Original proposition vs. inverse negative proposition, what is inverse proposition, what is not prop

Guo Dunyun: Do you think that the proposition of inverse negation has two meanings.

1.If a+b is not even, then a,b are odd and even (correct) – this conclusion is correct.

2.If a+b is not even, then both a and b are even (false) – this is redundant.

Inverse Negative Proposition: If a+b is not even, then a, b are not both odd numbers - the point is:

a, b – not all – are odd numbers. "is odd" affirms that there are "odd" parts, and "not all" is the division of a whole into two (in some cases, more than two) different components, in this case odd and even. It's not all the same ingredient, or odd or even.

The realization in your 2 that "if a + b is not even, then a and b are both even (wrong)", and of course "then a and b are both even numbers are wrong" is not wrong in itself, and the mistake is that there should be no such "option", this option is redundant. You did not judge according to the original meaning of "inverse negative proposition", and the original meaning of "not all odd numbers" became "all even numbers (wrong)" became a redundant option in your judgment. And in fact, 1 covers 2.

2. Regarding the first half of the inverse negative proposition in the question: "If a+b is not an even number", it means "a+b is an odd number" (so that one of them is an odd number and the other is an even number).

3. The second half of the inverse negative proposition: "a, b are not all odd numbers" means "a and b are not both odd numbers, at least one of them is odd".

4. The conclusion is that the inverse proposition is correct.

Theorem: If the original proposition is true, then the inverse negative proposition must be true.

Original proposition: If a and b are both odd numbers, then a + b is an even true proposition.

Inverse negative proposition: If a + b is not an even number, then a and b are not both odd numbers "Individually it is also a true proposition.

Reason: (a+b) is not an even number, then ab must be 1 odd and 1 even, so a and b are not both odd numbers, and the proposition is true.

If you don't understand, you can ask Helpful, I wish you progress in your studies, thank you.

You see, let's say 2+1=3, 1 is an odd number and 2 is an even number, and their sum of 3 is an odd number.

Either one odd or even is sure to be odd.

Let x be an even number.

The result of x+(x+1) is 2x+1 Since x is even, then x+1 is odd and 2x+1 is also odd.

I think his proposition is wrong, a+b is not even, then he may be 0 and odd, a and b may be one is zero, one is odd, and one may be odd and even.

If a+b is not even, then a and b can never be two even numbers, because the sum of two even numbers is still even.

Not all odd numbers are the conclusion, and this conclusion is completely correct.

Because the original proposition is not complete, because it is both even and even, and it is only a sufficient condition for all odd numbers, and the inverse proposition is also a sufficient and not necessary condition for the former to be the latter.

The inverse negative proposition is equivalent to the original proposition. The middle one is right for the other. This is how it is understood.

1 A declarative sentence that can judge whether it is true or false is called a proposition, a correct proposition is called a true proposition, and a false proposition is called a false proposition. 2 In the form of "if p, then q" p is called the condition of the proposition, and q is the conclusion of the proposition. 3 Classification of propositions:

Original proposition: A proposition itself is called an original proposition, e.g., if x>1, then f(x)=(x-1) 2 increases monotonically.

Inverse proposition: A new proposition that inverts the conditions and conclusions of the original proposition, e.g., if f(x)=(x-1) 2 increases monotonically, then x>1.

Negative proposition: A new proposition that negates all the conditions and conclusions of the original proposition, but does not change the order of the conditions and conclusions, e.g., if x《1, then f(x)=(x-1) 2 does not increase monotonically.

Inverse negative proposition: A new proposition that reverses the conditions and conclusions of the original proposition, and then negates the conditions and conclusions in full, for example: if f(x)=(x-1) 2 does not increase monotonically, then the negation of the proposition of x" The negation of the proposition is a new proposition that only negates the conclusion of the proposition, which is different from the negative proposition.

5 The four kinds of propositions and the relationship between the truth and falsehood of the negation of propositions.

Original proposition: If p, then q

No proposition: If it is not p, it is not q

Inverse proposition: If q, then p

Inverse negative proposition: If it is not q, then it is not p

Inverse proposition: Every proposition has an inverse proposition, as long as the question of the original proposition is changed to a conclusion, and the conclusion is changed to a question, the inverse proposition of the original proposition can be obtained. 、

No proposition: A no proposition is a concept in mathematics. In general, in mathematics, declarative sentences expressed in language, symbols, or formulas that can be judged to be true or false are called propositions.

