Logic and proof. Proof: direct, reverse, by contradiction. Method of mathematical induction

The lesson can start with the teacher's story.

Vashchenko N.M., at the lesson

V Ancient Greece all speakers were taught geometry. On the door of the school was written: "He who does not know geometry, let him not enter here." Why? Yes, because geometry teaches to prove. A person's speech is convincing only when he proves his conclusions. In their reasoning, people often use the method of proof, which is called "by contradiction".

Let us give examples of such proofs.

Example 1 The scouts were given the task of finding out if there was an enemy tank column in the given village. The reconnaissance commander reports: if there was a tank column in the village, then there would be traces of caterpillars, but we did not find them.

Reasoning scheme. It is required to prove: there is no column. Suppose there is a column. Then there must be traces. Contradiction - there are no traces. Conclusion: the assumption is incorrect, which means that there is no tank column.

Example 2 The doctor after examining a sick child says:

“The child does not have measles. If he had measles, then there would be a rash on his body, but there is no rash.”

The doctor's reasoning was also carried out according to the above scheme.

The question is asked: “What is the essence of the method of proving by contradiction?” - and a table is posted (Table 5).

By contradiction it is possible to solve previously known problems.

1. Given: a||b, lines c and a intersect. Prove: lines c and b intersect.

Proof.

1) Assume that b||c.

2) Then it turns out that two different lines a and b pass through the point O (the point of intersection of lines a and c), which are parallel to line b.

3) This contradicts the axiom of parallel lines.

Conclusion: it means that our assumption is wrong, but what was required to be proved is true, i.e., that the lines bis intersect.

2. Given: A, B, C - points of the line a, AB = 5 cm, AC = 2 cm, BC = 7 cm. Prove:

Proof.

1) Suppose point C lies between points A and B.

2) Then, according to the axiom of measuring the segments AB = AC + CBA

3) This contradicts the condition: AB \u003d AC + CB, since AB \u003d 5 cm, AC + C5 \u003d 9 cm.

Conclusion: point C does not lie between points A and B.

3. Given: AB - half-line, C AB, AC< АВ. Prove:

Proof.

1) Suppose point B lies between points A and C.

2) Then, according to the axiom of measuring the segments AB + BC = AC, i.e. AB

3) This contradicts the condition of the problem: AS<АВ.

Conclusion: point B does not lie between points A and C.

Problem solving is written in notebooks. In order for students to learn the essence of the method of proving by contradiction, as well as in order to save time when solving problems, you can use hint cards that are made of thick paper and inserted into plastic bags. The student must fill in the missing places on the plastic wrap. The tape records are easily erased, and therefore the cards can be used repeatedly.

The card looks like:

Assume the opposite of what is required to be proved, i.e.

It follows from the assumption that (based on ……

We get a contradiction.

This means that our assumption is wrong, but what was required to be proved is true, i.e.

Homework:

n. "Proof by contradiction" § 2 to the words: "Let's explain this ...".

1. Prove that if MN = 8 m, MK = 5 m, NK- 10 m, then the points M, N and K do not lie on one straight line.

2. Prove that if<(ab) = 100°, <(be) - 120°, то луч с не проходит между сторонами угла (ab).

3. Prove Theorem 1.1 by contradiction.

What is the method of proof "by contradiction"?

The essence of the method of proof by contradiction consists in two stages. The first in the proof of the EXISTENCE of the proof and the second in the proof of the UNIQUENESS of the proof. Clumsily described, but wanted to say the following. When proving theorems using this method, you need to show that there is a solution to a given problem or theorem, and then prove that this solution will be unique. This is not the only method used in proving theorems, but as a mathematical and logical tool it is not without interest.

The method of proof by contradiction is used not only in mathematics, although it has become quite widespread there as a tool for proving individual problems and theorems.

In fact, this is a logical method of proving any statements, which can be applied in any field of knowledge. Even in the humanities and social sciences. Simply, in the technical sciences we are dealing with numbers, and many people are convinced just by the presence of these icons, and in the world of logic we operate with conclusions that can never be considered absolute truth.

