The theorem is the converse theorem, proof by contradiction. Proof by contradiction
Proof by contradiction is a powerful and frequently used method in mathematics. Having assumed that some fact (object) is true (exists), and having come to a contradiction, we conclude that the fact is false (the object does not exist). Let's look at a few examples.
Euclid's theorem about infinity prime numbers is the classic and simplest argument by contradiction:
There is no largest prime number.
: Let this not be so, and the largest prime number exists. Let's build a number. It is not divisible by any or more than. We have arrived at a contradiction; therefore, the largest prime number (as an object!) does not exist and there are infinitely many prime numbers.
Note that it is not necessarily prime, since its prime factor may be between and , but will still be large.
Irrationality theoremThere are no natural and such that
.
: Let it not be so. Let's reduce the common factors of , , and square everything: . It follows that it is even number, therefore it is also even and representable using some natural number, like . Substituting into the original relation, we get , and, therefore, even. But this contradicts the fact that we have reduced all common factors, which means that such factors do not exist.
The psychological persuasiveness of both evidence is beyond doubt. However, it must be remembered that having received a contradiction, we do not always prove that we want prove. A contradiction does not necessarily indicate that the original premise is incorrect. It can be given by any of the statements used in the proof. There are especially many of them in the irrationality theorem. However, they are so “obvious” that we consider the initial premise to be erroneous.
It can be seen that the proof scheme for the above theorems is the same. We show that some object does not exist if the assumption of its existence leads to a contradiction.
Barber's Problem. In a certain village, all men shave either themselves or have a barber. A barber (male) shaves only those who do not shave themselves. Let us formulate the theorem:
The barber shaves himself.
Let this not be so, and the barber does not shave himself. Then he must be shaved by a barber. So the barber shaves himself.
Having made a negation of the theorem and received a contradiction, we must come to the conclusion that the theorem is true. But it is absolutely clear that this is not so, and we can construct not only the opposite proof, but also a direct one: “if the barber shave himself, then he cannot shave at the barber...”. In this case, we again get a contradiction.
The above description of a village with strict rules is due to Bertrand Russell, as a popular formulation of the problems that arise in trying to define"the set of all those sets that do not contain themselves as their element." We deliberately presented an obvious paradox in the form of a theorem to demonstrate a simple fact:
Obtaining a contradiction in a proof by contradiction may indicate not the truth of the theorem, but the inconsistency of the objects that participate in its formulation.In other words, you cannot say: “let’s take the set of all sets...” and prove “the theorem that...” First, you need to make sure that the object that will be discussed in the theorem exists. In particular, the village described by Russell cannot exist. Of course, the question arises - “what does it mean to exist or not to exist, and where not to exist?” There is an object defined above, and we can use it when constructing new objects and theorems about them...
The point is that mathematical reasoning explicitly or implicitly proceeds from certain axioms. It is the axioms that define the properties of an object. If you change at least one axiom in a fixed system of axioms, you may end up with an object with completely different properties. It is clear that it is impossible to set axioms arbitrarily. They shouldn't be contradictory, otherwise no object will be defined. Or, in other words, an object defined by contradictory axioms does not exist.
We will discuss the elements of formal axiomatic systems in more detail in the next section, where we will again analyze the barber's problem. Now let's look at another version of the same paradox.
The Librarian's Problem. There is a library with books. Any book within its text can mention itself (for example, give its title in the list of references). Accordingly, all books can be divided into two groups. The first includes books that do not refer to themselves, and the second includes books that refer to themselves. In addition, there are two books that are catalogs of all the books in the Library. The first catalog lists all those books that do not refer to themselves, and the second, on the contrary, lists all the books that refer to themselves:
Let us now formulate the theorem:
The first directory containsin the book list itself.
Let it not be so. Then the first directory is contained in the second (all books are listed in both directories and the directory is a book). But the second directory only lists self-referencing books, and the first directory cannot be there. We have reached a contradiction, therefore the theorem is true.
If we stop at this stage, we will get a deliberately incorrect conclusion. It is clear that the first directory cannot refer to itself (it is a directory of non-self-referencing books). As in the case of the barber, we can conduct both a reverse proof (by contradiction) and a direct one. And both times you get a contradiction.
What does it say? It is clear that it is not about the truth or falsity of the theorem. Believing that two different proofs must always lead to the same thing, we are forced to conclude: Library object, with specified properties, cannot exist.
Any reference to the “naturalness” or “apparent non-contradiction” of the original definitions is not worthy of a mathematician, since these are already emotions. The only way is to try to move away from psychological formulations and evidence to formal ones.
