Some readers familiar with the nature of the previous teaching of logic at school may question the appropriateness of entertaining logic. However, the reader will probably agree that everyone should be able to think consistently, judge with evidence, refute incorrect conclusions: 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 various fields: in geography, politics, in public life.

Automatic fuel sorter.
The warehouse, which has two rooms for storing large quantities of two types of fuel - coal and coke, each separately, receives trucks, each time with one of these fuels. The mine opening mechanism is required to open the mine to the coal room if a truck with this fuel arrives, and the mine to the coke room if a coke truck arrives. To ensure good sorting of the fuel, an additional requirement was made: each time only one truck is allowed into the warehouse and only one shaft is opened.

The question is whether this mechanism also has the following property: if the truck with coal has not entered the warehouse, then the coal mine will not open, and if the truck with coke has not entered, the coke mine will not open.

Note. This problem can be solved without the means of the logic of statements, by simple reasoning. A more difficult, and possibly speculatively impracticable, solution will be the case when the number of types of fuel exceeds two and when several trucks can enter the warehouse at the same time. Let the reader try to solve this problem for three fuels as well.

Free download the e-book in a convenient format, watch and read:
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• Mathematics and Design, Grade 1, Textbook for educational organizations, Volkova S.I., 2016
• Mathematics, Oral exercises, grade 1, Textbook for educational organizations, Volkova S.I., 2016
• Course of lectures on the theory and technology of teaching mathematics in primary grades, Part 2, V.P. Ruchkina, 2019

The following tutorials and books:

• Mathematics, Algebra and the beginnings of mathematical analysis, grade 11, Vilenkin N.Ya., Ivashev-Musatov O.S., Shvartsburd S.I., 2014

The words of Sherlock Holmes: “How many times have I told you, drop everything impossible, then what remains will be the answer, no matter how incredible it may seem,” could serve as an epigraph to this chapter.

If solving a puzzle requires only the ability to think logically and does not need to perform arithmetic calculations at all, then such a puzzle is usually called a logical problem. Logic problems, of course, are among the mathematical ones, since logic can be viewed as very general, fundamental mathematics. It is nevertheless convenient to isolate and study logical puzzles separately from their more numerous arithmetic sisters. In this chapter, we will outline three common types of logic problems and try to figure out how to approach them.

The most common type of problem that puzzle lovers sometimes call the "Smith - Jones - Robinson problem" (by analogy with the old puzzle invented by G. Dudeny).

It consists of a series of parcels, usually giving some information about the characters; on the basis of these premises, it is required to draw certain conclusions. For example, here is what the latest American version of the Dudeny problem looks like:

1. Smith, Jones and Robinson work in the same train crew as a machinist, conductor and fireman. Their professions are named not necessarily in the same order as their surnames. There are three passengers with the same surnames on the train, which is served by the brigade.

From now on, we will respectfully call each passenger "mister" (mr).

2. Mr. Robinson lives in Los Angeles.

3. The conductor lives in Omaha.

4. Mr. Jones has long forgotten all the algebra he was taught in college.

5. The passenger - the conductor's namesake lives in Chicago.

6. The conductor and one of the passengers, a well-known specialist in mathematical physics, go to the same church.

7. Smith always wins against the stoker when he happens to meet for a game of billiards.

What is the name of the driver?

These problems could be translated into the language of mathematical logic, using its standard notation, and seek solutions using appropriate methods, but this approach would be too cumbersome. On the other hand, it is difficult to understand the logical structure of the problem without abbreviations of one kind or another. It is most convenient to use the table, in the empty cells of which we will write all possible combinations of elements of the sets under consideration. In our case, there are two such sets, so we need two tables (Fig. 139).

Rice. 139 Two tables for the problem of Smith, Jones and Robinson.

In each cell we write 1 if the corresponding combination is admissible, or 0 if the combination contradicts the conditions of the problem. Let's see how this is done. Condition 7, obviously, excludes the possibility that Smith is a stoker, so we write 0 in the cell in the upper right corner of the left table. Condition 2 tells us that Robinson lives in Los Angeles, so in the lower left corner of the table we write 1, and all other cells in the bottom row and left column are 0 to indicate that Mr. Robinson does not live in Omaha or Chicago, and Mr. Smith and Mr. Jones do not live in Los Angeles.

