Syllogisms Once an investigator had to interrogate three witnesses at the same time: Claude, Jacques and Dick. Their testimonies contradicted each other, and each of

Once the investigator had to simultaneously interrogate three witnesses: Claude, Jacques and Dick. Their testimonies contradicted each other, and each of them accused someone of lying. Claude claimed that Jacques was lying, Jacques accused Dick of lying, and Dick persuaded the investigator not to believe either Claude or Jacques. But the investigator quickly brought them to clean water, without asking them a single question. Which of the witnesses spoke the truth

Ilya Muromets, Dobryna Nikitich and Alyosha Popovich were given 6 coins for their faithful service: 3 gold and 3 silver. Each got two coins. Ilya Muromets does not know which coins went to Dobryna and which ones to Alyosha, but he knows which coins he got himself. Come up with a question to which Ilya Muromets will answer "yes", "no" or "I do not know", and by the answer to which you can understand what coins he got

The rules of syllogisms 1. In a syllogism there should be only three statements and only three terms. WG All excursionists scattered in different directions, Petrov excursionist, it means that he fled in different directions. 3. If both premises are private statements, then it is impossible to draw a conclusion. 2. If one of the premises is a private statement, then the conclusion must be private. 4. If one of the premises is a negative statement, then the conclusion is also a negative statement. 5. If both premises are negative statements, then the conclusion is impossible 6. The middle term must be distributed in at least one of the premises. 7. A term cannot be distributed in the conclusion if it is not distributed in the premise.

All cats have four legs. All dogs have four legs. All dogs are cats. All people are mortal. All dogs are not human. Dogs are immortal (not mortal). Ukraine occupies a huge territory. Crimea is part of Ukraine. Crimea occupies a huge territory

... 18 years.

Solution

The first way ... According to the condition of the problem, you can make an equation. Let Dima's age be x years, then the age of the sister is x / 3, and the age of the brother is x / 2; (x + x / 3 + x / 2): 3 = 11. After solving this equation, we find that x = 18. Dima is 18 years old. It will be useful to give a slightly different solution, "in parts".

Second way ... If the ages of Dima, his brother and sister are depicted by segments, then the “Dima's segment” consists of two “brother's segments” or three “sister's segments”. Then, if Dima's age is divided into 6 parts, then the sister's age is two such parts, and the brother's age is three such parts. Then the sum of their ages is 11 such parts. On the other hand, if the average age is 11 years, then the sum of the ages is 33 years. Whence it follows that in one part - three years. This means that Dima is 18 years old.

Test criteria .

Complete correct solution - 7 points.

The equation is correct, but errors were made in the solution - 3 score .

The correct answer is given and the check is performed - 2 score .

0 points .

Answer ... Sam Gray.

Solution .

It is clear from the condition of the problem that the statements of each of the witnesses were made about the statements of the other two witnesses. Consider Bob Black's statement. If what he says is true, then Sam Gray and John White are lying. But from the fact that John White is lying, it follows that not all of Sam Gray's testimony is a complete lie. And this contradicts the words of Bob Black, whom we decided to believe and who claims that Sam Gray is lying. So, Bob Black's words cannot be true. It means that he lied, and we must admit that Sam Gray's words are true, and, therefore, John White's statements are a lie. Answer: Sam Gray didn't lie.

Test criteria .

A complete correct analysis of the problem situation is given and the correct answer is given - 7 points .

A complete correct analysis of the situation is given, but for some reason an incorrect answer is given (for example, instead of the one who did NOT lie, the answer indicates those who lied) - 6 points .

The correct analysis of the situation was given, but for some reason the correct answer was not given (for example, it was proved that Bob Black lied, but no further conclusions were drawn) - 4 score .

The correct answer is given and it is shown that it satisfies the condition of the problem (a check is carried out), but it has not been proven that the only answer is 3 score .

1 score .

0 points .

Answer ... One number 175.

