Entertaining logic in mathematics. Entertaining logic Math logic questions

1. Explanatory note
1.1 Relevance
1.2 Purpose of the program
1.3 Program objectives
1.4 Terms of the program implementation, age of children, forms of conducting classes
1.5 Stages of program implementation
1.6 Program content
1.7 Expected results

2. Methodological support
2.1 Perspective-thematic plan of the circle " Entertaining logic»

3. Diagnostic program for the logical thinking of older preschool children.

5. Information resources

1. Explanatory note.
Why logic for a little preschooler?
According to L.A. Wenger, “for five-year-old children, the external properties of things alone are clearly not enough. They are quite ready to gradually get acquainted not only with external, but also with internal, hidden properties and relationships that underlie scientific knowledge about the world ... All this will be beneficial mental development child only if the training is aimed at developing mental abilities, those abilities in the field of perception, imaginative thinking, imagination, which are based on the assimilation of samples of the external properties of things and their varieties ... "
The skills acquired by the child in the preschool period will serve as the foundation for gaining knowledge and developing abilities at an older age - at school. And the most important among these skills is the skill of logical thinking, the ability to "act in the mind." It will be more difficult for a child who has not mastered the methods of logical thinking to solve problems; performing exercises will require a lot of time and effort. As a result, the child's health may suffer, interest in learning may weaken or even fade away.
Having mastered logical operations, the child will be more attentive, learn to think clearly and clearly, and be able to concentrate on the essence of the problem at the right time. It will become easier to learn, which means that the learning process, and herself school life will bring joy and satisfaction.
This program shows how through special games and exercises it is possible to form the ability of children to independently establish logical relationships in the surrounding reality.
Working with preschoolers on the development of cognitive processes, you come to the conclusion that one of the necessary conditions for their successful development and learning is consistency, i.e. a system of special games and exercises with consistently developing and becoming more complex content, with didactic tasks, game actions and rules. Separately taken games and exercises can be very interesting, but using them outside the system, one cannot achieve the desired learning and developmental result.
1.1 Relevance
For the successful development of the school curriculum, the child needs not only to know a lot, but also to think consistently and conclusively, to guess, to show mental tension, to think logically.
Teaching the development of logical thinking is of no small importance for the future student and is very relevant today.
Mastering any method of memorization, the child learns to single out a goal and carry out certain work with the material to achieve it. He begins to understand the need to repeat, compare, generalize, group material for the purpose of memorization.
Teaching children about classification contributes to the successful mastery of a more complex way of remembering - the semantic grouping that children encounter at school.
Using the opportunities for the development of logical thinking and memory of preschoolers, it is possible to more successfully prepare children for solving the problems that school education sets before us.
The development of logical thinking includes the use of didactic games, ingenuity, puzzles, solving various logic games and labyrinths and is of great interest to children. In this activity, important personality traits are formed in children: independence, resourcefulness, ingenuity, perseverance is developed, and constructive skills are developed. Children learn to plan their actions, think about them, guess in search of a result, while showing creativity.
While working with children, you can notice that many children do not cope with seemingly simple logical tasks. For example, most children of older preschool age cannot correctly answer the question of what is more: fruits or apples, even if they have a picture in their hands on which fruits are drawn - many apples and several pears. Children will answer that there are more pears. In such cases, he bases his answers on what he sees with his own eyes. They are “let down” by imaginative thinking, and by the age of 5 children do not yet have logical reasoning. In senior preschool age they begin to show elements of logical thinking, characteristic of schoolchildren and adults, which must be developed in identifying the most optimal methods for the development of logical thinking.
Games of logical content help to cultivate cognitive interest in children, contribute to research and creative search, desire and ability to learn. Didactic games as one of the most natural activities of children and contributes to the formation and development of intellectual and creative manifestations, self-expression and independence. The development of logical thinking in children through didactic games is important for the success of subsequent schooling, for the correct formation of the personality of the student and in further education will help to successfully master the basics of mathematics and computer science.
1.2 Purpose of the program: creating conditions for the maximum development of the logical thinking of preschoolers in preparation for successful schooling.
1.3 Program objectives:

• teach children basic logical operations: analysis, synthesis, comparison, negation, classification, systematization, limitation, generalization, inference
• teach children to navigate in space
• develop in children higher mental functions, the ability to reason, prove
• to cultivate the desire to overcome difficulties, self-confidence, the desire to help a peer

1.4 Terms of the program implementation, age of children, forms of conducting classes
Program implementation terms – 1-2 years
The program is designed for children 5-7 years old.
The program provides for conducting circle classes in various forms:

• Individual independent work children.
• Work in pairs.
• Group forms of work.
• Differentiated.
• Frontal check and control.
• Self-assessment of the work done.
• Didactic game.
• Competition.
• Contests.