Now we have to think a little. From conditions 3 and 6, it is known that a mathematical physicist lives in Omaha, but we do not know his last name. He cannot be neither Mr. Robinson nor Mr. Jones (after all, he forgot even elementary algebra).

Therefore, it must be Mr. Smith. We note this circumstance by placing 1 in the middle cell of the upper row of the right table and 0 in the remaining cells of the same row and empty cells in the middle column. The third unit can now be inscribed in only one cell: this proves that Mr. Jones lives in Chicago. From condition 5, we learn that the conductor also has the surname Jones, and we enter 1 in the central cell of the left table and 0 in all the other cells in the middle row and middle column. After that, our tables take the form shown in Fig. 140.

Rice. 140 Table The eggs shown in Fig. 139, after pre-filling.

Now it is no longer difficult to continue the reasoning leading to the final answer. In the column with the inscription "Fireman", the unit can only be placed in the lower cell. It immediately follows that in the lower left corner there should be 0. Only the cell in the upper left corner of the table remains empty, where you can put only 1. So, the driver's surname is Smith.

Lewis Carroll was fond of inventing extremely complex and cunning problems of this kind. John J. Kemeny, the dean of the mathematics department at Dortmouth College, programmed one of the monstrous (with 13 variables and 12 conditions, from which it follows that "no judge sniffs tobacco") Carroll problems for the IBM-704 computer. The machine coped with the solution in about 4 minutes, although it would take 13 hours to print the complete "truth table" of the problem (a table showing whether the possible combinations of truth values ​​of the problem variables are true or false)!

For readers who want to try their luck in solving a problem that is more difficult than the Smith-Jones-Robinson problem, we offer a new puzzle. Its author is R. Smullian of Princeton University.

1. In 1918 the First World War ended. On the day of the signing of the peace treaty, three married couples gathered to celebrate the event at a festive table.

2. Each husband was the brother of one of the wives, and each wife was the sister of one of the husbands, that is, among those present it was possible to indicate three related pairs "brother and sister."

3. Helen is exactly 26 weeks older than her husband, who was born in August.

4. Mr. White's sister is married to Brother Helen's brother-in-law and married him on her birthday in January.

5. Margaret White is shorter than William Blake.

6. Arthur's sister is prettier than Beatrice.

7. John is 50 years old.

What's Mrs Brown's name?

Another type of logical problems is no less widespread, which, by analogy with the following well-known example, can be called problems of the "problem of colored caps" type. Three people (let's call them A, B and WITH) are blindfolded and said that each of them had either a red or a green cap on their heads. Then their eyes are untied and asked to raise their hand if they see a red cap, and leave the room if they are sure that they know what color the cap is on their head. All three caps turned out to be red, so all three raised their hand. Several minutes passed and WITH who is more clever than BUT and IN, left the room. How WITH was able to establish what color the cap is on it?

[The problem of the wise men in green caps is formulated in the text in such a way that it cannot have a solution. This is especially evident when the number of sages is large. How long does it take for the first sage to guess the true situation?

At the end of the forties, this problem was intensively discussed in Moscow in school mathematics circles, and a new version of it was invented, in which discrete time was introduced. This task looked like this.

In ancient times, wise men lived in the same city. Each of them had a wife. In the morning they came to the bazaar and learned all the city gossip there. They were gossips themselves. It gave them great pleasure to learn about the infidelity of any of the wives - they knew about it immediately. However, one unspoken rule was strictly observed: the husband was never informed about his wife, since each of them, having learned about their own shame, would have kicked their wife out of the house. So they lived, enjoying intimate conversations and remaining completely unaware of their own affairs.

But one day a real gossip came to town. He came to the market and publicly declared: "But not all wise men have faithful wives!" It would seem that the gossip had not said anything new - and so everyone knew it, every sage knew it too (only with malice he thought not about himself, but about something else), so none of the residents paid attention to the words of the gossip. But the sages began to think - that's what they are sages - and on n the day after the arrival of the gossip, the sages drove out the unfaithful wives (if there were n).

