a′+b′⩽ α + β ⩽ a′′+b′′)
The operation of multiplying infinite decimal fractions is defined similarly.
Find the points on the number circle with the given abscissa. Coordinates. Property of point coordinates. Center of the number circle. From circle to trigonometer. Find the points on the number circle. Dots with abscissa. Trigonometer. Mark a point on the number circle. Number circle on the coordinate plane. Number circle. Points with ordinate. Give the coordinate of the point. Name the line and coordinate of the point.
““Derivatives” 10th grade algebra” - Application of derivatives to study functions. The derivative is zero. Find the points. Let's summarize the information. The nature of the monotonicity of the function. Application of the derivative to the study of functions. Theoretical warm-up. Complete the statements. Select true statement. Theorem. Compare. The derivative is positive. Compare the formulations of the theorems. The function increases. Sufficient conditions for an extremum.
““Trigonometric equations” grade 10” - Values from the interval. X= tan x. Provide roots. Is the equality true? Series of roots. Equation cot t = a. Definition. Cos 4x. Find the roots of the equation. Equation tg t = a. Sin x. Does the expression make sense? Sin x =1. Never do what you don't know. Continue the sentence. Let's take a sample of the roots. Solve the equation. Ctg x = 1. Trigonometric equations. The equation.
“Algebra “Derivatives”” - Tangent equation. Origin of terms. Solve a problem. Derivative. Material point. Differentiation formulas. Mechanical meaning of derivative. Evaluation criteria. Derivative function. Tangent to the graph of a function. Definition of derivative. Equation of a tangent to the graph of a function. Algorithm for finding the derivative. An example of finding the derivative. Structure of the topic study. The point moves in a straight line.
“Shortest path” - A path in a digraph. An example of two different graphs. Directed graphs. Examples of directed graphs. Reachability. The shortest path from vertex A to vertex D. Description of the algorithm. Advantages of a hierarchical list. Weighted graphs. Path in the graph. ProGraph program. Adjacent vertices and edges. Top degree. Adjacency matrix. Path length in a weighted graph. An example of an adjacency matrix. Finding the shortest path.
"The History of Trigonometry" - Jacob Bernoulli. Operating technique with trigonometric functions. The doctrine of measuring polyhedra. Leonard Euler. The development of trigonometry from the 16th century to the present day. The student has to meet trigonometry three times. Until now, trigonometry has been formed and developed. Construction common system trigonometric and related knowledge. Time passes, and trigonometry returns to schoolchildren.
Understanding numbers, especially natural numbers, is one of the oldest math "skills." Many civilizations, even modern ones, have attributed certain mystical properties to numbers due to their enormous importance in describing nature. Although modern science and mathematics do not confirm these “magical” properties, the importance of number theory is undeniable.
Historically, a variety of natural numbers appeared first, then fairly quickly fractions and positive irrational numbers were added to them. Zero and negative numbers were introduced after these subsets of the set of real numbers. The last set, the set of complex numbers, appeared only with the development of modern science.
In modern mathematics, numbers are not entered into historical order, although quite close to it.
Natural numbers $\mathbb(N)$
The set of natural numbers is often denoted as $\mathbb(N)=\lbrace 1,2,3,4... \rbrace $, and is often padded with zero to denote $\mathbb(N)_0$.
$\mathbb(N)$ defines the operations of addition (+) and multiplication ($\cdot$) with the following properties for any $a,b,c\in \mathbb(N)$:
1. $a+b\in \mathbb(N)$, $a\cdot b \in \mathbb(N)$ the set $\mathbb(N)$ is closed under the operations of addition and multiplication
2. $a+b=b+a$, $a\cdot b=b\cdot a$ commutativity
3. $(a+b)+c=a+(b+c)$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ associativity
4. $a\cdot (b+c)=a\cdot b+a\cdot c$ distributivity
5. $a\cdot 1=a$ is a neutral element for multiplication
Since the set $\mathbb(N)$ contains a neutral element for multiplication but not for addition, adding a zero to this set ensures that it includes a neutral element for addition.
