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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Target: study various methods of evidence (direct reasoning, the method of "by contradiction" and reverse reasoning), illustrating the methodology of reasoning. Consider Method mathematical induction.
Theoretical material Proof methods
When proving theorems, logical reasoning is used. Proofs in computer science are an integral part of checking the correctness of algorithms. The need for proof arises when we need to establish the truth of a statement of the form (AB). There are several standard types of evidence, including the following:
Direct reasoning (proof).
We assume that statement A is true and show the validity of B. This method of proof excludes the situation when A is true and B is false, since it is in this and only in this case that the implication (AB) takes on a false value (see Table).
Thus, direct proof goes from considering the arguments to proving the thesis, i.e., the truth of the thesis is directly substantiated by the arguments. The scheme of this proof is as follows: from the given arguments (a, b, c,...) a provable thesis must necessarily follow q.
This type of evidence is carried out in judicial practice, in science, in controversy, in the writings of schoolchildren, in the presentation of material by a teacher, etc.
Examples:
1. The teacher in the lesson at direct evidence the thesis “The people - the creator of history”, shows; Firstly that the people are the creator of material wealth, Secondly, substantiates the enormous role of the popular masses in politics, explains how in the modern era the people are actively fighting for peace and democracy, thirdly, reveals its great role in the creation of spiritual culture.
2. In chemistry lessons, direct evidence of the combustibility of sugar can be presented in the form of a categorical syllogism: All carbohydrates are combustible. Sugar is a carbohydrate. Sugar is flammable.
In the modern fashion magazine “Burda”, the thesis “Envy is the root of all evil” is substantiated with the help of direct evidence with the following arguments: “Envy not only poisons people’s daily life, but can also lead to more serious consequences, therefore, along with jealousy, anger and hatred, undoubtedly one of the worst character traits. Creeping up imperceptibly, envy hurts painfully and deeply. A person envies the well-being of others, suffers from the consciousness that someone is more fortunate.
2. Reverse reasoning(proof) . We assume that statement B is false and show the fallacy of A. That is, in fact, we directly check the truth of the implication ((not B) (not A)), which, according to the table, is logically equivalent to the truth of the original statement (A B).
3. The method "by contradiction".
This method is often used in mathematics. Let a- a thesis or theorem to be proved. We assume by contradiction that a false, i.e. true nope(or ). From the assumption 
we deduce consequences that contradict reality or previously proven theorems. We have 
, wherein 
-
false, therefore, its negation is true, i.e. 
,
which, according to the law of two-valued classical logic ( 
→a) gives a. So it's true a, which was to be proved.
There are a lot of examples of proof “by contradiction” in the school mathematics course. So, for example, the theorem is proved that from a point lying outside a straight line, only one perpendicular can be dropped to this straight line. By contradiction, the following theorem is also proved: “If two lines are perpendicular to the same plane, then they are parallel.” The proof of this theorem begins directly with the words: “Assume the contrary, i.e., that the lines AB and CD not parallel."
Proof "from the contrary" (in Latin "reductio ad absurdum") is characterized by the fact that the very process of proving an opinion is carried out by refuting the opposite judgment. An antithesis can be proven false by establishing the fact that it is incompatible with a true proposition.
Usually such a method is visually demonstrated using a formula where A is the antithesis and B is the truth. If the solution turns out that the presence of variable A leads to results different from B, then A is proved to be false.
Proof "by contradiction" without the use of truth
There is also an easier proof of the falsity of the "opposite" - the antithesis. Such a formula-rule says: “If a contradiction arose in the formula when solving with variable A, A is false.” It does not matter whether the antithesis is negative or affirmative. In addition, a simpler way of proving by contradiction contains only two facts: the thesis and antithesis, truth B is not used. This greatly simplifies the proof process.
Apagogue
In the process of proving by contradiction (which is also called "reduction to absurdity"), apagogy is often used. This logical device, the purpose of which is to prove the inaccuracy of any judgment so that a contradiction is revealed directly in it or in the consequences arising from it. The contradiction can be expressed in the identity of obviously different objects or as conclusions: conjunction or pairs B and not B (true and not true).