The reasoning of the sages is not difficult to restore. It is more difficult to answer the question: what information did the gossip add to the one that was known to the sages without him?

This problem has been repeatedly encountered in the literature].

S asks himself if his cap could be green. If this were the case, then BUT would immediately have known that he was wearing a red cap, because only a red cap on his head could make IN raise a hand. But then BUT would leave the room. IN would think in exactly the same way and also leave the room. Since neither one nor the other came out, WITH concluded that his own cap should be red.

This problem can be generalized to the case when there are any number of people and all of them are wearing red caps. Suppose there is a fourth character in the problem D, even more insightful than C. D could reason like this: “If my cap were green, then A, B and WITH would find themselves in exactly the same situation as just described, and in a few minutes the smartest of the trio would certainly leave the room.

But already five minutes have passed, and none of them come out, therefore, my cap is red. "

If a fifth participant appeared, even smarter than D, then he could come to the conclusion that he was wearing a red cap after waiting ten minutes. Of course, our reasoning loses its credibility due to assumptions about varying degrees of ingenuity. A, B, C... and rather vague considerations as to how long the most ingenious person should wait before he can confidently name the color of his cap.

Some other colored cap problems contain less uncertainty. Such, for example, is the following problem, also invented by Smullian. Each of the three - A, B and WITH- is fluent in logic, that is, he can instantly extract all the consequences from a given set of premises and knows that others also have this ability.

We take four red and four green stamps, blindfold our "logicians" and stick two stamps on each of them on the forehead. Then we remove the blindfolds from their eyes and in turn set A, B and WITH the same question: "Do you know what color the stamp is on your forehead?" Each of them replies in the negative. Then we ask again BUT and again we get a negative answer. But when we ask the same question a second time IN, he answers in the affirmative.

What color are the marks on the forehead IN?

The third type of popular logic puzzles is the problems of liars and those who always tell the truth. In the classical version of the problem, we are talking about a traveler who found himself in a country inhabited by two tribes. Members of one tribe always lie, members of the other only tell the truth. The traveler meets two natives. "Do you always tell only the truth?" he asks the tall native. He answers: "Tarabara". "He said yes," explains the smaller native who knows English, "but he's a terrible liar." Which tribe does each of the natives belong to?

A systematic approach to the solution would consist in writing out all four possibilities: AI, IL, LI, LL (I means “true”, L- “false”) - and excluding those from them that contradict the given problem. The answer can be received much more quickly if you notice that the tall native must answer in the affirmative, regardless of whether he is lying or telling the truth. Since the smaller native told the truth, he must belong to the tribe of the truthful, and his tall friend - to the tribe of liars.

The most famous problem of this type, complicated by the introduction of probabilistic weights and not very clear formulation, can be found rather unexpectedly in the middle of the sixth chapter of the book by the English astronomer A. Eddington "New Pathways in Science". "If A, B, C and D tell the truth in one out of three cases (independently of each other) and BUT States that IN denies that WITH says like D liar, then what is the probability that D told the truth? "

Eddington's reply, 25/71, was met with a barrage of protests from readers and sparked a funny and confusing controversy that was never resolved. The English astronomer G. Dingle, the author of a review of Eddington's book, published in the journal Nature (March 1935), believed that the problem did not deserve attention at all as meaningless and only testified that Eddington had not sufficiently thought out the basic ideas of the theory of probability. The American physicist T. Stern (Nature, June 1935) objected to this, stating that, in his opinion, the problem was by no means meaningless, but the data were insufficient to solve it.

In response, Dingle noted (Nature, September 1935) that if we take Stern's point of view, then there is enough data for a solution and the answer will be 1/3. Eddington then got into the fray, publishing (Mathemetical Gazette, October 1935) an article detailing how he got his answer. The controversy culminated in two more articles appearing in the same journal, one of which defended Eddington, while the other advanced a different point of view.