In addition to these two operations, the “less than” relations ($
1. $a b$ trichotomy
2. if $a\leq b$ and $b\leq a$, then $a=b$ antisymmetry
3. if $a\leq b$ and $b\leq c$, then $a\leq c$ is transitive
4. if $a\leq b$ then $a+c\leq b+c$
5. if $a\leq b$ then $a\cdot c\leq b\cdot c$
Integers $\mathbb(Z)$
Examples of integers:
$1, -20, -100, 30, -40, 120...$
Solving the equation $a+x=b$, where $a$ and $b$ are known natural numbers, and $x$ is an unknown natural number, requires the introduction of a new operation - subtraction(-). If there is a natural number $x$ satisfying this equation, then $x=b-a$. However, this particular equation does not necessarily have a solution on the set $\mathbb(N)$, so practical considerations require expanding the set of natural numbers to include solutions to such an equation. This leads to the introduction of a set of integers: $\mathbb(Z)=\lbrace 0,1,-1,2,-2,3,-3...\rbrace$.
Since $\mathbb(N)\subset \mathbb(Z)$, it is logical to assume that the previously introduced operations $+$ and $\cdot$ and the relations $ 1. $0+a=a+0=a$ there is a neutral element for addition
2. $a+(-a)=(-a)+a=0$ there is an opposite number $-a$ for $a$
Property 5.:
5. if $0\leq a$ and $0\leq b$, then $0\leq a\cdot b$
The set $\mathbb(Z)$ is also closed under the subtraction operation, that is, $(\forall a,b\in \mathbb(Z))(a-b\in \mathbb(Z))$.
Rational numbers $\mathbb(Q)$
Examples of rational numbers:
$\frac(1)(2), \frac(4)(7), -\frac(5)(8), \frac(10)(20)...$
Now consider equations of the form $a\cdot x=b$, where $a$ and $b$ are known integers, and $x$ is an unknown. For the solution to be possible, it is necessary to introduce the division operation ($:$), and the solution takes the form $x=b:a$, that is, $x=\frac(b)(a)$. Again the problem arises that $x$ does not always belong to $\mathbb(Z)$, so the set of integers needs to be expanded. This introduces the set of rational numbers $\mathbb(Q)$ with elements $\frac(p)(q)$, where $p\in \mathbb(Z)$ and $q\in \mathbb(N)$. The set $\mathbb(Z)$ is a subset in which each element $q=1$, therefore $\mathbb(Z)\subset \mathbb(Q)$ and the operations of addition and multiplication extend to this set according to the following rules, which preserve all the above properties on the set $\mathbb(Q)$:
$\frac(p_1)(q_1)+\frac(p_2)(q_2)=\frac(p_1\cdot q_2+p_2\cdot q_1)(q_1\cdot q_2)$
$\frac(p-1)(q_1)\cdot \frac(p_2)(q_2)=\frac(p_1\cdot p_2)(q_1\cdot q_2)$
The division is introduced as follows:
$\frac(p_1)(q_1):\frac(p_2)(q_2)=\frac(p_1)(q_1)\cdot \frac(q_2)(p_2)$
On the set $\mathbb(Q)$, the equation $a\cdot x=b$ has a unique solution for each $a\neq 0$ (division by zero is undefined). This means that there is an inverse element $\frac(1)(a)$ or $a^(-1)$:
$(\forall a\in \mathbb(Q)\setminus\lbrace 0\rbrace)(\exists \frac(1)(a))(a\cdot \frac(1)(a)=\frac(1) (a)\cdot a=a)$
The order of the set $\mathbb(Q)$ can be expanded as follows:
$\frac(p_1)(q_1)
The set $\mathbb(Q)$ has one important property: between any two rational numbers there are infinitely many other rational numbers, therefore, there are no two adjacent rational numbers, unlike the sets of natural numbers and integers.
Irrational numbers $\mathbb(I)$
Examples of irrational numbers:
$\sqrt(2) \approx 1.41422135...$
$\pi\approx 3.1415926535...$
Since between any two rational numbers there are infinitely many other rational numbers, it is easy to erroneously conclude that the set of rational numbers is so dense that there is no need to expand it further. Even Pythagoras made such a mistake in his time. However, his contemporaries already refuted this conclusion when studying solutions to the equation $x\cdot x=2$ ($x^2=2$) on the set of rational numbers. To solve such an equation, it is necessary to introduce the concept of a square root, and then the solution to this equation has the form $x=\sqrt(2)$. An equation like $x^2=a$, where $a$ is a known rational number and $x$ is an unknown one, does not always have a solution on the set of rational numbers, and again the need arises to expand the set. A set of irrational numbers arises, and numbers such as $\sqrt(2)$, $\sqrt(3)$, $\pi$... belong to this set.