1.5 Stages of program implementation
The technology of activity is built in stages:

1. Diagnosis of the initial level of development of cognitive processes and control over their development.
2. Planning the means by which one or another quality can be developed (attention, memory, imagination, thinking), taking into account the individuality of each child and the available knowledge
3. Building an interdisciplinary (integral) basis for training in a developing course.
4. Gradual complication of the material, a gradual increase in the amount of work, increasing the level of independence of children.
5. Acquaintance with the elements of theory, teaching methods of reasoning, self-argumentation of choice.
6. Integration of knowledge and methods cognitive activity, mastering its generalized techniques.
7. Evaluation of the results of the developmental course according to the developed criteria, which should include the child (self-esteem, self-control, mutual control).

1. 6 Program content
Short description sections and topics of classes (sections correspond to a certain logical operation that children will learn in class):

1. Analysis - synthesis.
The goal is to teach children to divide the whole into parts, to establish a connection between them; learn to mentally combine parts of an object into a single whole.
Games and exercises: finding a logical pair (cat - kitten, dog - ? (puppy)). Complementing the picture (pick up a patch, draw a pocket to the dress). Search for opposites (light - heavy, cold - hot). Work with puzzles of varying complexity. Laying out pictures from counting sticks and geometric shapes.

2. Comparison.
The goal is to teach to mentally establish the similarities and differences of objects according to essential features; develop attention, perception of children. Improve orientation in space.
Games and exercises: consolidation of concepts: big - small, long - short, low - high, narrow - wide, higher - lower, further - closer, etc. Operating with the concepts "same", "most". Search for similarities and differences in 2 similar pictures.

3. Restriction.
The goal is to teach to single out one or more objects from a group according to certain characteristics. Develop children's observation skills.
Games and exercises: “circle only red flags with one line”, “find all non-circular objects”, etc. Exclusion of the fourth superfluous.

4. Generalization.
The goal is to teach to mentally combine objects into a group according to their properties. Contribute to the enrichment of vocabulary, expand everyday knowledge of children.
Games and exercises for operating with generalizing concepts: furniture, dishes, transport, vegetables, fruits, etc.

5. Systematization.
The goal is to teach to identify patterns; expand the vocabulary of children; learn to tell from a picture, retell.
Games and exercises: magic squares (pick up the missing part, picture). Drawing up a story based on a series of pictures, arranging the pictures in a logical sequence.

6. Classification.
The goal is to teach to distribute objects into groups according to their essential characteristics. Consolidation of generalizing concepts, free operation with them.

7. Inference.
The goal is to teach with the help of judgments to make a conclusion. Contribute to the expansion of household knowledge of children. Develop imagination.
Games and exercises: search for positive and negative in phenomena (for example, when it rains, it nourishes the plants - this is good, but the bad thing is that in the rain a person can get wet, catch a cold and get sick). Evaluation of the correctness of certain judgments (“the wind blows because the trees sway.” Right?). Solution logical tasks.

1.7 Expected results
Planned results:
Children should know:

• principles of constructing patterns, properties of numbers, objects, phenomena, words;
• the principles of the structure of puzzles, crosswords, chainwords, labyrinths;
• antonyms and synonyms;
• names of geometric shapes and their properties;
• the principle of programming and drawing up an algorithm of actions.

Children should be able to:

• determine patterns and perform a task according to this pattern, classify and group objects, compare, find common and particular properties, generalize and abstract, analyze and evaluate their activities;
• through reasoning, solve logical, non-standard problems, perform creative search, verbal-didactic, numerical tasks, find the answer to mathematical riddles;
• respond quickly and correctly during the warm-up to the questions posed;
• perform tasks to train attention, perception, memory
• perform graphic dictations, be able to navigate in a schematic representation of graphic tasks;
• be able to set a goal, plan the stages of work, achieve results with your own efforts.

