Direct prove



in mathematics, proof is in a specific axiom system, according to certain rules or standards, The process of deriving certain propositions is derived from the axiom and theorem. Compared to evidence, mathematical proof is generally relying on interpretation, rather than relying on natural summary and empirical ideas. This proposition is also called the theorem in the system.

Mathematical proof is established on logic, but usually contains some degree of natural language, so some vague parts may be generated. In fact, mathematical proof in writing in text, can be considered as non-form logic in most situations. Within the context of proven, consider the proofs written in a pure formal language. This difference has led to most of the experimental experience and mathematical experience and folk mathematics of past mathematics. Mathematics philosophy of philosophy focuses on the role of language and logic in mathematical proof, and as a language.


Mathematical proof includes two different concepts. The first is a non-formulated proof: a rigorous argument written in natural language, used to convince audience or readers to accept the trueity of a certain americ or argument. Since this certificate uses a natural language, it will depend on the criteria on non-formalization proven to be in terms of listeners or readers' understanding of the topics. Non-formal proof has occurred in most applications, such as some parts of science lectures, verbal debates, primary education or higher education. Sometimes non-formal proof is called "formal", but this is only the stringency of the argument. When the word "formal proof" is used, it refers to another completely different proof - formal proof.

In the logic, formal proven is not written in natural language, but writes in formalized language: this language contains characters composed of characters in a given alphabet string. It is proven to be a sequence of finite lengths composed of these strings. This definition allows people to talk about the "proven" in strict sense, without involving any logic blur. The formalization and theoretical theory of research proven is called the proof. Although theoretically, each non-formal proof can be transformed into formal proof, but in practice, this will be done. The main application of formal certification is to explore the general nature of proven, or to illustrate certain propositions.

Common proof Tips

Direct proven

Direct proven also known as logic interpretation, refers to the use of logical deduction from recognized facts or axiom The authenticity method of propositions need to be proven. Direct proven method generally uses predicate logic, and uses existence of quantifiers or full weighing quantifiers. The main proof mode has affirmative frontal equivalents, negative reusstal words, false words, three-segment expression, and words three-stage formula, etc. For example, it is necessary to prove proposition: "Any odd multiplication is still odd", you can directly prove that as follows:

Any odd number can be written into

, where
is an integer. Two odd numbers can be written in
, where
, and
are integers. They have the product of
. All numbers of double plus 1 that can be written into an integer is odd.
is an integer, so
is an odd number. Certificate.

Configuration method

Construction method is generally used to demonstrate the presence orifold, and the demonstration of the constructive method is called constructive proof. The specific practice is to construct an example of the specific properties required in the proposition to show the presence of objects or concepts of this property. You can also construct an antique example to prove that the proposition is wrong. For example, prove the proposition "2" 2 "Power Decimation is not always the number of" "," can be used:

only need to prove that there is a certain number of "section> , make 2

The power is not the number of prime numbers. To this end, the number of pilotes 11.2 is 11.2 power minus one equals
is not the number of prime. Therefore the proposition certificate.

Direct prove

Some constructs demonstrate that the examples of the proposition requirements are not directly constructed, but construct some auxiliary tools or objects make the problem easier to solve. A typical example is the structure of the Li Yapovino function in the stability theory of normal differential equations. Another way to add auxiliary lines or auxiliary graphics in many geometric certificates.

Non-constructor proves

and constructance proves that relative is a proven method for proven that the existence of the proposition requirements is proved to prove the presence of the proposition. For example, the following example:

proposition: There are two unreasonable numbers and
so that
is a quoter.

Proof: Consider

, if it is a rational number, the proposition is certified. If
is not a counted number, it must be unreasonable. Consider its

is a constant, the proposition is still correct.

, in any case, there is a non-generous number that meets the proposition requirements.

does not give {\ displayStyle X ^ {y}} in this proven to two specific unreasonable numbers.

Voluntary Law

Voluntary Law is a method that lists all the situations contained in propositions to prove propositions. For example, "the square of only 25 and 76 in all two digits is to be used as the mantissa", only the square of the two digits: 10 to 99, one by one, can be verified. Obviously, the conditions for using the exhaustive law are the possible conditions contained in propositions, otherwise they cannot be listed one by one.