Real numbers $\mathbb(R)$
The union of the sets of rational and irrational numbers is the set of real numbers. Since $\mathbb(Q)\subset \mathbb(R)$, it is again logical to assume that the introduced arithmetic operations and relations retain their properties on the new set. The formal proof of this is very difficult, so the above-mentioned properties of arithmetic operations and relations on the set of real numbers are introduced as axioms. In algebra, such an object is called a field, so the set of real numbers is said to be an ordered field.
In order for the definition of the set of real numbers to be complete, it is necessary to introduce an additional axiom that distinguishes the sets $\mathbb(Q)$ and $\mathbb(R)$. Suppose that $S$ is a non-empty subset of the set of real numbers. An element $b\in \mathbb(R)$ is called the upper bound of a set $S$ if $\forall x\in S$ holds $x\leq b$. Then we say that the set $S$ is bounded above. The smallest upper bound of the set $S$ is called the supremum and is denoted $\sup S$. The concepts of lower bound, set bounded below, and infinum $\inf S$ are introduced similarly. Now the missing axiom is formulated as follows:
Any non-empty and upper-bounded subset of the set of real numbers has a supremum.
It can also be proven that the field of real numbers defined in the above way is unique.
Complex numbers$\mathbb(C)$
Examples of complex numbers:
$(1, 2), (4, 5), (-9, 7), (-3, -20), (5, 19),...$
$1 + 5i, 2 - 4i, -7 + 6i...$ where $i = \sqrt(-1)$ or $i^2 = -1$
The set of complex numbers represents all ordered pairs of real numbers, that is, $\mathbb(C)=\mathbb(R)^2=\mathbb(R)\times \mathbb(R)$, on which the operations of addition and multiplication are defined as follows way:
$(a,b)+(c,d)=(a+b,c+d)$
$(a,b)\cdot (c,d)=(ac-bd,ad+bc)$
There are several forms of writing complex numbers, of which the most common is $z=a+ib$, where $(a,b)$ is a pair of real numbers, and the number $i=(0,1)$ is called the imaginary unit.
It is easy to show that $i^2=-1$. Extending the set $\mathbb(R)$ to the set $\mathbb(C)$ allows us to define Square root of negative numbers, which was the reason for the introduction of the set of complex numbers. It is also easy to show that a subset of the set $\mathbb(C)$, given by $\mathbb(C)_0=\lbrace (a,0)|a\in \mathbb(R)\rbrace$, satisfies all the axioms for real numbers, therefore $\mathbb(C)_0=\mathbb(R)$, or $R\subset\mathbb(C)$.
The algebraic structure of the set $\mathbb(C)$ with respect to the operations of addition and multiplication has the following properties:
1. commutativity of addition and multiplication
2. associativity of addition and multiplication
3. $0+i0$ - neutral element for addition
4. $1+i0$ - neutral element for multiplication
5. Multiplication is distributive with respect to addition
6. There is a single inverse for both addition and multiplication.
Number is an abstraction used to quantify objects. Numbers arose in primitive society in connection with the need of people to count objects. Over time, as science developed, number turned into the most important mathematical concept.
To solve problems and prove various theorems, you need to understand what types of numbers there are. Basic types of numbers include: natural numbers, integers, rational numbers, real numbers.
Integers- these are numbers obtained by natural counting of objects, or rather by numbering them (“first”, “second”, “third”...). The set of natural numbers is denoted by a Latin letter N (you can remember based on English word natural). It can be said that N ={1,2,3,....}
Whole numbers– these are numbers from the set (0, 1, -1, 2, -2, ....). This set consists of three parts - natural numbers, negative integers (the opposite of natural numbers) and the number 0 (zero). Integers are denoted by a Latin letter Z . It can be said that Z ={1,2,3,....}.
Rational numbers are numbers represented as a fraction, where m is an integer and n is a natural number. The Latin letter is used to denote rational numbers Q . All natural numbers and integers are rational.