Way to check the results of work : generalizing classes after each section and 2 diagnostics (initial (September) and final (May)) of the level of mastering the operations of logical thinking.

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 considered as very general, fundamental mathematics. Nevertheless, it is convenient to single out and study logical puzzles separately from their more numerous arithmetic sisters. In this chapter, we will outline three common types of logical 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. Dudeni).

It consists of a series of parcels, usually reporting certain information about the characters; On the basis of these assumptions, certain conclusions must be drawn. For example, here is what the latest American version of the Dudeney problem looks like:

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

In the future, we will respectfully call each passenger "Mr" (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. Passenger - 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 beats the stoker when they happen 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 a solution could be sought using appropriate methods, but such an approach would be too cumbersome. On the other hand, without abbreviations of one kind or another, it is difficult to understand the logical structure of the problem. It is most convenient to use a table, in the empty cells of which we will enter 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 enter 1 if the corresponding combination is admissible, or 0 if the combination contradicts the conditions of the problem. Let's see how it's done. Condition 7 obviously excludes the possibility that Smith is a stoker, so we enter 0 in the box 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 enter 1, and 0 to all other cells in the bottom row and left column to show 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 we know that the mathematical physicist lives in Omaha, but we do not know his last name. He cannot be either Mr. Robinson or Mr. Jones (after all, he has forgotten even elementary algebra).

Therefore, it must be Mr. Smith. We note this circumstance by putting 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 entered in only one cell: this proves that Mr. Jones lives in Chicago. From condition 5, we learn that the conductor also has the last name Jones, and we enter 1 in the central cell of the left table and 0 in all other cells of the middle row and middle column. After that, our tables take the form shown in Fig. 140.

Rice. 140 Table eggs shown in fig. 139, after prefilling.

Now it is not difficult to continue the reasoning leading to the final answer. In the column labeled "Stoker", a unit can only be placed in the bottom cell. It immediately follows from this that 0 should be in the lower left corner. Only the cell in the upper left corner of the table remains empty, where only 1 can be put. So, the name of the driver is Smith.

Lewis Carroll liked to invent extremely complex and ingenious problems of this kind. The dean of mathematics at Dortmouth College, John J. Kemeny, 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 completed the solution in about 4 minutes, although printing out the complete "truth table" of the problem (a table showing whether the possible combinations of truth values ​​of the problem's variables are true or false) would have taken 13 hours!

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

1. In 1918, the first World War. On the day of the signing of the peace treaty, three married couples gathered to celebrate this event at the 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, three related pairs of “brother and sister” could be indicated.

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

4. Mr. White's sister is married to Ellen'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 is Mrs Brown's name?

No less common is another variety of logical problems, which, by analogy with the following well-known example, can be called problems of the “colored caps problem” type. Three people (let's call them A, B and FROM) blindfold and say that each of them was put on either a red or a green cap. Then their eyes are untied and they are asked to raise their hand if they see a red cap, and to leave the room if they are sure that they know what color the cap is on their head. All three hats turned out to be red, so all three raised their hands. Several minutes passed and FROM, which is more intelligent than BUT and AT, left the room. How FROM was able to determine what color the hat 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 wise men is large. How long will it take the first wise man to guess the true situation?

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

In ancient times, wise men lived in one city. Each of them had a wife. In the mornings they came to the market and found out all the gossip of the city there. They were gossipers themselves. It gave them great pleasure to learn about the infidelity of any of the wives - they found out about it immediately. However, one unspoken rule was strictly observed: nothing was ever reported to the husband about his wife, since each of them, having learned about his own shame, would have driven his wife out of the house. So they lived, enjoying intimate conversations and remaining completely ignorant of their own affairs.

But one day a real gossip came to town. He came to the bazaar and publicly declared: “But not all wise men have faithful wives!” It would seem that the gossip did not say anything new - and so everyone knew it, every sage knew it (only with malice he thought not about himself, but about the other), so none of the residents paid any attention to the gossip's words. But the wise men thought - that's why they are wise men - and n-th day after the arrival of the gossip n wise men were expelled n unfaithful wives (if there were n).

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

This problem has been repeatedly encountered in the literature].