Solution . The first way . As part of the digits that write the number, there is no digit 0, otherwise the condition of the task cannot be met. This three-digit number is obtained by multiplying the product of its digits by 5, therefore, it is divisible by 5. This means that its record ends with the digit 5. We get that the product of digits multiplied by 5 must be divisible by 25. Note that even digits in the number are cannot, otherwise the product of digits would be equal to zero. Thus, the three-digit number must be divisible by 25 and not contain even digits. There are only five such numbers: 175, 375, 575, 775 and 975. The product of the digits of the required number must be less than 200, otherwise, multiplied by 5, will give a four-digit number. Therefore, the numbers 775 and 975 are obviously not suitable. Among the remaining three numbers, only 175 satisfy the condition of the problem. Second way. Note (similarly to the first solution) that the last digit of the required number is 5. Leta , b , 5 - consecutive digits of the required number. By the condition of the problem, we have: 100a + 10 b + 5 = a · b · 5 · 5. Dividing both sides of the equation by 5, we get: 20a + 2 b + 1 = 5 ab ... After subtracting from both sides of equality 20a and taking out the common factor on the right side from the brackets, we get: 2b + 1 = 5 a (b – 4 a) (1 ). Considering that a and b can take natural values from 1 to 9, we get that the possible values of a are only 1 or 2. But a = 2 does not satisfy the equality (1 ), on the left side of which there is an odd number, and on the right side, when a = 2 is substituted, an even number is obtained. So, the only possibility is a = 1. Substituting this value in (1 ), we get: 2 b + 1 = 5 b- 20, from where b = 7. Answer: the only number you are looking for is 175.

Test criteria .

Complete correct solution - 7 points .

The correct answer is received and there are arguments that significantly reduce the enumeration of options, but there is no complete solution - 4 score .

The equation is correctly drawn up and the transformations and reasoning are given that allow you to solve the problem, but the solution is not completed - 4 score .

Enumeration of options is shortened, but there is no explanation why, and the correct answer is indicated - 3 score .

The equation is correct, but the problem is not solved - 2 score .

There is reasoning in the solution that allows you to exclude any numbers from consideration or consider numbers with certain properties (for example, ending in the number 5), but there is no further significant progress in the solution - 1 score .

Only correct answer or answer with validation is given - 1 score .

Answer ... 75 ° .

Solution . Consider a triangle AOC, where O is the center of the circle. This triangle is isosceles, since OS and OA are radii. Hence, by the property of an isosceles triangle, angles A and C are equal. Let's draw a perpendicular CM to the AO side and consider a right-angled triangle OMC. According to the condition of the problem, the SM leg is half of the OS hypotenuse. This means that the SOM angle is 30 °. Then, by the theorem on the sum of the angles of a triangle, we find that the angle CAO (or CAB) is equal to 75 °.

Test criteria .

The correct reasoned solution to the problem - 7 points.

The correct reasoning is given, which is a solution to the problem, but for some reason the wrong answer is given (for example, the angle COA is indicated instead of the angle SAO) - 6 points.

On the whole, correct reasoning is presented, in which mistakes were made that do not have a fundamental decision in essence, and the correct answer is given - 5 points.

The correct solution of the problem is given in the absence of justifications: all intermediate conclusions are indicated without indicating the connections between them (references to theorems or definitions) - 4 points.

Additional constructions and designations in the drawing are made, from which the course of the solution is clear, the correct answer is given, but the reasoning itself is not given - 3 points.

The correct answer is given in case of incorrect reasoning - 0 points.

Only the correct answer is given - 0 points.

Answer ... See picture.

Solution . We transform this equation by selecting a full square under the root sign:. The expression on the right-hand side makes sense only for x = 9. Substituting this value into the equation, we get: 9 2 – y 4 = 0. Factor the left side: (3 -y)(3 + y)(9 + y 2 ) = 0. Whence y= 3 or y = –3. This means that the coordinates of only two points (9; 3) or (9; –3) satisfy this equation. The equation graph is shown in the figure.

Verification criteria.

The correct transformations and reasoning were carried out and the graph is correctly built - 7 points.

Correct conversions performed, but meaning is lost y = –3; one point is indicated as a graph -3 points.

One or two suitable points indicated, possibly with verification, but without other explanations, or after incorrect transformations -1 score.