Changes in the case

In the predicate logic, if a proportion of the priival and predicate are deny, the result is called the original question Changes . If the position of the subject is exchanged and the predicate, the result is called transposition . First change replacement position is called Change position , and the first translocation replacement is referred to as transposition . For example, "all S is P" exchange plays "all not p is not S". The challenge method refers to the use of a change and transposition, and a proposition is changed to a proposition with its logic equivalent, so as long as the latter proves the original proposition. For example, to prove that the pigeon cage prototype: "If there is more than N pigeons in n pigeon cage, then there are at least one or two or more pigeons in the cage," can be reached with the equivalent of its equivalent: "If there is one of the N pigeon cages, there is a pigeon, then N pigeon cages have N a pigeon." The latter is obvious.

Case Analysis

Case analysis or classification discussion refers to the method of dividing the conclusions into limited cases, and then proof one by one.

Calculated twice

is two times a two kinds of two kinds of different numbers, although there are different but correct analysis, obtaining two methods of different but equal expressions Commonly used to prove constant equations.


Anti-confidence is an ancient certificate of proof, its idea is: I want to prove that a proposition is a holiday question, then it is true that the proposition is true. In this case, if the logically contradiction can be caused by the correct and effective reasoning (such as the proposition itself is false, then the proposition is both true and false contradictions), it can prove that the original proposition is false. Contaminaries and row of row is the logical basis of anti-confidence law. The benefits of the anti-counterfeit law are in turn assume that the proposition is true, equal to a known condition, so that the proven to the topic is often helpful.

example: Proof Proposition "

is not a rationale".


is not a quota.

Proof: Suppose

is a rigor number, then there is a positive integer
as an integer. You may wish to set
to the smallest (depending on the principle of good sequence, there must be minimal
). Consider
is a positive integer than
, but
is also an integer. This is the smallest contradiction with {\ DisplayStyle A}. So the root number 2 is not a rational number.

Mathematical Senament method

mathematical induction method is a skill that proves that the number of infinite propositions. To demonstrate a bunch of propositions in natural number n first prove that proposition 1 was established, and proved that the proposition n was established n +1 Established, it is true for all propositions. In Piyano axiom system, the axioming definition of natural number set includes mathematical inductance. There are many variants of mathematics, such as the number of natural numbers other than 0, prove that the proposition n +1 is also established when the proposition is set to less than or equal to n Reverse induction method, decrease in definition method, etc. The broad mathematical inductive method can also be used to demonstrate a general basis structure, such as a tree in a collection. In addition, the over-limiting method provides a skill that handles an infinite proposition, is the promotion of mathematics.

example: Proof for all natural numbers

, proposition

when n = 1, left = 1, right =

assumes a natural number

, proposition
, the following proof
Established, ie:
< Section>

So, the number of natural numbers

, there are

other prove

intuitive certificate

Intuitive proof or visualization proof is a method of applying an intuitive means of image or form to prove propositions. This type of prove can reach the effects that do not have proven to be proven. A icon of the challenge proof.

Computer-assisted proofs

Computer Assistance proof is a proof method in the twentieth century. Until the twentieth century, people have always believed that any mathematical proof should be inspected by a level of mathematicians to confirm its correctness. However, today's mathematicians have been able to use a computer to prove the theorem, and complete the difficult calculations in humans. In 1976, the four-color theorem proved to be a classic example of computer-aided proof. The approach method is to reduce the infinite species of the map to 1936 states, and verify each possible situation by a computer. Many mathematicians have cautious attitudes for computer prove, because many proves too long, can't verify directly by human hand. In addition, the error on the algorithm, the mistakes when the input, even the errors that occur during the computer operation may result in errors.

Proof of

Sometimes the end of the certificate will add Q.E.D. three letters, which is the abbreviation of the Romance Quod Erat Demonstrandum, meaning "prove finishes". The current certificate is now the symbol, usually ■ (solid black square), called "Tombstone" or "Halmos Symbol" (because Paul Harmos first uses this). The tombstone is sometimes hollow. Another simple method is to write "Proven", "Shown" or "prove" and the like.

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