Reception of evidence "by contradiction" is often used. In many cases, it is not possible to prove the incorrectness of a judgment in any other way. In addition to apagogy, there is also a paradoxical form of proof by contradiction. This form was used in Euclid's "Elements" and represents the following rule: A is considered proven if it is possible to demonstrate the "true falsity" of A.
Thus, the process of evidence by contradiction (it is also called indirect and apogogical evidence) is as follows. An opposite opinion is put forward, consequences are deduced from this antithesis, among which the false is sought. They find evidence that among the consequences there is indeed a false one. From this it is concluded that the antithesis is wrong, and since the antithesis is wrong, the logical conclusion follows that the truth is contained in the thesis.
Theorem is a statement whose validity is established by reasoning. The reasoning itself is called the proof of the theorem.
Theorem inverse to this is a theorem in which the condition is the conclusion of this theorem, and the conclusion is its condition. For instance: Theorem: In an isosceles triangle, the angles at the base are equal. Inverse theorem: If two angles are equal in a triangle, then it is isosceles.
Consequence is a statement that is derived directly from the theorem. For instance: consequence from the isosceles triangle height theorem is: The median of an isosceles triangle drawn to the base is the height and the bisector.
Proof by contradiction is as follows:
1) An assumption is made opposite to what needs to be proved.
2) Then, starting from the assumption, by reasoning, they come to a contradiction either with the condition or with the known fact.
3) On the basis of the contradiction obtained, it is concluded that the assumption is false, which means that what was required to be proved is true.
A sign of equality of right triangles along the hypotenuse and leg.
If the hypotenuse and leg of the same right triangle are respectively equal to the hypotenuse and leg of another right triangle, then such triangles are equal.
Given :
DABC - right / corner
BC=B 1 C 1
Prove:
DABC = DA 1 B 1 C 1
Proof:
1. Let's apply to DABC to DA 1 B 1 C 1, so that vertex A is aligned with vertex A 1, vertex B is aligned with vertex B 1, and vertices C and C 1 are on different sides from straight line AB.
2. Since AB \u003d A 1 B 1 Þ, they will coincide.
3. ÐSA 1 С 1 = 90 0 + 90 0 = 180 0 ÞÐSA 1 С 1 – developed and points С, А 1 and С 1 – lie on one straight line.
4. Consider DСВС 1 – r/b (ВС= В 1 С 1 by condition)Þ РС = РС 1 (by property)
5. Thus, DABC \u003d DA 1 B 1 C 1 - along the hypotenuse and the acute angle. (h.t.d.)
Ticket number 9.
Perpendicular lines. Perpendicular to a line.
Perpendicular lines- these are two straight lines that, when intersected, form four right angles. (Show in the figure)
Perpendicular to a straight line is a line segment drawn from a point to a line at a right angle. The point of intersection of the segment and the line is called the base of the perpendicular (shown in the figure)
Theorems:
1) From a point not lying on a line, one can draw a perpendicular to this line, and moreover, only one.
2) Two lines perpendicular to the same line do not intersect.
Sign of an isosceles triangle.
If two angles are equal in a triangle, then it is isosceles.
Given:
РА = ∠С
Prove:
DABC - r / w
Proof:
1. Mentally copy DABC and turn the copy over - we get DABC.
2. Let's superimpose DCBA on DABC, so that the vertex C of the copy is aligned with the vertex A of DABC.
3. Since РА = РС (by condition) Þ РА of the copy and РС of the triangle will coincide when superimposed, so also РС of the copy and РА of the triangle will coincide when superimposed.
4. The segment CB of the copy will be superimposed on the ray AB of the triangle and the segment AB of the copy will be superimposed on the ray CB of the triangle.