Correct transformations were performed, but it was declared that the expression under the root (or on the right side after squaring) is negative and the graph is an empty set of points - 1 score.

Reasoning has been carried out that led to the indication of two points, but these points are somehow connected (for example, by a segment) - 1 score.

Two points are indicated without explanation, which are somehow connected - 0 points.

In other cases - 0 points.

Answers to the tasks of the second stage of the Olympiad

Answer . They can.

Solution . If a =, b = -, then a = b + 1 and a 2 = b 2

You can also solve a system of equations:

Verification criteria.

Correct answer with numbers a and b – 7 points .

A system of equations was compiled, but an arithmetic error was made when solving it - 3 score .

Only the answer is - 1 score .

Answer . In 12 seconds .

Solution . There are 3 flights between the first and fourth floors, and between the fifth and the first - 4. According to the condition, Petya runs 4 flights 2 seconds longer than his mother takes the elevator, and three flights are 2 seconds faster than his mother. This means that Petya runs one flight in 4 seconds. Then Petya runs from the fourth floor to the first (i.e., 3 flights) in 4 * 3 = 12 seconds.

Verification criteria.

Correct answer with complete solution - 7 points .

Explained that one hop takes 4 seconds, the answer says 4 seconds - 5 points .

A correct justification assuming that the path from the fifth floor to the first is 1.25 times longer than the path from the fourth floor to the first and the answer is 16 seconds - 3 score .

Only the answer is - 0 points .

Answer . See picture.

Solution . Because NS 2 =| NS | 2 , then at =| NS |, moreover, x ≠ 0.

It is also possible, using the definition of the modulus, to obtain that (for x = 0 function not defined).

Verification criteria.

Correct graph with explanation - 7 points .

Correct graph without any explanation - 5 points .

Function graph y = | x | no punctured point -3 score .

Answer . Yes .

Solution . We divide this square with side 5 straight lines parallel to its sides into 25 squares with side 1 (see fig.). If in each such square there were no more than 4 marked points, then in total no more than 25 * 4 = 100 points would be marked, which contradicts the condition. Therefore, at least one of the resulting squares must contain 5 of the marked points.

Verification criteria.

The right decision - 7 points .

Only the answer is - 0 points .

Answer . Eight ways.

Solution . From point a) it follows that the coloring of all points with integer coordinates is uniquely determined by the coloring of the points corresponding to the numbers 0, 1, 2, 3, 4, 5 and 6. Point 0 = 14-2 * 7 should be colored the same way as 14, those. red. Similarly, point 1 = 71-107 should be colored blue, point 3 = 143-20 * 7 blue, and 6 = 20-2 * 7 red. Therefore, it remains only to calculate how many different ways you can color the points corresponding to the numbers 2, 4 and 5. Since each point can be colored in two ways - red or blue - then there are 2 * 2 * 2 = 8 ways in total. Note. When counting the number of ways to color points 2, 4 and 5, you can simply list all the ways, for example, in the form of a table:

Test criteria .

The correct answer with the correct rationale is 7 points .

The problem is reduced to counting the number of ways to color 3 dots, but the answer is 6 or 7 - 4 score .

The task is reduced to counting the number of ways to color 3 dots, but there is no counting of the number of ways, or an answer is obtained that is different from the ones indicated earlier - 3 score .

The answer (including the correct one) without justification is 0 points .

Answer . 4 times.

Solution .

Let's draw segments of MK and AS . The MVKE quadrangle consists of

triangles MVK and MKE , and the quadrangle AECD - from triangles

1 way . Triangles MVK and ASD - rectangular and the legs of the first are 2 times smaller than the legs of the second, so they are similar and the area of the triangle ACD 4 times the area of the MVK triangle. Because M and K – the middle AB and BC, respectively, then MK– , therefore MK || AS and MK = 0,5АС . From the parallelism of the straight lines MK and AS follows the similarity

triangles MKE and AEC, and since similarity coefficient is 0.5, then the area of the AEC triangle is 4 times the area of the MKE triangle. Now: S AEC D = SAEC + SACD = 4 SMKE + 4 SMBK = 4 (SMKE + SMBK) = 4 SMBKE.

