If Xiaoming had been vaccinated against influenza, he wouldn't have caught a cold.

If Xiao Li studies hard, he won't take the single-digit exam.

If Xiao Zhang is rich in Gao Shuai, then Xiao Zhang won't use Nokia.

The logic used here is that if p is not q, it is equivalent to its negative proposition, and q is not p.

If Xiao Ming has a cold, he must have never been vaccinated.

If Xiao Li got a single-digit exam, he certainly didn't study hard.

If Xiao Zhang uses Nokia, then Xiao Zhang must not be rich in Gao Shuai.

After the logical language is clear, let's take a look at how the statistical language describes the above problems:

If Xiao Li studies hard, he is 95% sure that he won't take the single digit exam.

The problem arises. There is no camera in Xiao Li's dormitory. How do you know if Xiao Li has studied hard? We can only observe the probability of Xiao Li failing the exam through the exam.

Then let's assume with the greatest goodwill that Xiao Li has really studied hard.

Write this process into statistical language:

Original hypothesis: Xiao Li studies very hard.

Alternative hypothesis: Xiao Li doesn't study hard

In fact, the test scores are extracted from a special distribution. The result now is that Xiao Li got a single-digit score. Although we can't observe whether Xiao Li studies hard, we can say that by observing his single-digit scores, we are 95% sure that Xiao Li doesn't study hard.

Why not say that Xiao Li must not have studied hard? Because Xiao Li may study hard, but because his uncle came on the day of the exam, he failed the exam. However, the probability of this happening is very small. If Xiao Li didn't do well in the exam because his uncle came, then we made the first mistake (also called abandoning the truth). Of course, we don't want to wronged Xiao Li, so we hope to control the probability of such mistakes. Generally speaking, we can control such errors below 5%.

How to assume that mathematics is reduced to absurdity? The reduction to absurdity belongs to the category of "indirect proof", which is a proof method of thinking about problems from a negative perspective, that is, affirming the topic and denying the conclusion, thus leading to contradictory reasoning. Hadamard, a French mathematician, summed up the essence of reduction to absurdity: "If we affirm the hypothesis of the theorem and deny its conclusion, it will lead to contradictions". Specifically, the reduction to absurdity starts with the conclusion of the negative proposition, takes the negation of the conclusion of the proposition as a known condition for reasoning, and carries out correct logical reasoning, thus using known conditions, known axioms, theorems, laws or propositions that have been proved to be correct. The reason for the contradiction is that the hypothesis is not established, so the conclusion of the proposition is affirmed and the proposition is proved. The reduction to absurdity is based on the "law of contradiction" and "law of excluded middle" in the laws of logical thinking. In the same thinking process, two contradictory judgments cannot be true at the same time, at least one of them is false, which is the "law of contradiction" in logical thinking; Two contradictory judgments cannot be false at the same time. Simply saying "one or not one" is the "law of excluded middle" in logical thinking. In the process of proving absurdity, contradictory judgments are obtained. According to the law of contradiction, these contradictory judgments cannot be true at the same time, but one of them is bound to be false, and the known conditions, known axioms, theorems, rules or propositions that have been proved to be correct are all true, so the "negative conclusion" is bound to be false. According to "law of excluded middle", the contradictory and mutually negative judgments of conclusion and negative conclusion cannot be false at the same time, and there must be a truth, so we get that the original conclusion must be true. Therefore, reduction to absurdity is based on the basic laws and theories of logical thinking, and reduction to absurdity is credible. The problem model of reduction to absurdity can be simply summarized as "negation → reasoning → negation". That is to say, starting from the negative conclusion, through correct reasoning, logical contradictions are led out and new negation is achieved. It can be considered that the basic idea of reduction to absurdity is "negation of negation". The three main steps of proof by reduction to absurdity are: denying the conclusion → deducing the contradiction → establishing the conclusion. The specific implementation steps are as follows: step 1, reverse assumption: make an assumption opposite to the verification conclusion; The second step is to return to absurdity: under the condition of reverse design, output contradictions through a series of correct reasoning; The third step, conclusion: it shows that the reverse hypothesis is not established, thus affirming the original proposition. When applying reduction to absurdity, we must use "counter-hypothesis" for reasoning, otherwise it is not reduction to absurdity. When using reduction to absurdity to prove a problem, if only one aspect of the proposition needs to be proved, then refute this situation, which is also called reduction to absurdity; If the conclusion is multifaceted, then all the negative situations must be refuted one by one in order to infer the original conclusion. This method of proof is also called "exhaustive method". Reduction to absurdity is often used to solve mathematical problems. Newton once said, "Reduction to absurdity is one of the most skilled weapons for mathematicians". Generally speaking, the problems commonly proved by reduction to absurdity are: the proposition that the conclusion appears in the form of "negative form", "at least" or "at most", "unique" and "infinite"; Or the negative conclusion is more obvious.