5. Since two lines can have only one common point of intersection ⇒
m. In 1 will coincide with point B and ⇒ AB will be combined with CB ⇒ AB = CB
6. From the fact that AB \u003d CB ⇒ by definition, ΔABC is isosceles (p.t.d.)
Ticket number 10.
Isosceles triangle.
Triangle whose two sides are equal is called isosceles. Equal sides are called sides, and the third party basis. (show in picture)
Property of an isosceles triangle: In an isosceles triangle, the angles at the base are equal. (Show in the figure)
Sign of an isosceles triangle: If two angles are equal in a triangle, then it is isosceles. (show in picture)
Isosceles triangle height theorem: The height of an isosceles triangle drawn to the base is the median and the bisector. (show in picture)
Consequences from the theorem on the height of an isosceles triangle:
1) The median of an isosceles triangle drawn to the base is the height and the bisector. (show in picture)
2) The bisector of an isosceles triangle drawn to the base is the height and the median. (show in picture)
The explanatory dictionary of mathematical terms defines the proof from contrary theorem, opposite to the converse theorem. “Proof by contradiction is a method of proving a theorem (sentence), which consists in proving not the theorem itself, but its equivalent (equivalent), opposite inverse (reverse to opposite) theorem. Proof by contradiction is used whenever the direct theorem is difficult to prove, but the opposite inverse is easier. When proving by contradiction, the conclusion of the theorem is replaced by its negation, and by reasoning one arrives at the negation of the condition, i.e. to a contradiction, to the opposite (the opposite of what is given; this reduction to absurdity proves the theorem.
Proof by contradiction is very often used in mathematics. The proof by contradiction is based on the law of the excluded middle, which consists in the fact that of the two statements (statements) A and A (negation of A), one of them is true and the other is false./ Explanatory dictionary of mathematical terms: A guide for teachers / O. V. Manturov [and others]; ed. V. A. Ditkina.- M.: Enlightenment, 1965.- 539 p.: ill.-C.112/.
It would not be better to declare openly that the method of proof by contradiction is not a mathematical method, although it is used in mathematics, that it is a logical method and belongs to logic. Is it valid to say that proof by contradiction is "used whenever a direct theorem is difficult to prove", when in fact it is used if, and only if, there is no substitute for it.
Deserves special attention and a characteristic of the relation to each other of direct and inverse theorems. “An inverse theorem for a given theorem (or to a given theorem) is a theorem in which the condition is the conclusion, and the conclusion is the condition of the given theorem. This theorem in relation to the converse theorem is called the direct theorem (initial). At the same time, the converse theorem to the converse theorem will be the given theorem; therefore, the direct and inverse theorems are called mutually inverse. If the direct (given) theorem is true, then the converse theorem is not always true. For example, if a quadrilateral is a rhombus, then its diagonals are mutually perpendicular (direct theorem). If the diagonals in a quadrilateral are mutually perpendicular, then the quadrilateral is a rhombus - this is not true, i.e., the converse theorem is not true./ Explanatory dictionary of mathematical terms: A guide for teachers / O. V. Manturov [and others]; ed. V. A. Ditkina.- M.: Enlightenment, 1965.- 539 p.: ill.-C.261 /.
This characterization of the relationship between direct and inverse theorems does not take into account the fact that the condition of the direct theorem is taken as given, without proof, so that its correctness is not guaranteed. The condition of the inverse theorem is not taken as given, since it is the conclusion of the proven direct theorem. Its correctness is confirmed by the proof of the direct theorem. This essential logical difference between the conditions of the direct and inverse theorems turns out to be decisive in the question of which theorems can and which cannot be proved by the logical method from the contrary.
Let's assume that there is a direct theorem in mind, which can be proved by the usual mathematical method, but it is difficult. Let's formulate it in general view v short form So: from A should E . Symbol A has the meaning given condition theorem accepted without proof. Symbol E is the conclusion of the theorem to be proved.
We will prove the direct theorem by contradiction, logical method. The logical method proves a theorem that has not mathematical condition, and logical condition. It can be obtained if the mathematical condition of the theorem from A should E , supplement with the opposite condition from A it does not follow E .