2 way . Let the area of the rectangle ABCD is equal to S. Then the area of the triangle ACD is equal to ( the diagonal of the rectangle divides it into two equal triangles), and the area of the triangle MVK is equal to MV × VK = T.k. M and K – the middle of the segments AB and BC, then AK and CM – medians of triangle ABC, therefore E – the point of intersection of the medians of the triangle ABC, those. the distance from E to AC ish, where h - height of triangle ABC, drawn from vertex B. Then the area of the triangle AEC is equal to. Then for the area of the quadrangle AECD, equal to the sum of the areas of the triangles AEC and ACD, we get: Further, since MK– middle line of triangle ABC, then the area of the MKE triangle is* h - * h) = h) = (AC * h) == S ... Therefore, for the area of the quadrangle MVKE, equal to the sum of the areas of the triangles MVK and MKE, we get:. Thus, the ratio of the areas of the quadrangles AECD and MVKE is equal.

Verification criteria.

Correct solution and correct answer -7 points .

Correct solution, but the answer is incorrect due to arithmetic error -5 points .

5. SUMMING UP AND AWARDING THE WINNERS

The final indicators of the performed competitive tasks are determined by the jury incompliance with the developed evaluation criteria;

For the winners of the Olympiad, determined by the highest number of points,three prize places are established;

The results of the competition are drawn up by the report of the organizer of the Olympiad.

The winners are awarded with certificates and valuable gifts.

In case of disagreement with the mark given by the jury, the participant can submitwritten appeal within an hour after the announcement of the results.

The publicity of the competition is ensured - the results of the competition are announcedprize-winners.

The following sequence of steps can be distinguished in solving logical problems.

1. Select elementary (simple) statements from the problem statement and designate them with letters.

2. Write down the condition of the problem in the language of logic algebra, combine simple statements into complex ones using logical operations.

3. Compose a single logical expression for the requirements of the problem.

4. Using the laws of logic algebra, try to simplify the resulting expression and calculate all of its values, or build a truth table for the expression under consideration.

5. Choose a solution - set of values simple statements, in which the constructed logical expression is true.

6. Check whether the obtained solution satisfies the condition of the problem.

Example:

Objective 1:“Trying to remember the winners of last year's tournament, five former spectators of the tournament stated that:

1. Anton was second and Boris was fifth.

2. Victor was second and Denis was third.

3. Gregory was the first and Boris the third.

4. Anton was the third, and Evgeniy - the sixth.

5. Victor was third and Evgeniy was fourth.

Subsequently, it turned out that each viewer made a mistake in one of his two statements. What was the true distribution of places in the tournament. "

1) Let us denote through the first letter in the name of the participant of the tournament, and - the number of the place that he has, i.e. we have.

2) 1. ; 3. ; 5. .

3) A single logical expression for all the requirements of the task:.

4) In the formula L carry out equivalent transformations, we get:.

5) From point 4 it follows:,.

6) Distribution of places in the tournament: Anton was the third, Boris - the fifth, Victor - the second, Grigory - the first, and Evgeniy - the fourth.

Objective 2:“Ivanov, Petrov, Sidorov were brought to trial on charges of robbery. The investigation established:

1. if Ivanov is not guilty or Petrov is guilty, then Sidorov is guilty;

2. If Ivanov is not guilty, then Sidorov is not guilty.

Is Ivanov guilty? "

1) Consider the statements:

A: "Ivanov is guilty" V: "Petrov is guilty" WITH: "Sidorov is guilty."

2) The facts established by the investigation:,.

3) Single logical expression:. It is true.

Let's compose a truth table for him.

A	V	WITH	L

To solve a problem means to indicate at what values of A the resulting complex statement L is true. If, but, then the investigation does not have enough facts to accuse Ivanov of a crime. Analysis of the table shows and, i.e. Ivanov is guilty of robbery.

Questions and tasks.