The difficulty lies mainly in understanding Eddington's formulation. If IN, expressing his denial, is telling the truth, then can we reasonably assume that WITH said that D spoke the truth? Eddington believed that there was insufficient basis for such an assumption. Likewise if BUT lies, then can we be sure that IN and WITH did you say anything at all? Fortunately, we can get around all these language difficulties by making the following assumptions (Eddington did not):

1. None of the four remained silent.

2. Statements A, B and WITH(each of them separately) either confirm or deny the following statement.

3. A false statement coincides with its denial, and a false negation coincides with a statement.

All four of them lie independently of each other with a probability of 1/3, that is, on average, any two of their three statements are false. If a true statement is denoted by the letter AND, and false - with the letter L then for A, B, C and D we get a table of eighty-one different combinations. From this number, one should exclude those combinations that are impossible due to the conditions of the problem.

Number of allowed combinations ending with a letter AND(that is, a truthful - true - statement D), should be divided by the total number of all valid combinations, which will give the answer.

The formulation of the problem of a traveler and two natives should be clarified. The traveler realized that the word "gibberish" in the language of the natives means either "yes" or "no", but he could not guess what exactly. This would allow warning of several letters, one of which I quote below.

The tall native apparently did not understand a word of what the traveler had said (in English) to him, and could not answer "yes" or "no" in English. So his "gibberish" means something like "I don't understand" or "Welcome to Bongo Bongo." Consequently, the little native lied, saying that his friend had answered "yes," and since the little one was a liar, he also lied when he called the tall native a liar. Therefore, the tall native should be considered truthful.

So the feminine logic has dealt a blow to my masculine vanity. Doesn't it hurt your author's pride a little too?

Answers

The first logical problem is best solved using three tables: one for combinations of first and last names of wives, the second for first and last names of husbands, and the third for family ties.

Since Mrs. White's name is Margaret (condition 5), we have only two options for the names of the other two wives: a) Helen Blake and Beatrice Brown, or b) Helen Brown and Beatrice Blake.

Let us assume that the second of the possibilities takes place. White's sister should be either Helen or Beatrice. But Beatrice cannot be Wayne's sister, because then Helen's brother would be Blake, and Blake's two brothers-in-law would be White (his wife's brother) and Brown (his sister's husband); Beatrice Blake is not married to any of them, which contradicts condition 4. Therefore, Helen must be White's sister. From this, in turn, we conclude that Brown's sister is named Beatrice, and Blake's sister is Margaret.

Condition 6 implies that Mr. White's name is Arthur (Brown cannot be Arthur, since such a combination would mean that Beatrice is more beautiful than herself, and Blake cannot be Arthur, since from condition 5 we know his name: William). So, Mr. Brown can only be John. Unfortunately, from condition 7 we see that John was born in 1868 (50 years before the signing of the peace treaty). But 1868 is a leap year, so Helen must be one day older than her husband than the 26 weeks mentioned in condition 3. (From condition 4 we know that she was born in January, and from condition 3 that her husband was born in August. She could be exactly 26 weeks older than her husband if her birthday was on January 31, and his - on August 1, and if between these dates was not February 29!) So, the second of the possibilities, with which we started out should be dropped, which allows us to name the wives: Margaret White, Ellen Blake, and Beatrice Brown. There is no contradiction here, since we do not know Blake's year of birth. From the conditions of the problem, we can conclude that Margaret is Brown's sister, Beatrice is Blake's sister, and Helen is White's sister, but the question of the names of White and Brown remains unresolved.

In the problem with stamps, IN there are three possibilities. His marks can be: 1) both red; 2) both are green; 3) one is green and the other is red. Suppose both stamps are red.

After all three have answered once, BUT may reason like this: “The marks on my forehead cannot be both red (because then WITH would see four red marks and would immediately know that he had two green marks on his forehead, and if WITH both stamps were green, then IN if he saw four green marks, he would have realized that he had two red marks on his forehead). Therefore, I have one green and one red stamp on my forehead. "

But when BUT asked a second time, he did not know what color his brand was. It allowed IN discard the possibility that both of his own stamps are red. Reasoning exactly the same as A, B ruled out the case when both of his brands are green. Consequently, he was left with only one option: one stamp is green, the other is red.