As a result, a logical contradictory condition of the new theorem was obtained, which includes two parts: from A should E and from A it does not follow E . The resulting condition of the new theorem corresponds to the logical law of the excluded middle and corresponds to the proof of the theorem by contradiction.
According to the law, one part of the contradictory condition is false, another part is true, and the third is excluded. The proof by contradiction has its own task and goal to establish exactly which part of the two parts of the condition of the theorem is false. As soon as the false part of the condition is determined, it will be established that the other part is the true part, and the third is excluded.
According to explanatory dictionary mathematical terms “proof is reasoning, during which the truth or falsity of any statement (judgment, statement, theorem) is established”. Proof contrary there is a discussion in the course of which it is established falsity(absurdity) of the conclusion that follows from false conditions of the theorem being proved.
Given: from A should E and from A it does not follow E .
Prove: from A should E .
Proof: The logical condition of the theorem contains a contradiction that requires its resolution. The contradiction of the condition must find its resolution in the proof and its result. The result turns out to be false if the reasoning is flawless and infallible. The reason for a false conclusion with logically correct reasoning can only be a contradictory condition: from A should E and from A it does not follow E .
There is no shadow of a doubt that one part of the condition is false, and the other in this case is true. Both parts of the condition have the same origin, are accepted as given, assumed, equally possible, equally admissible, etc. In the course of logical reasoning, not a single logical feature has been found that would distinguish one part of the condition from the other. Therefore, to the same extent, from A should E and maybe from A it does not follow E . Statement from A should E may be false, then the statement from A it does not follow E will be true. Statement from A it does not follow E may be false, then the statement from A should E will be true.
Therefore, it is impossible to prove the direct theorem by contradiction method.
Now we will prove the same direct theorem by the usual mathematical method.
Given: A .
Prove: from A should E .
Proof.
1. From A should B
2. From B should V (according to the previously proved theorem)).
3. From V should G (according to the previously proved theorem).
4. From G should D (according to the previously proved theorem).
5. From D should E (according to the previously proved theorem).
Based on the law of transitivity, from A should E . The direct theorem is proved by the usual method.
Let the proven direct theorem have a correct converse theorem: from E should A .
Let's prove it by ordinary mathematical method. The proof of the inverse theorem can be expressed in symbolic form as an algorithm of mathematical operations.
Given: E
Prove: from E should A .
Proof.
!. From E should D
1. From D should G (by the previously proved inverse theorem).
2. From G should V (by the previously proved inverse theorem).
3. From V it does not follow B (the converse is not true). That's why from B it does not follow A .
In this situation, it makes no sense to continue the mathematical proof of the inverse theorem. The reason for the situation is logical. It is impossible to replace an incorrect inverse theorem with anything. Therefore, this inverse theorem cannot be proved by the usual mathematical method. All hope is to prove this inverse theorem by contradiction.
In order to prove it by contradiction, it is required to replace its mathematical condition with a logical contradictory condition, which in its meaning contains two parts - false and true.
Inverse theorem claims: from E it does not follow A . Her condition E , from which follows the conclusion A , is the result of proving the direct theorem by the usual mathematical method. This condition must be retained and supplemented with the statement from E should A . As a result of the addition, a contradictory condition of the new inverse theorem is obtained: from E should A and from E it does not follow A . Based on this logically contradictory condition, the converse theorem can be proved by the correct logical reasoning only, and only, logical opposite method. In a proof by contradiction, any mathematical actions and operations are subordinate to logical ones and therefore do not count.
In the first part of the contradictory statement from E should A condition E was proved by the proof of the direct theorem. In the second part from E it does not follow A condition E was assumed and accepted without proof. One of them is false and the other is true. It is required to prove which of them is false.
We prove with the correct logical reasoning and find that its result is a false, absurd conclusion. The reason for a false logical conclusion is the contradictory logical condition of the theorem, which contains two parts - false and true. The false part can only be a statement from E it does not follow A , in which E accepted without proof. This is what distinguishes it from E statements from E should A , which is proved by the proof of the direct theorem.