1. Make up the RCC for the formulas:

2. To simplify the RCS:

3. Based on the given switching scheme, construct a logical formula corresponding to it.

4. Check the equivalence of the DCS:

5. Construct a circuit of three switches and a light bulb in such a way that the light will light up only when exactly two switches are in the "on" position.

6. Using this conductivity table, construct a circuit of functional elements with three inputs and one output, which implements the formula.

x	y	z	F

7. Analyze the diagram shown in the figure and write down the formula for the function F.

8. Objective: “Once the investigator had to interrogate three witnesses at the same time: Claude, Jacques, Dick. Their testimonies contradicted each other, and each of them accused someone of lying.

1) Claude claimed that Jacques was lying.

2) Jacques accused Dick of lying.

3) Dick tried to persuade the investigator not to trust either Claude or Jacques.

But the investigator quickly brought them to clean water, without asking them a single question. Which of the witnesses spoke the truth?

9. Determine which of the four students passed the exam, if it is known that:

1) If the first passed, then the second passed.

2) If the second passed, then the third passed or the first did not pass.

3) If the fourth did not pass, then the first passed, and the third did not pass.

4) If the fourth passed, then the first passed.

10. When asked which of the three students studied logic, the answer was received: if he studied the first, then he studied the third, but it is not true that if he studied the second, then he studied the third. Who studied logic?

1.a) ( commutative disjunction );

b)

(commutativity of conjunction );

2.a) ( disjunction associativity );

b) ( conjunction associativity );

3.a) ( distributiveness of disjunction with respect to conjunction );

b) ( distributiveness of a conjunction with respect to disjunction );

4.

and

–de Morgan's laws .

5.

;

;

;

6.

(or

) (excluded third law );

(or

(law of contradiction );

7.

(or

);

(or

);

(or

);

(or

).

The listed properties are commonly used to transform and simplify Boolean formulas. Here are the properties of only three logical operations (disjunction, conjunction and negation), but it will be shown later that all other operations can be expressed through them.

With the help of logical connectives, you can compose logical equations, and solve logical problems in the same way as arithmetic problems are solved using systems of ordinary equations.

Example. Once the investigator had to simultaneously interrogate three witnesses: Claude, Jacques and Dick. Their testimonies contradicted each other, and each of them accused someone of lying. Claude claimed that Jacques was lying, Jacques accused Dick of lying, and Dick persuaded the investigator not to believe either Claude or Jacques. But the investigator quickly brought them to clean water, without asking them a single question. Which of the witnesses spoke the truth?

Solution. Consider the statements:

(Claude is telling the truth);

(Jacques is telling the truth);

(Dick is telling the truth.)

We do not know which of them are correct, but we do know the following:

1) either Claude told the truth, and then Jacques lied, or Claude lied, and then Jacques told the truth;

2) either Jacques told the truth, and then Dick lied, or Jacques lied, and then Dick told the truth;

3) either Dick told the truth, and then Claude and Jacques lied, or Dick lied, and then it is not true that both other witnesses lied (i.e. at least one of these witnesses told the truth).

Let us express these statements in the form of a system of equations:

The condition of the problem will be fulfilled if these three statements are simultaneously true, which means that their conjunction is true. Let's multiply these equalities (i.e. take their conjunction)

But

if and only if

, a

... Therefore, Jacques is telling the truth, and Claude and Dick are lying.

Any -term operation, denoted, for example,

, will be fully determined if it is established for what values of the statements

the result will be true or false. One of the ways to specify such an operation is to fill in the table of values:

In the table of meanings of a statement formed from the simplest statements

, there is lines. The value column also has positions. Therefore, there is

different options for filling it out, and, accordingly, the number of all -term operations is

... At

the number of single-term operations is 4, for

the number of binomials is 16, for

the number of triples is 256, etc.

Let's consider some special types of formulas.

The formula is called elementary conjunction if it is the conjunction of variables and negation of variables. For example, the formulas ,

,

,

- elementary conjunctions.

A formula that is a disjunction (possibly one-term) of elementary conjunctions is called disjunctive normal form (dn. f.). For example, the formulas ,

,

.

Theorem 1(on the reduction to dn. f.). For any formula , which is d. n. f. ...