The Liar Paradox. All mathematics consists of logical statements. Moreover, the logic of mathematics is binary. The statement "" is either true or false. There is no third option. It is this binarity that gives the mathematical proof that wonderful persuasiveness for which everything was started. Let us introduce the designation that a certain logical statement is true:
.In fact, the designation is unnecessary, since by writing down some statement as an axiom or premise, we assume its truth. However, this notation will be convenient for what follows. Let's define saying:
where "" is a logical negation sign, and after the colon comes definition approvals It is a variant of the liar's paradox: "-true if not true." Let us formulate the following theorem:Statement L is true: L=I.let L=L => True(L)=L => L=True(L)=I.
(Hereinafter “” means logical conclusion; “I” is true, “L” is false). In proof by contradiction, we have arrived at a contradiction. Therefore, the initial premise is not true and, therefore, the theorem is true. However, it is clear that this is not the case. We can carry out the proof in the forward direction.
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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Target: study various methods of evidence (direct reasoning, the method of contradiction and backward reasoning) illustrating the methodology of reasoning. Consider the method of mathematical induction.
Theoretical material Methods of evidence
When proving theorems, logical argumentation 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 truth 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 justified by the arguments. The scheme of this proof is as follows: from the given arguments (a, b, c,...) it is necessary to prove the thesis q.
This type of evidence is used in judicial practice, in science, in polemics, in the essays of schoolchildren, when a teacher presents material, etc.
Examples:
1. Teacher in class direct evidence the thesis “The people are the creator of history”, shows; Firstly that the people are the creators of material wealth, secondly, justifies the enormous role of the masses in politics, explains how in the modern era the people are actively fighting for peace and democracy, thirdly, reveals it big role in creating spiritual culture.
2. In chemistry lessons, direct evidence of the flammability of sugar can be presented in the form of a categorical syllogism: All carbohydrates are flammable. 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 everyday lives, but can also lead to more serious consequences, therefore, along with jealousy, malice and hatred, undoubtedly belongs to the worst character traits. Creeping up unnoticed, envy hurts painfully and deeply. A person envies the well-being of others, and is tormented by the knowledge 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, in a direct way we verify the truth of the implication ((not B)(not A)), which, according to the table, is logically equivalent to the truth of the original statement (AB).
3. The “by contradiction” method.
This method is often used in mathematics. Let A- a thesis or theorem that needs to be proven. We assume by contradiction that A false, i.e. true nope(or ). From the assumption 
we derive consequences that contradict reality or previously proven theorems. We have 
, while 
-
is false, that means 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 what needed to be proven.
There are many examples of proof by contradiction in the school mathematics course. Thus, for example, the theorem is proved that from a point lying outside a line, only one perpendicular can be lowered onto this line. The following theorem is also proved using the “by contradiction” method: “If two straight lines are perpendicular to the same plane, then they are parallel.” The proof of this theorem directly begins with the words: “Let us assume the opposite, i.e., that the straight lines AB And CD not parallel."
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.
____________________________________
Vasily Kozmich Nasty, outstanding Ukrainian educator (1513 - 1613)
I express my gratitude
lat. reductio ad absurdum) is a type of proof in which the validity of a certain judgment (thesis of the proof) is carried out through the refutation of a judgment that contradicts it - the antithesis. Refutation of the antithesis is achieved by establishing its incompatibility with a known true proposition. Often proof by contradiction is based on the double-valued principle.
Excellent definition
Incomplete definition ↓
EVIDENCE TO THE CONTRARY
substantiation of a judgment by refuting, by the method of “reducing to absurdity” (reductio ad absurdum), some other judgment, namely that which is a negation of the one being justified (D. from item 1 of the type) or that which is the negation of justified (D. from item 2 of the type); “reduction to absurdity” consists in deducing a s.-l. from a refuted proposition. an obviously false conclusion (for example, a formal logical contradiction), which indicates the falsity of this judgment. The need to distinguish two types of D. from clause follows from the fact that in one of them (namely, in D. from clause of the 1st type) there is a logical transition from the double negation of a judgment to the affirmation of this judgment (i.e. the so-called rule for removing double negations, allowing the transition from A to A, see Double negation laws), while in the other there is no such transition. The course of reasoning in D. from item 1 of the type: it is required to prove proposition A; for the purpose of proof, we assume that judgment A is false, i.e. that his denial is true: ? (not-A), and, based on this assumption, we logically deduce k.-l. false judgment, e.g. contradiction, – we carry out “reduction to absurdity” of judgment A; this indicates the falsity of our assumption, i.e. proves the truth of the double negative: A; application of the rule for removing double negation to A completes the proof of A’s proposition. The course of reasoning in D. from item 2 of the 2nd type: is it required to prove the proposition?; for the purpose of proof, we assume that judgment A is true and reduce this assumption to absurdity; on this basis we conclude that A is false, i.e. what's true?. Distinguishing between two types of logic from p. is important because in the so-called intuitionistic (constructive) logic the law of removing double negation does not take place, due to which reasoning from p., essentially related to the application of this logical law, is not allowed. See also Circumstantial Evidence. Lit.: Tarski?., 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?., Introduction to mathematics. logic, trans. from English, [vol.] 1, M., 1960.