Absurd mathematical proof

Can be set by the meaning of the question.

a(n+ 1)=qan，b(n+ 1)=pbn，p≠q

=an+bn

c(n+ 1)= a(n+ 1)+b(n+ 1)= qan+pbn

Suppose:/c (n-1) = (qan+pbn)/(an+bn) = k (fixed value) … ①

So (q-k)an+(p-k)bn = 0.

An/BN = (k-p)/(q-k) = (a1/b1) (q/p) (n-1) holds for any n ∈ n

(q/p) (n-1) = (b1/a1) [(k-p)/(q-k)] (fixed value)

Thus q/p = 1, but this is contradictory to p ≠ Q.

So the assumption is not true, so {} is not a geometric series.

Suppose there are two non-coincident straight lines, A and B, both passing through the point P outside L and perpendicular to L.

Because a is perpendicular to l and b is perpendicular to l.

So a||b

So a and b don't intersect.

Contradict with the hypothesis

So the assumption doesn't hold.

So crossing a point outside the straight line can only be a straight line perpendicular to the known straight line.

Suppose AB and CD have more than one intersection, that is, there are two or more intersections. According to the principle that two points determine a straight line, if these two straight lines have two or more intersections, then these two straight lines overlap, which contradicts the condition that AB and CD are two non-overlapping straight lines, so the assumption is not established, so the two non-overlapping straight lines AB and CD only intersect at one intersection.

You must master the basic rules proved by reduction to absurdity:

It should be assumed that the result is incorrect.

Deduce with incorrect results and get an inevitable and unique result.

And this inevitable and unique result is contrary to the given conditions.

Extrapolation assumption is incorrect.

The key is to master the only result.

If any two of the four coins have different denominations, all four coins have different denominations.

Therefore, a * * * has four different denominations of coins, which contradicts the topic, so the assumption is not established.

So at least two of the four coins have the same face value.

Simple!

Let me first introduce the reduction to absurdity: in mathematics, only yes or no, assuming no, and the result is inconsistent with the law, then the reduction to absurdity is successful. So proof is another answer.

Specific:

Suppose a triangle can have two right angles.

So the sum of the internal angles of the triangle must be greater than 180 degrees.

And because the sum of the internal angles of the triangle must be 180 degrees,

So a triangle can't have two right angles or obtuse angles.

How does the reduction to absurdity assume that the conclusion to be proved is negative, and then substitute it as a known condition until the contradiction eases, thus explaining that the conclusion is true?

What is reduction to absurdity and how to apply it to the hypothesis test of an argument? Independent variable test: based on the assumption of known distribution (such as normal distribution), estimate or test the overall independent variable.

Demonstrate the advantages and disadvantages of the test: the advantage is that the test efficiency is high when the conditions are met; Its disadvantage is that it requires strict data, such as grade data and uncertain data (> > 50mg), and the distribution type of data is required to be known and the total square.