This theorem and the following Theorem 2 will be proved in the next subsection. Applying these theorems, one can standardize the form of logical formulas.

The formula is called elementary disjunction if it is a disjunction of variables and negation of variables. For example, the formulas

,

,

etc.

A formula that is a conjunction (possibly one-term) of elementary disjunctions is called conjunctive normal form (Ph.D.). For example, the formulas

,

.

Theorem 2(on the reduction to c. n. f.). For any formula you can find an equivalent formula , which is c. n. f.

Assignment 35

One person went to work with a salary of $ 1,000 a year. During the discussion of the conditions for admission, he was promised that in case of a good job, an increase would be made to his salary. Moreover, the amount of the increase can be chosen from two options at your discretion: in one case, an increase of $ 50 every six months, starting from the second half, was offered, in the other - $ 200 every year, starting from the second. Having given freedom of choice, employers wanted to not only try to save on salaries, but also check how quickly the new employee was thinking. Thinking for a minute, he confidently named the conditions for the increase.

Which option was preferred?

Assignment 36

Once the investigator had to simultaneously interrogate three witnesses: Claude, Jacques and Dick. Their testimonies contradicted each other, and each of them accused someone of lying. Claude claimed that Jacques was lying. Jacques accused Dick of lying, and Dick persuaded the investigator not to believe either Claude or Jacques. But the investigator quickly led them to clean water, without asking them a single question.

Which of the witnesses spoke the truth?

Assignment 37

Terrible misfortune, inspector, said the museum employee. “You can't imagine how excited I am. I'll tell you everything in order. I stayed at the museum today to work and put our financial affairs in order. I was just sitting at this desk and looking through the accounts, when suddenly I saw a shadow on the right side. The window was open.

And you didn't hear any rustle? the inspector asked.

Absolutely none. The radio was playing music, and besides, I was too keen on what I was doing. Taking my eyes off the heat and, I saw that a man jumped out of the window. I immediately turned on the overhead light and found that two boxes with the most valuable collection of coins, which I took to my office for work, had disappeared. In a terrible state: after all, this collection is valued at 10 thousand marks.

You believe that I really am; believe your fabrications?

The inspector said irritably. “No one has ever misled me, and you will not be the first.

How did the inspector know that they were trying to trick him?

Assignment 38

The body of the missing person was found wrapped in a sheet bearing a laundry number tag. A family that used such tags was identified, however, during the verification process it turned out that the members of this family did not know and did not have any contact with the deceased and his relatives. No other evidence of their involvement in the murder was established.

Did you make mistakes in the completeness and correctness of the information received when checking?

Assignment 39

Potapov, Shchedrin, Semenov serve in the aviation unit. Konovalov and Samoilov. Their specialties are: pilot, navigator, flight mechanic, radio operator and weather forecaster.

Determine what specialty each of them has if the following facts are known.

Shchedrin and Konovalov are not familiar with the control of the aircraft;

Potapov and Konovalov are preparing to become navigators; the apartments of Shchedrin and Samoilov are located next to the apartment of the radio operator;

Semyon, while in the rest house, met Shchedrin and the forecaster's sister: Potapov and Shchedrin in their free time play chess with a flight mechanic and a pilot; Konovalov, Semyonov and the forecaster are fond of boxing; the radio operator is not fond of boxing.

Assignment 40

The aunt, who was waiting for her nephew, the inspector, rushed to meet him, not hiding her impatience.

Some woman just now; she snatched my purse with money and disappeared at once.

Most likely she hid in the very savings bank where you were, - said the inspector. - Let's try to find her.

Indeed, the aunt immediately saw her bag, which was on the bench between the two women. She was revealed. When the inspector took a close look at the bag, both women, noticing this, got up and walked to the other end of the room. The purse remained on the bench.

But I don't know which one stole my bag. I didn’t have time to see her, ”said my aunt.

Well, that's nonsense, ”said the nephew. - We'll interrogate both of them, but I think that the one who stole your bag was ...

Which?

Assignment 41

Having received a message that a gray Chevrolet with a number starting with a six had hit a woman and disappeared, the inspector and his assistant drove to the villa of the gentleman, whose car seemed to match the description. In less than half an hour they were there.