Several readers quickly noticed that a problem can be solved very quickly without going into question and answer analysis. Here is what one of the readers wrote about this: “The conditions of the problem are completely symmetric with respect to the red and green stamps.

Therefore, distributing marks between A, B and WITH observing all the conditions of the problem and replacing the red marks with green ones and, conversely, green ones with red ones, we will arrive at a different distribution, for which all conditions will also be fulfilled. It follows that if the solution is unique, then it should be invariant (should not change) when replacing green marks with red ones, and red ones with green ones. Such a solution can only be such a distribution of stamps, in which B will have one green and one red stamp ”.

As the Dean of the Department of Mathematics of Brooklyn College W. Manheimer put it, this elegant solution comes from the fact that they are not fluent in logic. A, B and WITH(as stated in the problem statement), and Raymond Smullian!

In Eddington's problem, the probability that D tells the truth, is 13/41. All combinations of truth and falsehood that contain an odd number of times false (or true) should be discarded as contradicting the conditions of the problem. As a result, the number of possible combinations decreases from 81 to 41, of which only 13 end with a truthful statement. D... Because the A, B and WITH tell the truth in cases that correspond to exactly the same number of acceptable combinations, the probability of telling the truth is the same for all four.

Using the equivalence symbol

meaning that the statements connected by him are either both true or both are false (then the false statement is true, otherwise it is false), and the negation symbol ~, Eddington's problem in the language of the propositional calculus can be written as follows:

or after some simplifications like this:

The truth table of this expression confirms the answer already received.

Notes:

Then frustrate- upset, do something in vain, hopeless, doom to failure (English).

See the chapter on Raymond Smullian in the book M. Gardner Time Travel (Moscow: Mir, 1990).

Eddington a... New Pathways in Science. - Cambridge: 1935; Michigan: 1959.

Logical tasks, as well as mathematics, is called "gymnastics of the mind." But, unlike mathematics, logic tasks- this is entertaining gymnastics, which in a fun way allows you to test and train thought processes, sometimes in an unexpected angle. To solve them, you need quick wits, sometimes intuition, but not special knowledge. Logic problem solving is to thoroughly analyze the condition of the problem, to unravel the tangle of contradictory connections between characters or objects. Logic tasks for children- these are, as a rule, whole stories with popular characters, in which you just need to get used to, feel the situation, visualize it and catch connections.

Even the most complex logic problems do not contain numbers, vectors, functions. But the mathematical way of thinking is necessary here: the main thing is to comprehend and understand the condition logical task... The most obvious solution on the surface is not always the right one. But more often than not, solving a logic problem turns out to be much easier than it seems at first glance, despite the confusing condition.

Interesting logic problems for children in a variety of subjects - mathematics, physics, biology - arouse their keen interest in these academic disciplines and help in their meaningful study. Logical tasks for weighing, transfusion, tasks for non-standard logical thinking will help to solve everyday problems in a non-standard way in everyday life.

In the process of solving logic tasks you will get acquainted with mathematical logic - a separate science, otherwise called "mathematics without formulas". Logic as a science was created by Aristotle, who was not a mathematician, but a philosopher. And logic was originally part of philosophy, one of the methods of reasoning. In his work "Analytics" Aristotle created 20 schemes of reasoning, which he called syllogisms. One of his most famous syllogisms is: “Socrates is a man; all people are mortal; means Socrates is mortal. " Logic (from Old Greek. Λογική - speech, reasoning, thought) is the science of correct thinking, or, in other words, "the art of reasoning."

There are certain tricks solving logical problems:

way of reasoning, with the help of which the simplest logical problems are solved. This method is considered the most trivial. In the course of the solution, reasoning is used that consistently takes into account all the conditions of the problem, which gradually lead to a conclusion and the correct answer.

way of tables, used in solving text logic problems. As the name suggests, solving logical problems consists in constructing tables that allow you to visualize the condition of the problem, control the process of reasoning and help you draw the correct logical conclusions.

way of graphs consists in enumerating possible options for the development of events and the final choice of the only correct solution.

block diagram method- a method widely used in programming and solving logical transfusion problems. It consists in the fact that, first, operations (commands) are allocated in the form of blocks, then the sequence of execution of these commands is established. This is a block diagram, which is essentially a program, the execution of which leads to the solution of the task.

billiards way follows from the theory of trajectories (one of the branches of the theory of probability). To solve the problem, it is necessary to draw a billiard table and interpret the actions by the movements of the billiard ball along different trajectories. In this case, it is necessary to keep records of possible results in a separate table.