Real numbers are numbers that are used to measure continuous quantities. The set of real numbers is denoted by the Latin letter R. Real numbers include rational numbers and irrational numbers. Irrational numbers are numbers that are obtained as a result of performing various operations with rational numbers (for example, taking roots, calculating logarithms), but are not rational.
1. Number systems.
A number system is a way of naming and writing numbers. Depending on the method of representing numbers, they are divided into positional - decimal and non-positional - Roman.
PCs use 2-digit, 8-digit and 16-digit number systems.
Differences: the recording of a number in the 16th number system is much shorter compared to another recording, i.e. requires less bit capacity.
In a positional number system, each digit retains its constant value regardless of its position in the number. In a positional number system, each digit determines not only its meaning, but also depends on the position it occupies in the number. Each number system is characterized by a base. The base is the number of different digits that are used to write numbers in a given number system. The base shows how many times the value of the same digit changes when moving to an adjacent position. The computer uses a 2-number system. The base of the system can be any number. Arithmetic operations on numbers in any position are performed according to rules similar to the 10 number system. For 2 number system used binary arithmetic, which is implemented in a computer to perform arithmetic calculations.
Addition of binary numbers:0+0=1;0+1=1;1+0=1;1+1=10
Subtraction:0-0=0;1-0=1;1-1=0;10-1=1
Multiplication:0*0=0;0*1=0;1*0=0;1*1=1
The computer widely uses the 8 number system and the 16 number system. They are used to shorten binary numbers.
2. The concept of set.
The concept of “set” is a fundamental concept in mathematics and has no definition. The nature of the generation of any set is diverse, in particular, surrounding objects, Live nature and etc.
Definition 1: The objects from which a set is formed are called elements of this set. To denote a set, capital letters of the Latin alphabet are used: for example, X, Y, Z, and its elements are written in curly brackets, separated by commas. lowercase letters, for example: (x,y,z).
An example of notation for a set and its elements:
X = (x 1, x 2,…, x n) – a set consisting of n elements. If the element x belongs to the set X, then it should be written: xÎX, otherwise the element x does not belong to the set X, which is written: xÏX. Elements of an abstract set can be, for example, numbers, functions, letters, shapes, etc. In mathematics, in any section, the concept of set is used. In particular, we can give some specific sets of real numbers. The set of real numbers x satisfying the inequalities:
· a ≤ x ≤ b is called segment and is denoted by ;
a ≤ x< b или а < x ≤ b называется half-segment and is denoted by: ;
· A< x < b называется interval and is denoted by (a,b).
Definition 2: A set that has a finite number of elements is called finite. Example. X = (x 1 , x 2 , x 3 ).
Definition 3: The set is called endless, if it consists of an infinite number of elements. For example, the set of all real numbers is infinite. Example entry. X = (x 1, x 2, ...).
Definition 4: A set that does not have a single element is called an empty set and is denoted by the symbol Æ.
A characteristic of a set is the concept of power. Power is the number of its elements. The set Y=(y 1 , y 2 ,...) has the same cardinality as the set X=(x 1 , x 2 ,...) if there is a one-to-one correspondence y= f(x) between the elements of these sets. Such sets have the same cardinality or are of equal cardinality. An empty set has zero cardinality.
3. Methods for specifying sets.
It is believed that a set is defined by its elements, i.e. the set is given, if we can say about any object: it belongs to this set or does not belong. You can specify a set in the following ways:
1) If the set is finite, then it can be defined by listing all its elements. So, if the set A consists of elements 2, 5, 7, 12 , then they write A = (2, 5, 7, 12). Number of elements of the set A equals 4 , they write n(A) = 4.
But if the set is infinite, then its elements cannot be enumerated. It is difficult to define a set by enumeration and a finite set with a large number of elements. In such cases, another method of specifying the set is used.
2) A set can be specified by indicating the characteristic property of its elements. Characteristic property- This is a property that every element belonging to a set has, and not a single element that does not belong to it. Consider, for example, a set X of two-digit numbers: the property that each element of this set has is “being a two-digit number.” This characteristic property makes it possible to decide whether an object belongs to the set X or does not belong. For example, the number 45 is contained in this set, because it is two-digit, and the number 4 does not belong to the set X, because it is unambiguous and not two-valued. It happens that the same set can be defined by indicating different characteristic properties of its elements. For example, a set of squares can be defined as a set of rectangles with equal sides and as a set of rhombuses with right angles.