A gray Chevrolet stood in front of the house. Seeing the police, the owner went down to them right in his pajamas.

Yanikuda didn’t leave today, ”he said after listening to the inspector. - Yes, and could not: yesterday I lost the ignition key, and the new one will be ready only on Friday.

The assistant, having managed to inspect the car in the meantime, whispered to the inspector:

Apparently, he is telling the truth. There are no signs of a collision on the car.

The inspector, leaning on the hood of the car, answered:

This does not mean anything, the blow was not strong, because the victim is alive. And your alibi, sir, seems extremely suspicious to me. Why are you trying to hide from me that you have just arrived here in this very car?

What gave the inspector a reason to suspect the master of a lie?

Assignment 42

The president of the firm informs the investigator about the theft committed from his house.

Arriving at work, I remembered that I had forgotten the necessary documents at home. I gave the key to the home safe to my assistant and sent him for a folder with documents. We have been working together for a long time, I have long trusted him, and often sent him home to take something from the safe. This time, shortly after leaving, he called me on the phone and said that upon entering the room, he saw that the door of the wall safe was open, and papers were scattered throughout the office. I arrived home and found that, in addition to the scattered documents, jewelry and money had disappeared from the safe.

The assistant's testimony: “When I arrived, the butler let me in and went up to the second floor of the apartment. Entering the office, he found papers scattered on the floor and an open safe door. I immediately called my boss on the phone and reported what I had seen. After that, I jumped out onto the landing of the stairs and called the butler. When I shouted, a maid appeared from the downstairs living room and asked what was the matter. I told her what I saw. At her call, the butler came running from the yard. To my question, they said that no one came to the apartment after the owner left and they did not hear any noise in the house. "

The butler explained: “After the owner left in the morning, I did my usual work on the ground floor and did not see anyone or hear anything unusual. The maid did not come out of the kitchen in front of me. When a long-familiar employee of our owner arrived, he went to the stairs to the second floor, and went out into the courtyard. A few minutes later the cook called me and I entered the house, where the assistant said about the theft from the owner's office. "

The maid said that after breakfast she was in the kitchen, did not go out anywhere, and only, hearing the cry of the assistant, went into the living room. The assistant said about the theft in the house and asked to know the butler.

To the investigator's question, the assistant replied that he did not touch anything in the office, except for the phone, and did not rearrange it. The butler and the maid said that they did not go to the office at all.

Upon examination in the office, the investigator did not find any fingerprints on the office door, safe door, objects and the telephone on the table. Having examined the lock of the safe door, the specialist did not find traces of any object or an extraneous key on its details.

The following sequence of steps can be distinguished in solving logical problems.

1. Select elementary (simple) statements from the problem statement and designate them with letters.

2. Write down the condition of the problem in the language of logic algebra, combine simple statements into complex ones using logical operations.

3. Compose a single logical expression for the requirements of the problem.

4. Using the laws of logic algebra, try to simplify the resulting expression and calculate all of its values, or build a truth table for the expression under consideration.

5. Choose a solution - set of values simple statements, in which the constructed logical expression is true.

6. Check whether the obtained solution satisfies the condition of the problem.

Example:

Objective 1:“Trying to remember the winners of last year's tournament, five former spectators of the tournament stated that:

1. Anton was second and Boris was fifth.

2. Victor was second and Denis was third.

3. Gregory was the first and Boris the third.

4. Anton was the third, and Evgeniy - the sixth.

5. Victor was third and Evgeniy was fourth.

Subsequently, it turned out that each viewer made a mistake in one of his two statements. What was the true distribution of places in the tournament. "

1) Let us denote through the first letter in the name of the participant of the tournament, and - the number of the place that he has, i.e. we have.

2) 1. ; 3. ; 5. .

3) A single logical expression for all the requirements of the task:.

4) In the formula L carry out equivalent transformations, we get:.

5) From point 4 it follows:,,,,.

6) Distribution of places in the tournament: Anton was the third, Boris - the fifth, Victor - the second, Grigory - the first, and Evgeniy - the fourth.