C asks himself if his cap can be green. If that were the case, then BUT would immediately recognize that he was wearing a red cap, because only a red cap on his head could make AT raise a hand. But then BUT would leave the room. AT would have begun to reason in exactly the same way and would also have left the room. Since neither one nor the other came out, FROM 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. Assume that a fourth actor has appeared in the problem D, even more insightful than C.D could reason like this: “If my cap were green, then A, B and FROM would find themselves in exactly the same situation that has just been described, and in a few minutes the most perceptive of the trio would certainly leave the room.

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

If there were a fifth member who was even smarter than D, he could have come to the conclusion that he was wearing a red cap after waiting ten minutes. Of course, our reasoning loses its persuasiveness due to assumptions about different degrees of ingenuity. A, B, C... and rather vague considerations as to how long the most perceptive person should wait before he can confidently name the color of his hat.

Some other "color cap" problems contain less uncertainty. Such, for example, is the following problem, also invented by Smullyan. Each of the three A, B and FROM- is fluent in logic, that is, he knows how to instantly extract all the consequences from a given set of premises and knows that the rest also have this ability.

We take four red and four green stamps, blindfold our “logicians” and stick two stamps on each of their foreheads. Then we remove the bandages from their eyes and, in turn, ask A, B and FROM the same question: "Do you know what color the stamps are on your forehead?" Each of them answers in the negative. We then ask again BUT and again we get a negative answer. But when we ask the same question a second time AT, he answers in the affirmative.

What color is the mark on the forehead AT?

The third type of popular logic puzzles are problems about liars and those who always tell the truth. AT classic version tasks we are talking about a traveler who finds himself in a country inhabited by two tribes. Members of one tribe always lie, members of another always tell the truth. The traveler meets two natives. "Do you always tell the truth?" he asks the tall native. He replies: "Tarabar". "He said yes," explains the smaller native who knows English, "but he's a terrible liar." To which tribe does each of the natives belong?

A systematic approach to solving would be to write out all four possibilities: AI, IL, LI, LL (I means "true", L - "false") - and exclude those that contradict the data of the problem. An answer can be obtained much more quickly if one observes that the tall native must answer in the affirmative 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 probability weights and a not very clear formulation, can be found quite unexpectedly in the middle of the sixth chapter of the book New Pathways in Science by the English astronomer A. Eddington. "If a A, B, C and D tell the truth one time out of three (independently) and BUT States that AT denies that FROM says as if D liar, what is the probability that D told the truth?"

Eddington's answer, 25/71, was met with a hail of protest from readers and gave rise to a ridiculous and confused dispute that was never finally 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 does not deserve attention at all as meaningless and only indicates that Eddington had not sufficiently thought through the basic ideas of probability theory. The American physicist T. Stern (Nature, June 1935) objected to this, stating that, in his opinion, the problem is by no means meaningless, but there is not enough data to solve it.

In response, Dingle remarked (Nature, September 1935) that if one takes Stern's point of view, then there is enough data for a decision and the answer will be 1/3. Here Eddington entered the fray, publishing (Mathemetical gazette, October 1935) an article explaining in detail how he got his answer. The dispute ended with two more articles that appeared in the same journal, the author of one of them defended Eddington, and the other put forward a point of view different from all the previous ones.

The difficulty lies mainly in understanding Eddington's formulation. If a AT, expressing his denial, speaks the truth, then can we reasonably assume that FROM said that D speak the truth? Eddington believed that there were not enough grounds for such an assumption. Likewise, if BUT lies, can we be sure that AT and FROM did they say anything at all? Fortunately, we can get around all these linguistic difficulties by making the following assumptions (Eddington did not make them):

1. None of the four remained silent.

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

3. A false assertion coincides with its negation, and a false negation coincides with an assertion.

All four 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 - letter L, then for A, B, C and D we get a table consisting 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 valid combinations ending with a letter And(i.e. 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 about 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 have alerted several emails, one of which I reproduce below.

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

So female logic dealt a blow to my male vanity. Doesn't it hurt your author's pride a little?

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 are left with only two possibilities 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 must be either Helen or Beatrice. But Beatrice cannot be Wyne's sister, because then Blake would be Helen's brother, 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 either of them, which contradicts condition 4. Therefore, White's sister must be Helen. From this, in turn, we conclude that Brown's sister is called Beatrice, and Blake's sister is Margaret.