In cases where the characteristic property of the elements of a set can be represented in symbolic form, a corresponding notation is possible. If the set IN consists of all natural numbers less than 10, then they write B = (x N | x<10}.
The second method is more general and allows you to specify both finite and infinite sets.
4. Numerical sets.
Numerical - a set whose elements are numbers. Numerical sets are specified on the axis of real numbers R. On this axis, the scale is chosen and the origin and direction are indicated. The most common number sets:
· - set of natural numbers;
· - set of integers;
· - set of rational or fractional numbers;
· - set of real numbers.
5. Power of the set. Give examples of finite and infinite sets.
Sets are called equally powerful or equivalent if there is a one-to-one or one-to-one correspondence between them, that is, a pairwise correspondence. when each element of one set is associated with a single element of another set and vice versa, while different elements of one set are associated with different elements of another.
For example, let's take a group of thirty students and issue exam tickets, one ticket to each student from a stack containing thirty tickets, such a pairwise correspondence of 30 students and 30 tickets will be one-to-one.
Two sets of equal cardinality with the same third set are of equal cardinality. If the sets M and N are of equal cardinality, then the sets of all subsets of each of these sets M and N are also of equal cardinality.
A subset of a given set is a set such that each element of it is an element of the given set. So the set of cars and the set of trucks will be subsets of the set of cars.
The power of the set of real numbers is called the power of the continuum and is denoted by the letter “alef” א . The smallest infinite domain is the cardinality of the set of natural numbers. The cardinality of the set of all natural numbers is usually denoted by (alef-zero).
Powers are often called cardinal numbers. This concept was introduced by the German mathematician G. Cantor. If sets are denoted by symbolic letters M, N, then cardinal numbers are denoted by m, n. G. Cantor proved that the set of all subsets of a given set M has a cardinality greater than the set M itself.
A set equal to the set of all natural numbers is called a countable set.
6. Subsets of the specified set.
If we select several elements from our set and group them separately, then this will be a subset of our set. There are many combinations from which a subset can be obtained; the number of combinations only depends on the number of elements in the original set.
Let us have two sets A and B. If each element of set B is an element of set A, then set B is called a subset of A. Denoted by: B ⊂ A. Example.
How many subsets of the set A=1;2;3 are there?
Solution. Subsets consisting of elements of our set. Then we have 4 options for the number of elements in the subset:
A subset can consist of 1 element, 2, 3 elements and can be empty. Let's write down our elements sequentially.
Subset of 1 element: 1,2,3
Subset of 2 elements: 1,2,1,3,2,3.
Subset of 3 elements: 1;2;3
Let's not forget that the empty set is also a subset of our set. Then we find that we have 3+3+1+1=8 subsets.
7. Operations on sets.
Certain operations can be performed on sets, similar in some respects to operations on real numbers in algebra. Therefore, we can talk about set algebra.
Association(connection) of sets A And IN is a set (symbolically it is denoted by ), consisting of all those elements that belong to at least one of the sets A or IN. In form from X the union of sets is written as follows
The entry reads: “unification A And IN" or " A, combined with IN».
Set operations are visually represented graphically using Euler circles (sometimes the term “Venn-Euler diagrams” is used). If all elements of the set A will be concentrated within the circle A, and the elements of the set IN- within a circle IN, the operation of unification using Euler circles can be represented in the following form
Example 1. Union of many A= (0, 2, 4, 6, 8) even digits and sets IN= (1, 3, 5, 7, 9) odd digits is the set = =(0, 1, 2, 3, 4, 5, 6, 7, 8, 9) of all digits of the decimal number system.
8. Graphic representation of sets. Euler-Venn diagrams.
Euler-Venn diagrams are geometric representations of sets. 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):
Cartesian (or direct) product of setsA And B such a resulting set of pairs of the form ( x,y) constructed in such a way that the first element from the set A, and the second element of the pair is from the set B. Common designation:
A× B={(x,y)|x∈A,y∈B}
Products of three or more sets can be constructed as follows:
A× B× C={(x,y,z)|x∈A,y∈B,z∈C}
Products of the form A× A,A× A× A,A× A× A× A etc. It is customary to write it as a degree: A 2 ,A 3 ,A 4 (the base of the degree is the multiplier set, the exponent is the number of products). They read such an entry as a “Cartesian square” (cube, etc.). There are other readings for the main sets. For example, R n It is customary to read as “er nnoe”.
