Some of the readers familiar with the character of the former teaching of logic in school may cast doubt on the expediency of entertaining logic. However, the reader will probably agree to think consistently, to judge the evidence, to refute the wrong conclusions should be able to be able to disprove any: a physicist and a poet, a tractor driver and a chemist. Especially in our time, constantly bringing many extraordinary and amazing discoveries and inventions in a variety of areas: in geography, politics, public life.

Automatic fuel sorter.
On a warehouse having two rooms for storing large quantities of two types of fuel - coal and coke, each separately, trucks come, each every time with one of these types of fuel. The mechanism that opens the mine is made a requirement to open a mine in the room for coal, if a truck comes with this fuel, and a coke room for coke, if a truck comes with a coke. To ensure good fuel sorting, an additional requirement was presented: every time the warehouse is admired only one truck and only one mine opens.

It is asked whether this mechanism also has the following property: if a truck with coal is not entered into the warehouse room, then the coal mine will not open, and if the truck did not enter the coke, then the coke mine will not appear.

Note. This task can be solved without means of logic of statements, simple reasoning. More difficult, and perhaps, and specifically impracticable will be the decision in the case when the number of fuels exceeds two and when several trucks can simultaneously enter the warehouse. Let the reader trial to solve this problem also for three types of fuel.

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• Mathematics and design, class 1, training manual for general education organizations, Volkova S.I., 2016
• Mathematics, oral exercises, grade 1, Tutorial for general education organizations, Volkova S.I., 2016
• Course of lectures on the theory and technology of learning mathematics in primary classes, part 2, Rumull V.P., 2019

The following textbooks and books:

• Mathematics, algebra and start of mathematical analysis, grade 11, Vilenkin N.Ya., Ivashev-Musatov O.S., Schwarzburg S.I., 2014

Introduction

Logic is the God of thinking.

L. Feichtvanger

The ability to correctly argue is necessary in any field of human activity: science and technology, justice and diplomacy, planning of the economy and military business. And this skill dates back to the oldest times, logic, i.e. The science of what form of reasoning is correct, there was only a little more than two thousand years ago. It was developed in the VI century. BC. In the works of the great ancient Greek philosopher Aristotle, his students and followers.

At some point, mathematics asked the question: "What, in fact, is mathematics, mathematical activity?" A simple answer lies in the fact that mathematics prove theorems, that is, they find out some of the truths about the real world and the "ideal mathematical world." An attempt to answer the question of what is a mathematical theorem, mathematical truth and what mathematical statement is true or provable, this is the network source point of mathematical logic. We must learn how to analyze, compare, allocate, to generalize and systematize, prove and refute, determine and explain the concepts, to put and solve problems. Mastering these methods and means the ability to think. In science, it falls through reasoning to derive a variety of formulas, numerical patterns, rules, to prove theorems. For example, in 1781, the planet Uranus was opened. Observations showed that the movement of this planet differs from the theoretically calculated movement. French scientist Leerier (1811-1877), logically arguing and fulfilling quite complex calculations, determined the influence on Uranus another planet and indicated the place of its location. In 1846, Astronomer Galle confirmed the existence of the planet, which was named Neptune. At the same time, they used the logic of mathematical reasoning and calculations.

The second source point of our consideration is to find out what means that the mathematical function is calculated and can be calculated using a certain algorithm, the formal rule, exactly described procedure. These two initial productions have a lot in common, they naturally unite under the common name "Mathematical Logic", where under mathematical logic it is understood primarily the logic of mathematical reasoning and mathematical actions.

I chose this particular topic, because the possession of elements of mathematical logic will help me in my future economic profession. After all, the marketer analyzes trendsmarket prices, turnover and marketing methods, collects data on competing organizations,gives recommendations. To do this, use knowledge of logic.

Purpose of work: Examine and use the possibilities of mathematical logic in solving problems in various fields and human activities.

Tasks:

1. Analyze the literature on the essence and the emergence of mathematical logic.

2. To study the elements of mathematical logic.

3. Pick up and solve problems with elements of mathematical logic.

Methods: Analysis of literature, concepts, analogy method in solving problems, self-surveillance.