It follows from condition 6 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 older than her husband by one day more than the 26 weeks stated 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 31st and his on August 1st, and if there was no February 29th between these dates!) So, the second of the possibilities, with which we started should be discarded, which allows us to name the wives: Margaret White, Helen Blake and Beatrice Brown. There is no contradiction here, since we do not know the year of Blake's birth. From the conditions of the problem, it can be concluded 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 AT there are three possibilities. His stamps can be: 1) both red; 2) both green; 3) one is green and the other is red. Let's assume that both stamps are red.

After all three have answered once, BUT can reason like this: “The marks on my forehead cannot be both red (because then FROM would have seen four red stamps and would have recognized at once that he had two green stamps on his forehead, and if FROM both stamps were green, then AT, seeing four green stamps, would have realized that he had two red stamps on his forehead). That is why I have one green and one red mark on my forehead.”

But when BUT asked a second time, he didn't know what color his brand was. It allowed AT discard the possibility that both of his own stamps are red. Arguing in exactly the same way as A, B ruled out the case when both of his stamps are green. Therefore, he was left with only one possibility: one stamp is green, the other is red.

Several readers quickly noticed that the problem can be solved very quickly without having to analyze the questions and answers. Here is what one of the readers wrote about this: “The conditions of the problem are completely symmetrical with respect to the red and green marks.

Therefore, by distributing stamps between A, B and FROM if all the conditions of the problem are met and replacing the red marks with green and, conversely, green with red, we will arrive at a different distribution, for which all conditions will also be met. It follows that if the solution is unique, then it must be invariant (should not change) when replacing green labels 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 W. Manheimer, Dean of the Department of Mathematics at Brooklyn College, put it, this elegant solution comes from the fact that not A, B and FROM(as stated in the condition of the problem), and Raymond Smullyan!

In the Eddington problem, the probability that D tells the truth, is 13/41. All combinations of true and false 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 is reduced from 81 to 41, of which only 13 end in a true statement. D. Because the A, B and FROM tell the truth in cases that correspond to exactly the same number of valid combinations, the probability of telling the truth is the same for all four.

Using the Equivalence Symbol

which means that the propositions connected by it are either both true or both false (then the false proposition is true, otherwise it is false), and the negation symbol ~, Eddington's problem in 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:

That's frustrating- upset, make something futile, hopeless, doom to failure (English).

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

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

Introduction

Logic is the God of the thinkers.

L. Feuchtwanger

The ability to reason correctly is necessary in any field of human activity: science and technology, justice and diplomacy, economic planning and military affairs. And this ability 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 VI century. BC. in the works of the great ancient Greek philosopher Aristotle, his students and followers.

At some point, mathematicians asked the question: “What, in fact, is mathematics, mathematical activity?” The simple answer is that mathematicians prove theorems, that is, find out some truths about real world and "ideal mathematical world". An attempt to answer the question of what is a mathematical theorem, mathematical truth, and what is a mathematical statement true or provable, this is also the network of 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 patterns, rules, and prove theorems by reasoning. For example, in 1781 the planet Uranus was discovered. Observations have shown that the motion of this planet differs from the theoretically calculated motion. The French scientist Le Verrier (1811-1877), logically reasoning and performing rather complex calculations, determined the influence of another planet on Uranus and indicated its location. In 1846 the astronomer Galle confirmed the existence of a planet which was named Neptune. In doing so, they used the logic of mathematical reasoning and calculations.

The second starting point of our considerations is to clarify 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 formulations 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. To do this, you need to use the knowledge of logic.

Objective: 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 studied 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 the exclusion of the third. Aristotle's contribution to logic is very great, not without reason its other name is Aristotelian logic. Even Aristotle himself noticed that between the science he created and mathematics (at that time it was called arithmetic) there is much in common. He tried to combine these two sciences, namely, to reduce reflection, or rather inference, to calculation on the basis of initial positions. In one of his treatises, Aristotle came close to one of the sections of mathematical logic - the theory of proofs.