Each of these methods is applicable to solving logical problems from different areas. These seemingly complex and scientific tricks can be easily used in solving logic problems for 1, 2, 3, 4, 5, 6, 7, 8, 9 classes.

We present you the most diverse logical tasks for 1, 2, 3, 4, 5, 6, 7, 8, 9 grades. We have selected the most interesting logic problems with answers, which will be interesting not only for children, but also for parents.

• pick up for the child logic tasks according to his age and development
• take your time to open the answer, let the child find himself logical solution tasks... Let him come to the right decision on his own and you will see what pleasure and delight he will have when his answer coincides with the given one.
• in the process solving problems on logic leading questions and indirect clues indicating the direction of thinking are permissible.

With our selection logical problems with answers you will really learn to solve logic problems, broaden your horizons and significantly develop logical thinking. Go for it !!!

Solving logical problems - the first step to child development.

E. Davydova

Logic is the art of coming to an unpredictable conclusion.

Samuel Johnson

Without logic, it is almost impossible to bring into our world ingenious finds of intuition.

Kirill Fandeev

A person who thinks logically stands out nicely against the background of the real world.

American dictum

Logic is the morality of thought and speech.

Jan Lukasiewicz

Introduction

Logic is the God of the thinkers.

L. Feuchtwanger

The ability to reason correctly is necessary in any area of ​​human activity: science and technology, justice and diplomacy, economic planning and military affairs. And this skill goes back to ancient times, logic, i.e. the science of which forms of reasoning are correct arose only a little over two thousand years ago. It was developed in the 6th century. BC. in the works of the great ancient Greek philosopher Aristotle, his students and followers.

At some point, mathematicians asked the question: "What exactly is mathematics, mathematical activity?" The simple answer is that mathematicians prove theorems, that is, find out some 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 is a mathematical statement is true or provable, it is also the starting point of mathematical logic. At school, we must learn to analyze, compare, highlight the main thing, generalize and systematize, prove and refute, define and explain concepts, pose and solve problems. Mastering these methods means the ability to think. In science, one has to deduce various formulas, numerical laws, rules, and prove theorems by reasoning. For example, in 1781 the planet Uranus was discovered. Observations have shown that the motion of this planet is different from the theoretically calculated motion. The French scientist Le Verrier (1811-1877), reasoning logically and performing rather complex calculations, determined the influence of another planet on Uranus and indicated its location. In 1846, the astronomer Halle confirmed the existence of a planet called Neptune. In doing so, they used the logic of mathematical reasoning and calculations.

The second starting point of our consideration is to find out what it means that a mathematical function is computable and can be calculated using some algorithm, a formal rule, a precisely described procedure. These two initial statements have much in common, they are naturally united under the general name "mathematical logic", where mathematical logic is understood primarily as the logic of mathematical reasoning and mathematical actions.

I chose this particular topic because mastering the elements of mathematical logic will help me in my future economic profession. After all, a marketer analyzes trendsmarket,prices, turnover and marketing methods, collects data on competing organizations,issues recommendations. For this you need to use the knowledge of logic.

Purpose of work: to study 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 origin of mathematical logic.

2. Study the elements of mathematical logic.

3. Select and solve problems with elements of mathematical logic.

Methods: analysis of literature, concepts, the method of analogies in solving problems, self-observation.

