The theorems are proved by contradiction. Logic and proof. Proof: direct, reverse, by contradiction. Method of mathematical induction

THE METHOD BY OPPOSITE (hereinafter referred to as MOP) is a scientific and applied method named after the outstanding Ukrainian educator, founder of a number of scientific schools and directions, Vasily Kozmich Opposite. V.K. Protivny was born on February 29, 1513 according to the old style in the village of Nizhnie Lopuhi near Chernigov. Since childhood, Vasya was a weak and frail boy and constantly, starting from kindergarten, was subjected to ridicule from his peers, which later predetermined his bad character.

Subsequently, the words “do everything to spite others” actually became the motto of V.K. Nasty’s life. So, to spite everyone, he left his native Kholmogory and entered Moscow State University. Lomonosov (and not to the Suvorov School, as his father wanted), to spite everyone he never married anyone (although his grandmother Vasilisa Opposite found him at least 14 brides in his entire life), to spite everyone, citing the mushroom season, he did not receive The Fields Medal is the highest award in mathematics.

The essence of the method from the opposite can be conveyed by the following points:
1. An incorrect assumption is made.
2. It turns out what follows from this assumption on the basis of known knowledge.
3. A dead end is reached.
4. The correct conclusion is drawn that the incorrect assumption is incorrect.

Many scientists, philosophers, researchers and even artists became ardent adherents of the ideas of the Ukrainian enlightener. For example, lobotomy was used for the first time in medical practice, when an attempt was made to resolve the eternal philosophical dispute about the primacy of matter or consciousness with the help of a medical experiment. Thus, V.K. Protivny’s student Lobachevsky created non-Euclidean geometry, so his admirer Tchaikovsky wrote a hymn to alternative love - the “Blue Danube” waltz, and so on.

The method from the opposite is often used nowadays in various areas of human life. For example, Moscow Mayor Luzhkov successfully uses it to cultivate the artistic taste of Muscovites by installing Tsereteli sculptures in the city. The leadership of the Central Internal Affairs Directorate, using this method, decided to find the killers of the famous journalist Politkovskaya, since other methods, due to the particular complexity of the case, did not produce results. Moscow police officers armed with MOP know that by consistently identifying all those not involved, they will automatically follow the trail of the murderers.

The whole life and even death of V.K. Opposite was a vivid illustration of his method. The scientist tragically died on February 29, 1613 at the age of 112, having hanged himself in spite of his grandmother Vasilisa Nastya, who did not allow Vasily Kozmich to try the jam from the refrigerator. Despite their ambivalent attitude towards V.K. Nasty because of his bad character, most scientists and researchers still consider MOP one of the most powerful weapons modern science in general and mathematics in particular.
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Vasily Kozmich Nasty, outstanding Ukrainian educator (1513 - 1613)

I express my gratitude

Often, when proving theorems, the proof method is used by contradiction. The essence of this method helps to understand the riddle. Try to solve it.

Imagine a country in which a person sentenced to death is asked to choose one of two identical-looking papers: one says “death”, the other says “life”. Enemies slandered one resident of this country. And so that he would have no chance of escape, they made it so that “death” was written on the back of both pieces of paper, from which he must choose one. Friends found out about this and informed the convict. He asked not to tell anyone about this. He pulled out one of the pieces of paper. And he stayed to live. How did he do it?

Answer. The condemned man swallowed the piece of paper he had chosen. To determine which lot fell to him, the judges looked at the remaining piece of paper. It said "death" on it. This proved that he was lucky, he pulled out a piece of paper on which was written: “life.”

As in the case described in the riddle, when proving, only two cases are possible: it is possible... or it is impossible... If you can be convinced that the first is impossible (on the piece of paper that the judges got, it is written: “death”), then you can immediately conclude that the second possibility is valid (on the second piece of paper it is written: “life”).

Proof by contradiction is carried out as follows.

1) Establish what options are in principle possible when solving a problem or proving a theorem. There can be two options (for example, are the lines in question perpendicular or perpendicular); There may be three or more answer options (for example, what kind of angle is obtained: acute, straight or obtuse).

2) They prove it. That none of the options that we need to discard can be fulfilled. (For example, if you need to prove that the lines are perpendicular, we look at what happens if we consider non-perpendicular lines. As a rule, it is possible to establish that in this case any of the conclusions contradicts what is given in the condition and is therefore impossible.

3) Based on the fact that all undesirable conclusions were discarded and only one (desirable) remained unexamined, we conclude that it is the correct one.

Let's solve the problem using proof by contradiction.

Given: lines a and b such that any line that intersects a also intersects b.

Using the method of proof by contradiction, prove that a ll b.

Proof.

Only two cases are possible:

1) straight lines a and b are parallel (life);

2) lines a and b are not parallel (death).

If we manage to exclude the undesirable case, then we can only conclude that the second of two possible cases occurs. To eliminate the undesirable case, let's think about what happens if lines a and b intersect:

By condition, any line that intersects a also intersects b. Therefore, if it is possible to find at least one line that intersects a, but does not intersect b, this case will need to be discarded. You can find as many such lines as you like: it is enough to draw through any point K a straight line a, except point M, a straight line KS parallel to b:

Since one of the two is discarded possible cases, one can immediately conclude that a ll b.