1. From the history of mathematical logic

Mathematical logic is closely connected with logic and is obliged to her appearance. The basics of logic, science on laws and forms of human thinking were laid down the greatest ancient Greek philosopher Aristotle (384-322 BC), which in his treatises thoroughly investigated the terminology of logic, dismantled in detail the theory of conclusions and evidence, described a number of logical operations, Formulated the basic laws of thinking, including the laws of contradiction and the exceptions of the third. The contribution of Aristotle to logic is very large, no wonder its other name is Aristotelian logic. An Aristotle himself noticed that between the science and mathematics created by him (then she was called arithmetic) a lot in common. He tried to connect two of these sciences, namely, reflement, or, rather, conclusion, to calculate on the basis of the initial positions. In one of his treatises, Aristotle came close to one of the sections of mathematical logic - the theory of evidence.

In the future, many philosophers and mathematicians have developed certain provisions of logic and sometimes even outlined the contours of the current estimation of statements, but at the second half of the XVII century, the outstanding German scientist Gotfride Wilhelm Leibniz (1646 - 1716), which indicated the paths for translation of logic approached the creation of mathematical logic "From the verbal kingdom, complete uncertainties, in the kingdom of mathematics, where the relationship between objects or statements is determined by quite accurate." Leibniz even hoped even that in the future the philosophers, instead of the fruitlessly argue, will take the paper and calculate which of them is right. At the same time, in their works, the leibies affected the binary number system. It should be noted that the idea of \u200b\u200busing two characters for encoding information is very old. Australian aborigines were considered twins, some tribes of the builders of New Guinea and South America also used the binary account system. Some African tribes transmit messages using drums in the form of combinations of ringing and deaf blows. An example of a two-grained coding is familiar - Alphabet of Morse, where the letters of the alphabet are represented by certain combinations of points and dash. After the leibher of the study, many outstanding scientists conducted in this area, but the real success came here to the English Mathematics-self-taught George Bul (1815-1864), his dedication did not know the borders.

The material situation of George's parents (whose father was a shoe master) allowed him to end only primary school for the poor. After some time, Boule, changing several professions, opened a small school, where he taught himself. He paid a lot of time to self-education and soon carried away the ideas of symbolic logic. In 1847, Bhul published an article "Mathematical analysis of logic, or experience of calculating deductive conclusions", and in 1854 there was the main work "The study of the laws of thinking, which are based on mathematical theories of logic and probabilities." Boulev invented a kind of algebra - a system of designations and rules applicable to all sorts of objects, from numbers and letters to proposals. Using this system, he could encode statements (statements, the truth or the falsity of which was required to prove) with the help of symbols of his language, and then manipulate them, just as in mathematics manipulate numbers. The main operations of the Boolean algebra are conjunction (s), disjunction (or) and denial (not). After some time, it became clear that the Bul system was well suited to describe electrical switchals. Current in the chain can either leak, or absent, just as the statement can be either true or false. And a few more decades later, already in the 20th century, scientists have combined the mathematical apparatus created by George Bull with a binary system, thereby laying the foundations for the development of a digital electronic computer. The individual provisions of the work of the bulb to one way or another were addressed before, and after it, other mathematicians and logic. However, today in this area, it is the works of George Bul that the Mathematical Classic is expected, and he himself is considered to be the founder of mathematical logic and the more important sections - the Logic algebras (Boolean algebra) and statement algebras.

Russian scientists P.S. contributed great contribution to the development of logic Poretsky (1846-1907), I.I. Zhegalkin (1869-1947).

In the XX century, a huge role in the development of mathematical logic was played

D. Hilbert (1862-1943), who proposed a program for the formalization of mathematics associated with the development of the foundations of mathematics itself. Finally, in recent decades of the XX century, the rapid development of mathematical logic was due to the development of the theory of algorithms and algorithmic languages, the theory of automata, theory of graphs (S.K. Klini, A. Church, A.A Markov, P.S. Novikov and many others) .

In the middle of the 20th century, the development of computing equipment led to the emergence of logical elements, logical blocks and devices of computing equipment, which was associated with the additional development of such logic areas, as problems of logical synthesis, logical design and logical modeling of logical devices and computing equipment. In the 80s of the 20th century, research began in the field of artificial intelligence on the basis of languages \u200b\u200band logical programming systems. The creation of expert systems using and the development of automatic proof by theorems, as well as methods of evidentiary programming to verify algorithms and computer programs. In the 80s, changes began in education. The emergence of personal computers in secondary schools led to the creation of computer science textbooks with the study of elements of mathematical logic to explain the logical principles of the logic schemes and devices of computing equipment, as well as the principles of logical programming for computers of the fifth generation and the development of computer science textbooks with the study of predicate based on the basis of knowledge .

1. Basics of set theory

The concept of multiple is one of those fundamental concepts of mathematics, which is difficult to give an accurate definition using elementary concepts. Therefore, we restrict ourselves to a descriptive explanation of the concept of set.