Inverse negative proposition: If the conditions and conclusion of one of the two propositions are the conclusion and negation of the condition of the other proposition, respectively, then the two propositions are said to be inverse negative propositions of each other.

The inverse proposition is the reversal of both ends of the sentence, the negative proposition is the negation of the meaning of the sentence, and the inverse proposition is the negation of the two ends of the sentence and the reversal. For example: q and p, inverse proposition:

p and q, no proposition: non-q and not p, inverse negative proposition: not p or non-q, remember:

Headwinds must be remembered to turn and into or, and vice versa.

What is an inverse proposition? An inverse proposition is one in which the conditions are opposite and the result is the same. A negative proposition is one in which the conditions are the same, but the results are opposite. The inverse of the negative proposition is that the result and the condition are opposite.

The inverse proposition is the opposite proposition, and the negative proposition is. A negative and negative proposition of your question is a judgment of the negative and negative of this proposition.

If a proposition is the original proposition, then the proposition that is the inverse of the proposition is the inverse of the original proposition.

The original proposition and the inverse proposition are equivalent propositions If the original proposition is true, the inverse negative proposition is true. The inverse proposition and the negative proposition are equivalent propositions, and if the inverse proposition is true, the negative proposition is true.

Logic holds that propositions are equivalent to inverse propositions, i.e., if the proposition is true, then the inverse proposition is also true. The equivalence of a proposition with its inverse proposition exists as an axiom, and you can neither prove it true nor false it. In fact, this thing can be considered an axiom.

It is equivalent to the axiomatic "law of contradiction". Our mathematical system is based on these axioms.

Abuse of the negative proposition

There are many abuses of the reverse negative logic in real life, and the following points should be paid attention to when using it:

1. The premise of the application of the concepts of inverse proposition, inverse proposition, and negative proposition is that the original proposition is a compound proposition, not a simple proposition. Compound propositions are made up of simple propositions that are connected to each other by logical connectors. It is difficult to distinguish between premises and conclusions in simple propositions, and their truth or falsity can only be judged through life experience and objective facts.

2. There must be an appropriate implication relationship in the original proposition (original compound proposition) of the inverse negative proposition. If there is no definite causal relationship, it is meaningless to seek the negative proposition and judge the truth or falsity from the negative proposition.

Inverse propositionThe relation to a proposition and a negative proposition is an equivalence relation. The original proposition and its inverse negation.

The truth is the same, and it is an equivalence relationship. Because the inverse proposition of the original proposition and the negative proposition are a pair of propositions that are inverse to each other, the inverse proposition of the original proposition and its negative proposition are equivalent, and the truth and falsehood are the same.

Proposition form

For two propositions, if the conditions and conclusions of one proposition are the conclusions and conditions of the other proposition, respectively, then the two propositions are called inverse propositions, one of which is called the original proposition, and the other proposition is called the inverse proposition of the original proposition.

For two propositions, if the conditions and conclusions of one proposition are the negation of the conditions of the other proposition and the negation of the conclusion, respectively, then the two propositions are called mutually negative propositions, one of which is called the original proposition, and the other proposition is called the negative proposition of the original proposition.

For two propositions, if the conditions and conclusions of one proposition are the negation of the conclusion of the other proposition and the negation of the condition, respectively, then the two propositions are called mutually inverse negative propositions, one of which is called the original proposition, and the other proposition is called the inverse of the original proposition.

A proposition with his inverse negative proposition.

It can be the same true or false proposition, for example, the original proposition and the inverse proposition have the same true or falsehood, and the inverse proposition.

Nor negative propositions are consistent with true or false.

The original proposition is true, and its inverse and negative propositions may not be true; The original proposition is false, and its inverse proposition and negative proposition are not necessarily false. Therefore, the inverse and negative propositions of a theorem must be proved logically to determine whether they are true or not. If true, they are called inverse theorems and negative theorems, respectively.

The two propositions that are mutually inverse and negative are true and false. It can be concluded from this that in order to prove a proposition to be true, if it is difficult or too complicated to prove it directly, it can be turned to prove that the proposition is true.