In the future, many philosophers and mathematicians developed certain provisions of logic and sometimes even outlined the contours of the modern propositional calculus, but the closest to the creation of mathematical logic came in the second half of the 17th century, the outstanding German scientist Gottfried Wilhelm Leibniz (1646 - 1716), who pointed out the ways for translating logic "from the verbal realm, full of uncertainties, to the realm of mathematics, where the relations between objects or statements are determined with perfect precision." 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, Leibniz also touched upon the binary number system in his works. It should be noted that the idea of ​​using two characters to encode information is very old. The Australian aborigines counted in deuces, some tribes of hunter-gatherers of New Guinea and South America also used a binary counting 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, many eminent scientists conducted research in this area, but the real success here came to the self-taught English mathematician George Boole (1815-1864), his determination knew no bounds.

Financial situation George's parents (whose father was a shoemaker) allowed him to graduate only primary 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 the article "Mathematical Analysis of Logic, or the Experience of the Calculus of Deductive Inferences", and in 1854 his main work "Investigation of the laws of thought on which the 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 needed to be proven true or false) using the symbols of his language, and then manipulate them in the same way that numbers are manipulated in mathematics. The basic operations of Boolean algebra are conjunction (AND), disjunction (OR), and negation (NOT). After some time, it became clear that Boole's system is well suited for describing electrical switching circuits. Current in a circuit can either flow or not, 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 foundation for the development of a digital electronic computer. Individual provisions of Boole's work were touched upon to some extent both before and after him by other mathematicians and logicians. However, today in this area, it is the works of George Boole that are considered mathematical classics, and he himself is rightfully considered the founder of mathematical logic and, all the 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 also made by Russian scientists P.S. Poretsky (1846-1907), I.I. Zhegalkin (1869-1947).

In the 20th 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 20th century, the rapid development of mathematical logic was due to the development of the theory of algorithms and algorithmic languages, automata theory, graph theory (S.K. Kleene, A. Church, A.A. Markov, P.S. Novikov and many others) .

In the middle of the 20th century, the development of computer technology led to the emergence of logical elements, logical blocks and computer technology devices, which was associated with the additional development of such areas of logic as the problems of logical synthesis, logical design and logical modeling of logical devices and computer technology. In the 1980s, research began in the field of artificial intelligence based on languages ​​and systems of logic programming. The creation of expert systems began with the use and development of automatic proof of theorems, as well as methods of evidence-based programming for the verification of algorithms and computer programs. Changes in education also began in the 1980s. The advent 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 work logic circuits and computing devices, 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 the predicate calculus language for designing knowledge bases.

1. Fundamentals of set theory

The concept of a set is one of those fundamental concepts of mathematics that are difficult to define precisely using elementary concepts. Therefore, we confine ourselves to a descriptive explanation of the concept of a set.

many called a set of certain quite distinct objects, considered as a single whole. The creator of set theory, 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 a set are called set elements.

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

It is customary to use the following notation:

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

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

∀ - quantifier of arbitrariness, generality, denoting "any", "whatever", "for all";

∃ - existence quantifier:∃ y∈ B - "there is (there is) an element y from the set B";

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

: - “such that; possessing the property";

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

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

Sets are finite and endless . The sets are called final , if the number of its elements is finite, i.e. if there is a natural number n, which is the number of elements of 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 elements -∅ . The two sets are called equal , if they consist of the same elements, i.e. are the same set. Sets are not equal to X ≠ Y if X has elements that do not belong to Y, or Y has elements that do not belong to X. The set equality symbol has the following properties:

X=X; - reflexivity

if X=Y, Y=X - symmetry

if X=Y,Y=Z, then X=Z is transitive.

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

Subsets. Inclusion relation.

A set X is a subset of a set Y if any element of the set X∈ and set Y. 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:

X ⊂ Y ⇔ X ⊆ Y and X≠Y

We note some properties of the subset that follow from the definition:

X⊆ X (reflexivity);

→ X⊆ Z (transitivity);

The original set A in relation to its subsets is called complete set and is denoted by I.

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

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

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

A set equivalent to the 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 correspondence with the natural series by assigning to each element the number of the number of the natural series, the square of which it is.

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 - indicates the characteristic properties that all elements of the set have.

A set is completely defined by its elements.

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

Ways to specify a set by description:

a) by specifying a generating procedurewith an indication of the set (sets) that the parameter (parameters) of this procedure runs through - 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.

(multiple elements x, such that x 1 \u003d 1, x 2 =1 and arbitrary x k+1 (for k=1,2,3,...) is calculated by the formula x k+2 \u003d x k + x k + 1) or X \u003d)