We studied this method of proof at school in the middle classes, when we take as a basis some statement that cannot be proven in any way, instead they take the directly opposite statement to it, prove that it is false, therefore, what we cannot prove is true, And this is the only correct solution to this issue.

In life we talk about something, we cannot prove it, but we give an opposite example and prove that it is wrong: money was stolen from the cache, Vasya and Petya knew about it, but Petya had an alibi - he went to the dacha for the whole week, which means , Vasya stole the money.

By the method of proof by contradiction called the way in which an unprovable truth becomes true, only because something else is always wrong - and this is exactly what is provable. Accordingly, as a result of this method, albeit indirectly, we proved unprovable truth

This law is based on the law of double negation, if A is not true, then A is true.

For example, you think you have an ulcer. Your doctor, in order to refute this judgment, proves to you by refuting what you are sure of, that is, your statement and says that you do not have an ulcer, since gastroscopy showed that there is no damage in the stomach cavity, you do not lose weight and you can eat everything whatever you want.

Standard technique, for example, in mathematics. We need to prove statement A. And this is difficult. Then they take the opposite statement B, and prove that it is false. It follows that A is true. The same is true in life. A simple example: someone says: Mr. X is a thiefquot ;. His opponent: But how to prove it? First: Suppose he is an honest manquot ;. Second: Yes, this is a mockery of chickens! First: So we proved that X is a thief :)))

The opposite method

Apagogue- a logical device that proves the inconsistency of an opinion in such a way that either in 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 proposition in order to show its inconsistency, and then, according to the law of elimination of the middle, concludes that what was required to be proved is correct. This kind of proof is also called reduction to absurdity. Its essential property is the argument that the third does not exist, i.e., that apart from the opinion, the validity of which must be proved, and the second, opposite to it, which serves as the starting point of the proof, no third fact is allowed. Therefore, circumstantial evidence comes from a fact that denies the proposition, the validity of which is required to be proved.

Examples

- 'BEING AND TIME' ('Sein und Zeit', 1927) Heidegger's main work. The creation of B.i.V. is traditionally believed to have been influenced by two books: Brentano's The Meaning of Being According to Aristotle and Husserl's Logical Investigations. The first of them…… History of Philosophy: Encyclopedia

- (from the late Latin intuitio, from the Latin intueor I look closely) a direction in the justification of mathematics and logic, according to which the ultimate criterion for the acceptability of the methods and results of these sciences is visually meaningful intuition. All math... Philosophical Encyclopedia

Mathematics is usually defined by listing the names of some of its traditional branches. First of all, this is arithmetic, which deals with the study of numbers, the relationships between them and the rules for working with numbers. The facts of arithmetic admit various ... ... Collier Encyclopedia

A term that previously united various sections of mathematics. analysis related to the concept of an infinitesimal function. Although the method of infinitesimals (in one form or another) was successfully used by scientists of ancient Greece and medieval Europe to solve ... ... Mathematical Encyclopedia

- (from lat. absurdus ridiculous, stupid) absurdity, contradiction. In logic, A. is usually understood as a contradictory expression. In such an expression, something is affirmed and denied at the same time, as, for example, in the statement “Vanity exists and vanity ... ... Philosophical Encyclopedia

Often when proving theorems, the method of proof is used. contrary. The essence of this method helps to understand the riddle. Try to unravel 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 inhabitant of this country. And so that he had no chance to escape, they made it so that on the back of both pieces of paper, from which he must choose one, “death” was written. Friends found out about this and informed the convict. He asked not to tell anyone about it. Pulled out one of the papers. And stayed to live. How did he do it?

Answer. The convict swallowed the piece of paper he chose. To determine which lot fell to him, the judges looked into the remaining piece of paper. On it was written: "death." This proved that he was lucky, he pulled out a piece of paper on which was written: "life."

