A concept of reduction to absurdity 2 Logical basis, types and steps of reduction to absurdity (1) Logical basis of reduction to absurdity (2) Types of reduction to absurdity (3) Steps of reduction to absurdity.

Scope of application of reduction to absurdity in middle school mathematics (1) Negative proposition (2) Finite proposition (3) Infinite proposition (4) Inverse proposition (5) Some existential propositions (6) Full name affirmative proposition (7) Proof of some inequality propositions (8) Basic proposition 4 Problems needing attention when using reduction to absurdity (1)

On the application of reduction to absurdity in middle school mathematics

This paper focuses on the absurd concept, whose logical basis is "Law of Contradiction" and "law of excluded middle". The types of reduction to absurdity include simple reduction to absurdity and exhaustive reduction to absurdity, and the general steps of reduction to absurdity (reversal, reduction to absurdity and conclusion). The practice of proof tells us that the following propositions are generally more convenient to prove by reduction to absurdity, negative proposition, finite proposition and infinite proposition. The problems that should be paid attention to in the application of reduction to absurdity must be correctly denied, the characteristics of reasoning must be clearly understood, and the types of contradictions must be understood. Keywords: disproof Keywords: contradictory conclusion of disproof hypothesis

There is a famous story of "bitter plum by the roadside": Once upon a time, there was a child named Wang Rong. One day, he and his children found a tree on the roadside covered with plums. The children scrambled to pick it, and only after tasting it did they know it was bitter. Only Wang Rong did not move. Wang Rong said: "If plums were not bitter, they would have been picked by passers-by, but this tree is full of plums, so plums must be bitter." In this story, Wang Rong uses a special method to discuss from the opposite side why plums are not sweet or delicious. This indirect proof is the reduction to absurdity that we will discuss below.

The concept of reduction to absurdity

Reduction to absurdity is a kind of proof method to think about problems from a negative perspective, which belongs to the category of "indirect proof", that is, affirming the topic and denying the conclusion, thus leading to contradictions and reasoning. Reduction to absurdity is one of the indirect proof methods commonly used in mathematics. The logical basis of reduction to absurdity is law of excluded middle in the basic law of formal logic. Usually, the reduction to absurdity is to deduce contradictions according to the opposite side of the proposition conclusion to be proved, so that the opposite side of the original conclusion is not established, but the original conclusion is affirmed to be true. In middle school algebra, some initial propositions, negative propositions, uniqueness propositions, inevitable propositions, propositions with conclusions in the form of "at most ..." and "at least ...", "infinite" propositions, and the proof of some inequalities can be proved by reduction to absurdity, which can achieve good results. Assuming that the opposite of proposition judgment holds, under the known conditions and the new conditions of "negative proposition judgment", through logical reasoning, we can draw a conclusion that contradicts axioms, theorems, assumptions and temporary assumptions, and thus draw the conclusion that the opposite of proposition judgment does not hold, that is, the conclusion of proposition must be correct. When a proposition is known and difficult to prove directly, the proof method to correct its inverse proposition is called reduction to absurdity. The block diagram is as follows: this theorem precedes the opposite of the problem, this problem precedes this axiom, and this definition precedes this axiom.

first

It is impossible to enumerate all individuals by exhaustive method, such as proving that there are infinite prime numbers; The number of irrational numbers is not less than that of irrational numbers.

second

If you can't prove a conclusion with what you have learned, for example, the two sides of a triangle are not equal, then the angles opposite to the two sides are not equal.

Because high school mathematics involves a wide range, this situation is more common.

third

The steps of direct proof are complicated and easy to make mistakes, which often appear on the conic curve of the solution.

reductio ad absurdum

Definition: the method of proving theorem. First, a hypothesis contrary to the conclusion in the theorem is put forward, and then a result contradictory to the known conditions is obtained from this hypothesis, thus denying the original hypothesis and affirming the theorem. Also called reduction to absurdity.

Scope of application: To prove some propositions, it is difficult to prove them positively, and the situation is more or more complicated, while the opposite is relatively simple.

Specific methods (for example):

Proposition r= under c, if a is b.

Counterevidence: if a, then B.

Prove the contradiction between b and a

For example, if you want to prove that "if P is Q" is a true proposition, you can start from denying its conclusion, that is, "non-Q", and through correct logical reasoning, you can deduce the contradiction and make "non-Q" false, that is, the original proposition is true. This method of proof is called reduction to absurdity.

First, a hypothesis contrary to the conclusion in the theorem is put forward, and then a result contradictory to the known conditions is obtained from this hypothesis.

An indirect argument of reducing to absurdity. First, the proposition that contradicts the original proposition is proved to be false, and then the original proposition is confirmed to be true according to law of excluded middle. The demonstration process can be expressed as follows:

[Verification] A (original title)

[Prove] (1) Let non-A be true (non-A is the antithesis)

(2) If it is not A, then B(B is a conclusion derived from non-A)

(3) Not B (known)

(4) Therefore, it is not non-A (negative ex post facto proof of hypothetical reasoning under sufficient conditions)

(5) So, a (non-non-non-non-A=A).

For example, when linguists think that there is no necessary connection between the sound of a language and what it represents, they use the following reduction to absurdity: "There is no necessary connection between the sound and what it represents, no"

A certain sound must represent an object. If there is any necessary connection between sound and things, then words representing the same thing in all languages in the world should have the same sound. Since the sounds of words expressing the same thing in the world are different, it can be seen that there is no necessary connection between the sounds of languages and the things they represent.

Yes "This paragraph discusses the disproof process analysis is as follows:

Title: There is no necessary connection between the sound of language and what it represents (put forward at the beginning and summarized at the end).

Counter-proposition: the combination of sound and matter is bound to be related.

Set the antithesis as true, and then deduce: "If there is any necessary connection between sound and things, all languages in the world express the same."

The sound of a word should be the same. Obviously, the latter cannot be established: "The sounds of words representing the same thing in the world are different". According to the negative formula of sufficient condition hypothesis reasoning, denying the latter will inevitably deny the former, thus proving the counter-proposition that "the combination of sound and object is inevitable."

Contact "is false. Then, according to law of excluded middle, the original proposition is proved to be true. It should be noted that the reduction to absurdity determines the truth of the original proposition by first proving the falsehood of the counter-proposition, and then deducing the truth from the falsehood. Therefore, the counter-proposition and the original proposition must be contradictory, not antagonistic. Because the judgments of opposites can be both false, that is, the falsehood of one judgment may not necessarily lead to the truth of another judgment.

Reduction to absurdity is often used in mathematics. When the topic is not easy or can't be proved from the front, we should use reduction to absurdity.

Hope to adopt