1. From the history of the emergence of mathematical logic

Mathematical logic is closely related to logic and owes its origin to it. The foundations of logic, the science of the laws and forms of human thinking were laid by the greatest ancient Greek philosopher Aristotle (384-322 BC), who in his treatises thoroughly investigated the terminology of logic, analyzed in detail the theory of inferences and proofs, described a number of logical operations, formulated the basic laws of thinking, including the laws of contradiction and exclusion of the third. Aristotle's contribution to logic is very great, no wonder its other name is Aristotle's logic. Even Aristotle himself noticed that between the science he created and mathematics (then it was called arithmetic) there is a lot in common. He tried to combine these two sciences, namely, to reduce thinking, or rather, inference, to a calculation based on the starting positions. In one of his treatises, Aristotle came close to one of the branches of mathematical logic - the theory of proofs.

Subsequently, many philosophers and mathematicians developed certain provisions of logic and sometimes even outlined the contours of modern propositional calculus, but the outstanding German scientist Gottfried Wilhelm Leibniz (1646 - 1716) came closest to the creation of mathematical logic in the second half of the 17th century, who showed the way to translate logic "From the realm of words, full of ambiguities, to the realm of mathematics, where the relationship between objects or statements is determined exactly." Leibniz even hoped that in the future philosophers, instead of fruitlessly arguing, would take paper and figure out which of them was right. At the same time, in his works, Leibniz also touched on the binary number system. It should be noted that the idea of ​​using two characters to encode information is very old. Australian aborigines counted as twos, some hunter-gatherer tribes in New Guinea and South America also used the binary system. In some African tribes, messages are transmitted using drums in the form of combinations of voiced and dull beats. A familiar example of two-character coding is Morse code, where the letters of the alphabet are represented by certain combinations of dots and dashes. After Leibniz, research in this area was carried out by many outstanding scientists, but the real success here came to the English self-taught mathematician George Boole (1815-1864), his determination knew no bounds.

The financial situation of George's parents (whose father was a shoemaker) allowed him to graduate only from elementary school for the poor. After some time, Buhl, having changed several professions, opened a small school, where he taught himself. He devoted a lot of time to self-education and soon became interested in the ideas of symbolic logic. In 1847, Boole published an article "Mathematical Analysis of Logic, or Experience of Calculating Deductive Inferences", and in 1854 his main work, "Investigation of the Laws of Thinking on which Mathematical Theories of Logic and Probability are Based", appeared. Boole invented a kind of algebra - a system of notation and rules applicable to all kinds of objects, from numbers and letters to sentences. Using this system, he could encode statements (statements that had to be proven true or false) with the symbols of his language, and then manipulate them, just as numbers are manipulated in mathematics. The main operations of Boolean algebra are conjunction (AND), disjunction (OR) and negation (NOT). After a while, it became clear that Boulle's system is well suited for describing electrical switching circuits. Current in a circuit can either flow or be absent, just as a statement can be either true or false. And a few decades later, already in the 20th century, scientists combined the mathematical apparatus created by George Boole with the binary number system, thereby laying the foundations for the development of a digital electronic computer. Certain provisions of Boole's works were, to one degree or another, touched upon both before and after him by other mathematicians and logicians. However, today in this area, it is the works of George Boole that are ranked among the mathematical classics, and he himself is rightfully considered the founder of mathematical logic, and even more so its most important sections - the algebra of logic (Boolean algebra) and the algebra of propositions.

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

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

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 the last 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, the theory of graphs (S.K. Kleene, A. Church, A.A. Markov, P.S.Novikov and many others) ...

In the middle of the 20th century, the development of computing technology led to the emergence of logical elements, logical blocks and computing devices, which was associated with the additional development of such areas of logic as problems of logical synthesis, logical design and logical modeling of logical devices and computer technology. In the 80s of the XX century, research began in the field of artificial intelligence based on languages ​​and logical programming systems. The creation of expert systems with the use and development of automatic theorem proving, as well as methods of demonstrative programming for the verification of algorithms and computer programs began. The 1980s also saw changes 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 operation of logical circuits and computing devices, as well as the principles of logical programming for fifth generation computers and the development of computer science textbooks with the study of the predicate calculus language for the design of knowledge bases. ...

1. Foundations of set theory

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

Many is called a set of certain completely distinguishable objects, considered as a whole. The creator of the theory of sets Georg Cantor gave the following definition of a set - "a set is a lot that we think of as a whole."