As in the case that the riddle tells about, only two cases are possible during the proof: it is possible ... or it is impossible ... If you can make sure that the first is impossible (on the piece of paper that the judges got, it is written: “death”), then we can immediately conclude that the second possibility is valid (on the second piece of paper it is written: "life").

The 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, whether the lines under consideration are perpendicular or not); There can be three or more answer options (for example, what angle is obtained: acute, straight or obtuse).

2) Prove. That none of the options that we need to reject can be performed. (For example, if it is necessary 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 therefore is impossible.

3) Based on the fact that all undesirable conclusions are discarded and only one (desirable) remains unconsidered, we conclude that it is he who is correct.

Let's solve the problem using proof by contradiction.

Given: lines a and b are 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) lines a and b are parallel (life);

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

If it is possible to exclude the undesirable case, then it remains to conclude that the second of the two possible cases takes place. To discard the undesirable case, let's think about what happens if lines a and b intersect:

By assumption, 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 must be discarded. You can find as many such lines as you like: it is enough to draw through any point K of the line a, except for the point M, the line KS parallel to b:

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

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False, we thereby substantiate the truth of the opposite position - the thesis. For example, a doctor, convincing a patient that he is not sick with the flu, may reason as follows: “If you really were sick with the flu, then you would have a fever, a stuffy nose, and so on. But there is none of that. Therefore, there is no flu." The proof of a certain proposition by contradiction is the truth of this proposition, based on the demonstration of the falsity of the "opposite" (contradictory) proposition and the excluded third.
General D. from p. is described as follows. It is necessary to prove some A. In the process of proof, the opposite proposition not-A is first formulated and it is assumed that it is true: suppose that A is false, then not-A must be true. Then, from this allegedly true antithesis, consequences are drawn - until either it turns out, or one that explicitly contradicts the known true statement. If it is shown that not-A is false, then the truth of the thesis A is justified ( cm. PROOF).

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

(lat. reduc-tio ad absurdum), type of proof, with chrome "proof" of a certain judgment (of proof thesis) is carried out through a judgment that contradicts it - antithesis. The refutation of the antithesis is achieved by establishing the fact of its incompatibility with c.-l. obviously true judgment. This form of D. from p. corresponds track. proof scheme: if B is true and A implies B is false, then A is false. Another, more general D. from p. is by refuting (reasons for falsehood) antithesis according to the rule: having admitted A, they deduced , therefore - not-A. Here A can be either affirmative or negative. In the latter case, D. from p. is based on and 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. editors: L. F. Ilyichev, P. N. Fedoseev, S. M. Kovalev, V. G. Panov. 1983 .

PROOF FROM THE CONTRARY

Lit.: Tarsky A., Introduction to the logic and methodology of deductive sciences, trans. from English, M., 1948; Asmus VF, The doctrine of logic about proof and refutation, [M.], 1954; Kleene S. K., Introduction to Metamathematics, trans. from English, M., 1957; A. Church, 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 "PROOF FROM THE CONTRARY" is in other dictionaries:

- (proof by contradiction) A proof in which the recognition of the initial premise as incorrect leads to a contradiction. That is, the assumption of the fallacy of the original premise allows you to simultaneously prove any statement and refute it; … Economic dictionary

One type of circumstantial evidence... Big Encyclopedic Dictionary

This article lacks links to sources of information. Information must be verifiable, otherwise it may be questioned and removed. You can ... Wikipedia

One of the types of circumstantial evidence. * * * PROOF FROM THE CONTRARY PROOF FROM THE CONTRARY, one of the types of circumstantial evidence (see INDIRECT PROOF) ... encyclopedic Dictionary

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

PROOF FROM THE CONTRARY- (lat. reductio ad absurdum) a type of evidence in which the validity of a certain judgment (proof thesis) is carried out through the refutation of the antithesis judgment that contradicts it. The refutation of the antithesis is achieved by ... ... Professional education. Dictionary

See: Circumstantial evidence... Glossary of Logic Terms

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

Logic and proof. Proof: direct, reverse, by contradiction. Method of mathematical induction

Examples

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