Objective 2:“Ivanov, Petrov, Sidorov were brought to trial on charges of robbery. The investigation established:

1. if Ivanov is not guilty or Petrov is guilty, then Sidorov is guilty;

2. If Ivanov is not guilty, then Sidorov is not guilty.

Is Ivanov guilty? "

1) Consider the statements:

A: "Ivanov is guilty" V: "Petrov is guilty" WITH: "Sidorov is guilty."

2) The facts established by the investigation:,.

3) Single logical expression:. It is true.

Let's compose a truth table for him.

A	V	WITH	L

Questions and tasks.

1. Make up the RCC for the formulas:

2. To simplify the RCS:

3. Based on the given switching scheme, construct a logical formula corresponding to it.

4. Check the equivalence of the DCS:

5. Construct a circuit of three switches and a light bulb in such a way that the light will light up only when exactly two switches are in the "on" position.

6. Using this conductivity table, construct a circuit of functional elements with three inputs and one output, which implements the formula.

x	y	z	F

7. Analyze the diagram shown in the figure and write down the formula for the function F.

1) Claude claimed that Jacques was lying.

2) Jacques accused Dick of lying.

3) Dick tried to persuade the investigator not to trust either Claude or Jacques.

But the investigator quickly brought them to clean water, without asking them a single question. Which of the witnesses spoke the truth?

9. Determine which of the four students passed the exam, if it is known that:

1) If the first passed, then the second passed.

2) If the second passed, then the third passed or the first did not pass.

3) If the fourth did not pass, then the first passed, and the third did not pass.

4) If the fourth passed, then the first passed.

Assignment 35

Which option was preferred?

Assignment 36

Once the investigator had to simultaneously interrogate three witnesses: Claude, Jacques and Dick. Their testimonies contradicted each other, and each of them accused someone of lying. Claude claimed that Jacques was lying. Jacques accused Dick of lying, and Dick persuaded the investigator not to believe either Claude or Jacques. But the investigator quickly led them to clean water, without asking them a single question.

Which of the witnesses spoke the truth?

Assignment 37

And you didn't hear any rustle? the inspector asked.

You believe that I really am; believe your fabrications?

The inspector said irritably. “No one has ever misled me, and you will not be the first.

How did the inspector know that they were trying to trick him?

Assignment 38

Did you make mistakes in the completeness and correctness of the information received when checking?

Assignment 39

Potapov, Shchedrin, Semenov serve in the aviation unit. Konovalov and Samoilov. Their specialties are: pilot, navigator, flight mechanic, radio operator and weather forecaster.

Determine what specialty each of them has if the following facts are known.

Shchedrin and Konovalov are not familiar with the control of the aircraft;

Potapov and Konovalov are preparing to become navigators; the apartments of Shchedrin and Samoilov are located next to the apartment of the radio operator;

Assignment 40

The aunt, who was waiting for her nephew, the inspector, rushed to meet him, not hiding her impatience.

Some woman just now; she snatched my purse with money and disappeared at once.

Most likely she hid in the very savings bank where you were, - said the inspector. - Let's try to find her.

But I don't know which one stole my bag. I didn’t have time to see her, ”said my aunt.

Well, that's nonsense, ”said the nephew. - We'll interrogate both of them, but I think that the one who stole your bag was ...

Which?

Assignment 41

A gray Chevrolet stood in front of the house. Seeing the police, the owner went down to them right in his pajamas.

Yanikuda didn’t leave today, ”he said after listening to the inspector. - Yes, and could not: yesterday I lost the ignition key, and the new one will be ready only on Friday.

The assistant, having managed to inspect the car in the meantime, whispered to the inspector:

Apparently, he is telling the truth. There are no signs of a collision on the car.

The inspector, leaning on the hood of the car, answered:

What gave the inspector a reason to suspect the master of a lie?

Assignment 42

The president of the firm informs the investigator about the theft committed from his house.

Syllogisms Once an investigator had to interrogate three witnesses at the same time: Claude, Jacques and Dick. Their testimonies contradicted each other, and each of

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