Set a combination of certain well-distinguished objects considered as a whole is called. The creator of the theory of sets Georg Kantor gave the following definition of the set - "Many there is a lot that we think as a whole."

Separate objects of which consists of many, are calledelements of set.

Sets are made by large letters of the Latin alphabet, and the elements of these sets are small letters of the Latin alphabet. Sets are recorded in curly brackets ().

It is customary to use the following notation:

a.∈ X - "The element A belongs to the set x";

a.∉ X - "Element A does not belong to the set X";

∀ - quantitor of arbitrariness, generality denoting "any", "no matter how", "for all";

∃ - quantitor of existence:∃ y.∈ B - "There is a (found) element Y from the set B";

∃! - Quantitor of existence and uniqueness:∃ ! B.∈ C - "There is a single element B from a set C";

: - "Such that; possessing property ";

→ - the symbol of the investigation means "entails";

⇔ - quantitor equivalence, equivalence - "Then and only then."

Sets arefinite and endless . Sets are calledend if the number of its elements is of course, i.e. If there is a natural number n, which is the number of elements of the set. A \u003d (a 1, a 2, a 3, ..., a n ). Many calledinfinite if it contains an infinite number of items. B \u003d (b 1, b 2, b 3 , ...). For example, a set of letters of the Russian alphabet - the final set. Many natural numbers are an infinite set.

The number of elements in the final set M is called the power of the set M and is indicated | M |.Empty the set is a set that does not contain a single element -∅ . Two sets are calledequal if they consist of the same elements, i.e. represent the same set. Sets are not equal to X ≠ Y, if there are elements that do not belong to Y, or in y there are elements that do not belong to H. The symbol of sets of sets has properties:

X \u003d x; - reflexiveness

if x \u003d y, y \u003d x - symmetry

if x \u003d y, y \u003d z, then x \u003d z - transitivity.

According to such a determination of sets of sets, we naturally get that all empty sets are equal to each other or that the same thing is that there is only one empty set.

Subsets. The inclusion ratio.

The set x is a subset of the set Y, if any element of the set x∈ and the set y. denotes x⊆ Y.

If it is necessary to emphasize that y contains other elements, except for elements from x, then use the strict inclusion symbol⊂ : X. ⊂ Y. Communication between characters⊂ and ⊆ Gives an expression:

X. ⊂ Y. ⇔ X. ⊆ Y and x ≠ y

We note some properties of subsets arising from the definition:

X.⊆ X (reflexiveness);

→ X.⊆ Z (transitivity);

The initial set A with respect to its subsets is calledfull a set and denotes I.

Any subset of A. I. sets and called own set A.

Sets consisting of all subsets of this set x and empty set∅ , called Bulean X and denotes β (x). Power Buleana | β (x) | \u003d 2 n.

Counting set - this is such a set A, all the elements of which can be amiced in the sequence (MB endless) 1, a 2, and 3, ..., and n , ... So that, with each element, you get one N number N and each natural number n would be given one and only one element of our set.

The set equivalent to the set of natural numbers is called a countable set.

Example. Many squares of integers 1, 4, 9, ..., n 2 it is only a subset of the set of natural numbers N. The set is countable, as it is provided in mutually unambiguous correspondences with a natural number by attributing to each element of the number of the number of natural rows, the square of which it is.

There are 2 main methods for setting sets.

listing (x \u003d (a, b), y \u003d (1), z \u003d (1,2, ..., 8), m \u003d (m 1 , M. 2 , M. 3 , .., m n. });

description - indicates the characteristic properties that all elements of the set are possessed.

The set is fully determined by their elements.

By listing, you can define only final sets (for example, many months per year). Infinite sets can be set only a description of the properties of its elements (for example, a plurality of rational numbers can be specified by the description Q \u003d (N / M, M, N∈ Z, m ≠ 0).

Methods for setting a set description:

but) setting the generating procedurewith the set of multiple (sets), which runs the parameter (parameters) of this procedure is recursive, inductive.

X \u003d (x: x 1 \u003d 1, x 2 \u003d 1, x k + 2 \u003d x k + x k + 1 , k \u003d 1,2,3, ...) - MN-in Fibonichchi numbers.

(MN-in elements x, such as x 1 \u003d 1, x 2 \u003d 1 and arbitrary x K + 1. (with K \u003d 1,2,3, ...) calculated by the formula x k + 2 \u003d x k + x k + 1) or x \u003d)