The individual objects that make up the set are called elements of the set.

Sets are usually denoted by capital letters of the Latin alphabet, and the elements of these sets - by small letters of the Latin alphabet. Sets are written in curly braces ().

It is customary to use the following notation:

a∈ X - "the element a belongs to the set X";

a∉ X - "the element a does not belong to the set X";

∀ - the quantifier of arbitrariness, generality, denoting “any”, “whatever it is”, “for all”;

∃ - existential quantifier:∃ y∈ B - “there is (is) an element y from the set B”;

∃! - quantifier of existence and uniqueness:∃ ! b∈ C - "there is only one element b from the set C";

: - "such that; possessing the property ";

→ - the symbol of the consequence, means "entails";

⇔ - the quantifier of equivalence, equivalence - "then and only then."

There are many finite and endless ... The sets are called the final if the number of its elements is finite, i.e. if there is a natural number n, which is the number of elements in the set. A = (a 1, a 2, a 3, ..., a n ). The set is called endless if it contains an infinite number of elements. B = (b 1, b 2, b 3 , ...). For example, the set of letters of the Russian alphabet is a finite set. The set of natural numbers is an infinite set.

The number of elements in a finite set M is called the cardinality of the set M and is denoted by | M |. Empty set - a set that does not contain any element -∅ ... The two sets are called equal if they consist of the same elements, i.e. are one and the same set. Sets are not equal to X ≠ Y if X contains elements that do not belong to Y, or Y contains elements that do not belong to X. The symbol of equality of sets has the following properties:

X = X; - reflexivity

if X = Y, Y = X - symmetry

if X = Y, Y = Z, then X = Z - transitivity.

According to this definition of equality of sets, we naturally obtain that all empty sets are equal to each other, or what is the same, that there is only one empty set.

Subsets. Inclusion ratio.

A set X is a subset of a set Y if any element of a set X∈ and the set Y. It is denoted by X⊆ Y.

If it is necessary to emphasize that Y contains other elements besides elements from X, then the strict inclusion symbol is used⊂ : X ⊂ Y. Relationship between symbols⊂ and ⊆ is given by the expression:

X ⊂ Y ⇔ X ⊆ Y and X ≠ Y

Let us note some properties of the subset that follow from the definition:

X⊆ X (reflexivity);

→ X⊆ Z (transitivity);

The original set A with respect to its subsets is called complete set and is denoted by I.

Any subset A i set A is called a proper set A.

The set consisting of all subsets of a given set X and an empty set∅ , is called a boolean X and is denoted by β (X). Boolean cardinality | β (X) | = 2 n.

Countable setis a set A, all elements of which can be numbered in a sequence (maybe infinite) and 1, а 2, а 3, ..., а n , ... so that in this case each element receives only one number n and each natural number n would be given as a number to one and only one element of our set.

A set equivalent to a set of natural numbers is called a countable set.

Example. Set of squares of integers 1, 4, 9, ..., n 2 represents only a subset of the set of natural numbers N. The set is countable, since it is brought into one-to-one correspondences with the natural series by assigning to each element the number of the number of the natural number of which it is a square.

There are 2 main ways to define sets.

enumeration (X = (a, b), Y = (1), Z = (1,2, ..., 8), M = (m 1 , m 2 , m 3 , .., m n });

description - specifies the characteristic properties that all elements of the set have.

The set is completely defined by its elements.

Only finite sets can be specified by enumeration (for example, many months in a year). Infinite sets can be specified only by describing the properties of its elements (for example, the set of rational numbers can be specified by describing Q = (n / m, m, n∈ Z, m ≠ 0).

Ways of setting a set by description:

but) by specifying the generative procedureindicating the set (s) that runs through the parameter (s) of this procedure - recursive, inductive.

X = (x: x 1 = 1, x 2 = 1, x k + 2 = x k + x k + 1 , k = 1,2,3, ...) - many Fibonicci numbers.

(many elements x such that x 1 = 1, x 2 = 1 and arbitrary x k + 1 (for k = 1,2,3, ...) is calculated by the formula x k + 2 = x k + x k + 1) or X =)