The common method of proving theorem brocade set

The following methods of proving theorems are mainly summed up in the following ways:

1) Direct proof: by proving that when P is true, Q must be true for the proof of p->q. 2) Contradiction: The rebuttal method is an indirect proof method, the use of conditional statement p->q equivalent to its inverted ￢q->￢p fact, in other words, by proving that Q is false when p must be false to prove that P->q is true. It is very effective to use the counter-evidence when it is not easy to find direct proofs. In the case of absurdity, it is assumed that the conclusion of the conditional statement is false and that the direct proof method indicates that the premise must be false. 3) to the fallacy proof: the attribution of fallacy is also an indirect proof method, assuming we want to prove that p is true, suppose we can find the Paradox Q makes ￢p->q true, because Q is False, ￢p->q is true, we can draw ￢p necessarily false, which means P is true. So our goal becomes how to find the Paradox Q, in order to help us prove that P is true. Because no matter what Proposition R is, R-^￢r is contradictory. In other words, if we can prove that a proposition r,￢p-> (R ^￢r) is true, it proves that P is true. This type of proof is called the proof of the fallacy. The fallacy can also be used to prove conditional statements. In such a proof, first of all assume the negation of the conclusion. Then apply theorem premise and conclusion negation to get a contradictory formula. Therefore, it is possible to rewrite the proof of the conditional statement into a fallacy. 4) Proof of exhaustion: The result obtained by examining the results established by a series of all cases. 5) Sub-case proof: decomposition of the situation into a proof covering all possible individual cases. A poor lift proves to be a special type of sub-case proof. 6) Without loss of generality: Suppose that a proof can be proved by reducing the need to prove the case to prove a law. That is, by proving the theorem in one of the cases, the other series of cases are demonstrated through simple changes. 7) Counter example: Make P (x) a false element x. 8) The existence of the tectonic proof: to prove that the element with a certain nature exists, by means of display to find such elements. 9) The existence of non-tectonic proof: to prove that the element with a specific nature exists, but not to look for such elements. A common method for proving non-structural proof is to use the normalized proof. 10) Proof of uniqueness: An element that proves to be of a particular nature exists only. In addition, there are many important proof methods are: Mathematical induction, Cantor Diagonalization method, counting demonstration method and so on. There is not much elaboration here.  Here are a few examples to rehearse the above methods: before giving the exercise, we give a couple of related definitions:An integer n is an even number, if there is an integer k so that n = 2k; integer n is odd, if there is an integer k makes n = 2k + 1. If there is an integer p and q (q≠0) that makes R = p/q, then the real r is the rational number. Real numbers that are not rational numbers are called irrational numbers. If there is an integer b, which causes a = B2, then the integer A is a full square number.  Exercise:1, Proof: if n is odd, then N2 is odd. 2, Proof: if M and n are both full squared, then MN is also a complete square number. 3, the proof: two rational number and is the rational number. 4, Proof: If M+n and n+p are even, where m, N and p are integers, then m+p is even. Direct. 5, Proof theorem: If 3n+2 is odd, then n is odd. 6, Proof: If n is an integer and N2 is odd, then n is an odd number. 7, prove: If N=ab, where A and B is a positive integer, then a≤√n or b≤√n. 8, Proof: If n is the total square, then the n+2 will not be completely squared. Contrary 9, 10, 11, 、、、:-)13, Proof: If x is irrational number, then 1/x is irrational number. Contrary 14. Proof: At least 4 days in any 22 days belong to the same day of one weeks. 15, Proof: √2 is irrational number. 16, Proof: When n is a positive integer, and n≤4, (n+1) ^3≥3^n. 17, Proof: Within 100, a continuous positive integer is the full power of only 8 and 9 (the full power number is that it can be written as Na, where A is greater than 1 integers). 18, Proof: When n is an integer, there is n2≥n. 19. Proof: The last digit of any complete square number is: 0, 1, 4, 5, 6 or 9. 20. Proof: There is no solution for integer x, y,x2 + 3y2 = 8. 21, Proof: (x + y) R < XR + yr. Here x, Y is the positive real number, R is the real number of the 0<r<1.  the workaround for each problem is not unique, and the available methods are:Direct: 1, 2, 3, 4 the contrary: 5, 6, 7, 8, 13 of the fallacy: 5, 14, 15 poor lifting: 16, 17: 18, 19, 20 without exception: Proof of existence:proof of the existence of tectonics:proves that there is a positive integer that can be represented in two different ways as a cubic and of a positive integer. after a lot of calculations, such as using computer search, you can find 1729 = 103 + 93 = 123 + 13. Because 1729 satisfies the question set request, the proof. proof of the existence of non-tectonic:proving that there are irrational numbers x and y makes xy a rational number.  Proof Strategy:forward and backward push (backtracking):given two different positive real numbers x and y, the arithmetic mean is (x + y)/2, whose geometric mean is √xy, proving that the arithmetic mean is always greater than the geometric average.  assume that the two people play a game, taking turns from the first 15 pieces of stone heap each fetch 1, 2 or 3 stones. The man who takes the last stone wins. Proof: No matter how the B is taken, the first armor can win the game.  modification of the existing proof: according to √2 is an irrational number of the proof process, speculated that √3 is irrational number (note: Need to use the knowledge of theory).  another: Some questions about the strategy of proof of action1. Can we fill the standard checkerboard (8x8) with a domino (a rectangle, consisting of two squares)? 2. Can we use dominoes to fill the standard chessboard with a corner/adjacent angle/diagonal?  Concepts related to terminology:theorem: A mathematical assertion that can be proved to be true. Axiom: Often as the basis of the proof theorem, and assumed to be a true proposition. cyclic argumentation or stealing a thesis: one or more steps based on the reasoning of the correctness of the proposition to be proved.

The common method of proving theorem brocade set