Therefore, the statement is true: from E should A , which was to be proved.
Conclusion: only that converse theorem is proved by the logical method from the contrary, which has a direct theorem proved by the mathematical method and which cannot be proved by the mathematical method.
The conclusion obtained acquires an exceptional importance in relation to the method of proof by contradiction of Fermat's great theorem. The overwhelming majority of attempts to prove it are based not on the usual mathematical method, but on the logical method of proving by contradiction. The proof of Fermat Wiles' Great Theorem is no exception.
In other words, Gerhard Frey suggested that the equation of Fermat's Last Theorem x n + y n = z n
, where n > 2
, has solutions in positive integers. The same solutions are, by Frey's assumption, the solutions of his equation
y 2 + x (x - a n) (y + b n) = 0
, which is given by its elliptic curve.
Andrew Wiles accepted this remarkable discovery of Frey and, with its help, through mathematical method proved that this finding, that is, Frey's elliptic curve, does not exist. Therefore, there is no equation and its solutions that are given by a non-existent elliptic curve. Therefore, Wiles should have concluded that there is no equation of Fermat's Last Theorem and Fermat's Theorem itself. However, he takes the more modest conclusion that the equation of Fermat's Last Theorem has no solutions in positive integers.
It may be an undeniable fact that Wiles accepted an assumption that is directly opposite in meaning to what is stated by Fermat's Last Theorem. It obliges Wiles to prove Fermat's Last Theorem by contradiction. Let's follow his example and see what happens from this example.
Fermat's Last Theorem states that the equation x n + y n = z n , where n > 2
According to the logical method of proof by contradiction, this statement is preserved, accepted as given without proof, and then supplemented with a statement opposite in meaning: the equation x n + y n = z n , where n > 2 , has solutions in positive integers.
The hypothesized statement is also accepted as given, without proof. Both statements, considered from the point of view of the basic laws of logic, are equally admissible, equal in rights and equally possible. By correct reasoning, it is required to establish which of them is false, in order to then establish that the other statement is true.
Correct reasoning ends with a false, absurd conclusion, the logical cause of which can only be a contradictory condition of the theorem being proved, which contains two parts of a directly opposite meaning. They were the logical cause of the absurd conclusion, the result of proof by contradiction.
However, in the course of logically correct reasoning, not a single sign was found by which it would be possible to establish which particular statement is false. It can be a statement: the equation x n + y n = z n , where n > 2 , has solutions in positive integers. On the same basis, it can be the statement: the equation x n + y n = z n , where n > 2 , has no solutions in positive integers.
As a result of the reasoning, there can be only one conclusion: Fermat's Last Theorem cannot be proven by contradiction.
It would be a very different matter if Fermat's Last Theorem were an inverse theorem that has a direct theorem proved by the usual mathematical method. In this case, it could be proven by contradiction. And since it is a direct theorem, its proof must be based not on the logical method of proving by contradiction, but on the usual mathematical method.
According to D. Abrarov, Academician V. I. Arnold, the most famous contemporary Russian mathematician, reacted to Wiles's proof "actively skeptical". The academician said: “this is not real mathematics - real mathematics is geometric and has strong links with physics.” The academician's statement expresses the very essence of Wiles' non-mathematical proof of Fermat's Last Theorem.
By contradiction, it is impossible to prove either that the equation of Fermat's Last Theorem has no solutions, or that it has solutions. Wiles' mistake is not mathematical, but logical - the use of proof by contradiction where its use does not make sense and does not prove Fermat's Last Theorem.
Fermat's Last Theorem is not proved even with the help of the usual mathematical method if in it given: the equation x n + y n = z n , where n > 2 , has no solutions in positive integers, and if required to prove: the equation x n + y n = z n , where n > 2 , has no solutions in positive integers. In this form, there is not a theorem, but a tautology devoid of meaning.