1, reasoning:

(1) Reasonable reasoning: inductive reasoning and analogical reasoning are based on the existing facts, and after observation, analysis, comparison and association, they are induced and analogized, and then a guess is put forward. This kind of reasoning is called reasonable reasoning.

(1) inductive reasoning:

Definition 1: Some objects of a certain food have certain characteristics, and all objects of this food have the inference of these characteristics, or the inference that some facts summarize a general conclusion, which is called inductive reasoning, or induction for short.

Two characteristics:

* induction is based on special phenomena to infer general phenomena, so the conclusion drawn by induction is beyond the scope of the premise;

* induction is based on a number of known and endless phenomena to infer unknown phenomena, so the conclusion is speculative;

* The premise of induction is special circumstances, so induction is based on observation, experience and experiment;

* Induction is to draw regular conclusions based on observation, experience, experiments and analysis of limited data.

Step 3:

* Observe, analyze and summarize limited data;

* put forward a conclusion with regularity, that is, conjecture;

* Test the conjecture.

② Analogical reasoning:

Definition 1: It is inferred that two kinds of objects have some known characteristics similar to one of them, and the other kind of objects also have these characteristics, which is called analogical reasoning, or analogy for short.

Two characteristics:

* Analogy is to infer the nature of things being studied from the nature of things that people have mastered, and to analogize new results on the basis of old knowledge;

* analogy is to infer the special attributes of another thing from the special attributes of one thing;

* The result of analogy is speculative, not necessarily reliable, but it has the function of discovery.

Step 3:

* Find out the similar features that can be accurately expressed between two objects;

* use the known characteristics of one object to infer the characteristics of another object, so as to get a guess;

* Test the conjecture.

(2) Deductive reasoning:

① Definition: Derive a special conclusion from the general principle, which is called deductive reasoning.

② Deductive reasoning is from general to special reasoning;

③? Syllogism? Is a general model of deductive reasoning, including:

General conclusion of known major premise;

The special situation studied by the minor premise;

Conclusion The judgment of special circumstances is based on general principles.

④? Syllogism? Understanding the basis of reasoning from the point of view of set;

If all elements in the set M have the attribute P and S is a subset of M, then all elements in S also have the attribute P. ..

(3) The differences and connections between perceptual reasoning and deductive reasoning:

(1) induction is the reasoning from special to general;

(2) Analogy is the reasoning from special to special;

Deductive reasoning is the reasoning from general to special.

④ From the conclusion of reasoning, the conclusion of reasonable reasoning is not necessarily correct and needs to be proved; The conclusion of deductive reasoning must be correct.

⑤ Deductive reasoning is an important thinking process to prove mathematical conclusions and establish a mathematical system; The discovery of mathematical conclusions and proof ideas mainly depends on reasonable reasoning.

The proof of high school mathematics (1) directly proves:

(1) synthesis method: using known conditions and some mathematical definitions, theorems, axioms, etc. After a series of reasoning and argumentation, the conclusion to be proved is finally drawn. This method of proof is called synthesis. Comprehensive method, also called forward deduction method, is characterized by: causality? .

(2) Analysis method: Starting from the conclusion to be proved, gradually seek the sufficient conditions for its establishment, until finally, reduce the conclusion to be proved to determine an obviously established condition (known conditions, definitions, theorems, axioms, etc.). ). This method of proof is called analysis. Analysis, also known as reduction to absurdity, is characterized by: holding the cause? .

③ Mathematical induction:

A, the axiom of mathematical induction:

When n takes the first value,

(e.g.

When the conclusion is correct;

2 suppose when

The conclusion is correct when n=k+ 1

So, from the proposition

All the first positive integers n hold.

2. Description:

* The two steps of mathematical induction are indispensable, and the problem must be proved in strict accordance with the steps when using mathematical induction;

The axiom of mathematical induction is the basis of proving the proposition of natural numbers.

(2) Indirect proof (reduction to absurdity): Assuming that the original proposition is not established, through correct reasoning, contradictions are finally drawn, which shows that the hypothesis is wrong, thus proving that the original proposition is established. This method of proof is called reduction to absurdity.

Prove a proposition by reducing to absurdity often adopts the following steps:

(1) The conclusion of the hypothetical proposition is not valid;

(2) Reasoning, in which one of the following situations occurs: contradiction with known conditions; Contradicting an axiom or theorem;

(3) Due to the appearance of the above contradictions, can we assert that the original hypothesis is valid? The conclusion is not valid? Is wrong;

④ Affirm that the conclusion of the original proposition is correct.

Namely. Reverse the absurd conclusion?

Four reasoning methods to solve the problem of proof in senior high school 1. Reasonable reasoning

1. inductive reasoning is reasoning from the local to the whole and from the individual to the general. In induction, we must first deform some known individuals and find out the relationship between them, so as to draw general conclusions;

2. Analogical reasoning is the reasoning from special to special, and it is the reasoning between two similar objects, one of which has certain properties and the other has similar properties. In analogy, we should fully consider the reasoning process of known object properties, and then deduce the properties of analogy objects through analogy.

Second, deductive reasoning.

Deductive reasoning is from general to special reasoning, and the mathematical proof process is mainly carried out through deductive reasoning. As long as the major premise, minor premise and reasoning form of deductive reasoning are correct, then its conclusion must be correct, so we must pay attention to the correctness and integrity of the reasoning process.

Third, direct proof and indirect proof.

Direct proof is relative to indirect proof, and synthesis method and analysis method are two common direct proofs. Generally speaking, the synthesis method uses known conditions and some mathematical definitions, theorems and axioms, and after a series of reasoning and argumentation, it finally deduces that the conclusion to be proved is valid. This method of proof is called synthesis (or forward deduction, from cause to effect). Generally speaking, the analysis method is to start from the conclusion to be proved and gradually seek the sufficient conditions for its establishment, until finally the conclusion to be proved is reduced to judging an obviously established condition (known conditions, theorems, definitions, axioms, etc.). ). This method of proof is called analysis.

Indirect proof is relative to direct proof, and reduction to absurdity is a common method of indirect proof. Assuming that the original proposition is not established, through correct reasoning, the contradiction is finally obtained, so the assumption is wrong, thus proving that the original proposition is established. This method of proof is called reduction to absurdity.

Fourthly, mathematical induction.

A special method to prove the proposition related to natural number n in mathematics is mainly used to study the mathematical problems related to positive integers, and is often used to prove the establishment of equality and the general term formula of sequence in high school mathematics.