1. Analysis: the cut-in point and requirements of the analysis diagram.

2. Proof: Make auxiliary lines, comprehensively apply theorems, find out the connection between the known and the unknown, or overturn the hypothesis that the negative proposition is not valid. 3. Finishing: Standardize the answer.

Common proof methods

Divided into direct proof and indirect proof.

reductio ad absurdum

Reduction to absurdity is an ancient proof method. Its idea is that if you want to prove a proposition is false, you must assume it is true. In this case, if the original proposition leads to logical contradiction through correct and effective reasoning (for example, the proposition itself is false, so it falls into the contradiction of both truth and falsehood), or runs counter to a fact or axiom, it can be proved to be false. The law of non-contradiction and law of excluded middle are the logical basis of the reduction to absurdity. The advantage of reduction to absurdity is that it assumes that the proposition is true in turn, which is equal to one more known condition, which is often helpful for the proof of the topic.

complete induction

Mathematical induction is a skill to prove countless propositions. In order to prove a series of propositions numbered by natural number n, firstly, it is proved that proposition 1 holds, and when proposition p(n) holds, it is proved that proposition p(n+ 1) holds, then it holds for all propositions. In piano's axiomatic system, natural number set's axiomatic definition includes mathematical induction. There are many variants of mathematical induction, such as induction from natural numbers other than 0, proving that proposition p(n+ 1) holds for natural numbers less than or equal to n, backward induction, decreasing induction and so on. Generalized mathematical induction can also be used to prove general well-founded structures, such as trees in set theory. In addition, transfinite induction provides a skill to deal with uncountable infinite propositions, which is a generalization of mathematical induction.

structured approach

Construction method is generally used to prove the existence theorem, and the proof using construction method is called constructive proof. The concrete way is to construct an example with the specific properties required in the proposition to show the existence of objects or concepts with this property. You can also construct counterexamples to prove that the proposition is wrong.

Some constructive proofs do not directly construct examples that meet the requirements of the proposition, but construct some auxiliary tools or objects to make the problem easier to solve. A typical example is the construction of Lyapunov function in the stability theory of ordinary differential equations. Another example is the method of adding auxiliary lines or auxiliary graphics, which is often used in many geometric proof problems.

Unstructured proof

The opposite of constructive proof is non-constructive proof, that is, a proof method to prove the existence of the object required by the proposition without giving a specific structure.

method of exhaustion

Exhaustion is a method to prove a proposition by enumerating all the situations contained in it. Obviously, the condition of using the exhaustive method is that the possible situations contained in the proposition are limited, otherwise they cannot be enumerated one by one. For example, to prove that "only the squares of 25 and 76 in all two digits have their own mantissa", we only need to calculate the squares of all two digits: 10 to 99 and verify them one by one.