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False, we thereby justify the truth of the opposite position - the thesis. For example, a doctor, convincing a patient that he does not have the flu, may reason as follows: “If you really were sick with the flu, you would have a fever, a stuffy nose, etc. But there is none of this. Therefore, there is no flu.” Proof of a certain position by contradiction is the truth of this position, based on the demonstration of the falsity of the “opposite” (contradictory) position and the excluded third.
The general D. from item is described as follows. It is necessary to prove some A. In the process of proof, the opposite statement not-A is first formulated and it is assumed to be true: let's say that A is false, then not-A must be true. Then, from this supposedly true antithesis, consequences are derived - until either one is obtained, or one that clearly contradicts the known true statement. If it is shown that not-A is false, then the truth of thesis A is thereby justified ( cm. PROOF).

Philosophy: Encyclopedic Dictionary. - M.: Gardariki. Edited by A.A. Ivina. 2004 .

(lat. reduction ad absurdum), a type of evidence, in addition to “proving” a certain proposition (thesis proof) is carried out through a judgment that contradicts it - an antithesis. Refutation of the antithesis is achieved by establishing the fact of its incompatibility with k.-l. obviously true judgment. This form of D. from item corresponds to track. proof scheme: if B is true and A implies the falsity of B, then A is false. Another, more general D. from item - this is by refutation (reasons for falsity) antithesis according to the rule: having admitted A, they deduced , therefore - not-A. Here A can be either an affirmative or a negative proposition. In the latter case, D. from n. is based on the law of double negation. In addition to those indicated above, there is a “paradoxical” form of D. from p., which was already used in Euclid’s “Elements”: A can be considered proven if it can be shown that A follows even from the assumption of the falsity of A.

Philosophical encyclopedic dictionary. - M.: Soviet Encyclopedia. Ch. editor: L. F. Ilyichev, P. N. Fedoseev, S. M. Kovalev, V. G. Panov. 1983 .

EVIDENCE TO THE CONTRARY

Lit.: Tarski A., Introduction to logic and methodology of deductive sciences, trans. from English, M., 1948; Asmus V.F., The doctrine of logic about proof and refutation, [M.], 1954; Kleene S.K., Introduction to Metamathematics, trans. from English, M., 1957; Church A., Introduction to mathematics. logic, trans. from English, [vol.] 1, M., 1960.

Philosophical Encyclopedia. In 5 volumes - M.: Soviet Encyclopedia. Edited by F. V. Konstantinov. 1960-1970 .

See what “EVIDENCE BY CONTRARY” is in other dictionaries:

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One type of indirect evidence. * * * EVIDENCE BY CONTRARY EVIDENCE BY CONTRARY, one of the types of indirect evidence (see INDIRECT EVIDENCE) ... Encyclopedic Dictionary

Proof by contradiction- (lat. reduction ad absurdum) a type of evidence in which the validity of a certain judgment (thesis of evidence) is carried out through a refutation of the antithesis judgment that contradicts it. Refutation of the antithesis is achieved by... ... Research activities. Dictionary

EVIDENCE TO THE CONTRARY- (lat. reductio ad absurdum) a type of evidence in which the validity of a certain judgment (thesis of evidence) is carried out through a refutation of the antithesis judgment that contradicts it. Refutation of the antithesis is achieved by... ... Vocational education. Dictionary

See: Indirect evidence... Dictionary of Logic Terms

- (lat. reductio ad absurdum) a type of Proof in which “proof” of a certain judgment (thesis of evidence) is carried out through a refutation of the antithesis judgment that contradicts it. In this case, the refutation of the antithesis is achieved... ... Great Soviet Encyclopedia

Often, when proving theorems, the proof method is used by contradiction. The essence of this method helps to understand the riddle. Try to solve it.

Proof by contradiction is carried out as follows.

3) Based on the fact that all undesirable conclusions were discarded and only one (desirable) remained unexamined, we conclude that it is the correct one.

Let's solve the problem using proof by contradiction.

Given: lines a and b such that any line that intersects a also intersects b.

Using the method of proof by contradiction, prove that a ll b.

Proof.

Only two cases are possible:

1) straight lines a and b are parallel (life);

2) lines a and b are not parallel (death).

Since one of the two possible cases is rejected, one can immediately conclude that a ll b.

Still have questions? Don't know how to prove the theorem?
To get help from a tutor -.
The first lesson is free!

blog.site, when copying material in full or in part, a link to the original source is required.

The opposite method

Apagogy- a logical technique that proves the inconsistency of an opinion in such a way that either in it itself, or in the consequences that necessarily follow from it, we discover a contradiction.

Therefore, apogogical proof is indirect proof: here the prover first turns to the opposite position to show its inconsistency, and then, according to the law of exclusion of the third, draws a conclusion about the validity of what was required to be proven. This type of proof is also called reduction to absurdity. Its essential component is the argument that the third does not exist, i.e., that apart from the opinion, the validity of which must be proven, and the second, opposite to it, which serves as the starting point of the proof, no third fact is allowed. Therefore, indirect evidence comes from a fact that denies the position, the validity of which must be proven.

The theorems are proved by contradiction. Logic and proof. Proof: direct, reverse, by contradiction. Method of mathematical induction

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Examples

See also

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