Properties
Let's consider several properties of the Cartesian product:
1. If A,B are finite sets, then A× B- final. And vice versa, if one of the factor sets is infinite, then the result of their product is an infinite set.
2. The number of elements in a Cartesian product is equal to the product of the numbers of elements of the factor sets (if they are finite, of course): | A× B|=|A|⋅|B| .
3. A np ≠(A n) p- in the first case, it is advisable to consider the result of the Cartesian product as a matrix of dimensions 1× n.p., in the second - as a matrix of sizes n× p .
4. The commutative law is not satisfied, because pairs of elements of the result of a Cartesian product are ordered: A× B≠B× A .
5. The associative law is not fulfilled: ( A× B)× C≠A×( B× C) .
6. There is distributivity with respect to basic operations on sets: ( A∗B)× C=(A× C)∗(B× C),∗∈{∩,∪,∖}
10. The concept of utterance. Elementary and compound statements.
Statement is a statement or declarative sentence that can be said to be true (I-1) or false (F-0), but not both.
For example, “It’s raining today,” “Ivanov completed laboratory work No. 2 in physics.”
If we have several initial statements, then from them, using logical unions or particles we can form new statements, the truth value of which depends only on the truth values of the original statements and on the specific conjunctions and particles that participate in the construction of the new statement. The words and expressions “and”, “or”, “not”, “if... then”, “therefore”, “then and only then” are examples of such conjunctions. The original statements are called simple , and new statements constructed from them with the help of certain logical conjunctions - composite . Of course, the word “simple” has nothing to do with the essence or structure of the original statements, which themselves can be quite complex. In this context, the word “simple” is synonymous with the word “original”. What matters is that the truth values of simple statements are assumed to be known or given; in any case, they are not discussed in any way.
Although a statement like “Today is not Thursday” is not composed of two different simple statements, for uniformity of construction it is also considered as a compound, since its truth value is determined by the truth value of the other statement “Today is Thursday.”
Example 2. The following statements are considered as compounds:
I read Moskovsky Komsomolets and I read Kommersant.
If he said it, then it's true.
The sun is not a star.
If it is sunny and the temperature exceeds 25 0, I will arrive by train or car
Simple statements included in compounds can themselves be completely arbitrary. In particular, they themselves can be composite. The basic types of compound statements described below are defined independently of the simple statements that form them.
11. Operations on statements.
1. Negation operation.
By negating the statement A ( reads "not A", "it is not true that A"), which is true when A false and false when A– true.
Statements that deny each other A And are called opposite.
2. Conjunction operation.
Conjunction statements A And IN is called a statement denoted by A B(reads " A And IN"), the true values of which are determined if and only if both statements A And IN are true.
The conjunction of statements is called a logical product and is often denoted AB.
Let a statement be given A- “in March the air temperature is from 0 C to + 7 C" and saying IN- “It’s raining in Vitebsk.” Then A B will be as follows: “in March the air temperature is from 0 C to + 7 C and it’s raining in Vitebsk.” This conjunction will be true if there are statements A And IN true. If it turns out that the temperature was less 0 C or there was no rain in Vitebsk, then A B will be false.
3 . Disjunction operation.
Disjunction statements A And IN called a statement A B (A or IN), which is true if and only if at least one of the statements is true and false - when both statements are false.
The disjunction of statements is also called a logical sum A+B.
The statement " 4<5 or 4=5 " is true. Since the statement " 4<5 " is true, and the statement " 4=5 » – false, then A B represents the true statement " 4 5 ».
4 . Operation of implication.
By implication statements A And IN called a statement A B("If A, That IN", "from A should IN"), whose value is false if and only if A true, but IN false.
In implication A B statement A called basis, or premise, and the statement IN – consequence, or conclusion.
12. Tables of truth of statements.
A truth table is a table that establishes a correspondence between all possible sets of logical variables included in a logical function and the values of the function.
Truth tables are used for:
Calculating the truth of complex statements;
Establishing the equivalence of statements;
Definitions of tautologies.
