The contradictory propositional relationship among original proposition, negative proposition and negative proposition is:

Original proposition: a = = > b is; If a holds, so does b.

No proposition: A bar = = = > B bar; If A doesn't hold, then B doesn't hold.

Inverse proposition: b = = > a is; If b holds, so does a.

Negative proposition: B bar = = = > A bar is; If b doesn't hold, so does a.

The original proposition and the negative proposition are equivalent propositions;

Inverse proposition and negative proposition are equivalent propositions;

Equivalence is also called equivalence. The equivalence of proposition A and proposition B can be deduced from each other and can be written as A.

The characteristics of equivalence proposition are: truth is the same as truth, and falsehood is false.

The equivalence between the original proposition and the negative proposition can be proved by reduction to absurdity as follows:

Known: a = = > B. Verification: non-b = = = = non-a.

Prove that it is incorrect to assume that it is not b = = = = not A,

Then, non-B = = = > A. (law of excluded middle)

But a = = = = > B, (known).

So it's not b = = = = > B (transitivity)

This contradiction (which violates the identity that B is B) proves that it is not B = = = = not A is correct.

On the other hand, when the negative proposition is correct, it can also prove that the original proposition is necessarily correct. It can be seen that the two propositions that are mutually negative are equivalent.

Tongxiang, the inverse proposition and the negative proposition are also reciprocal propositions, so they are equivalent propositions. So there are essentially only two propositions, namely (1) and (2). Propositions (3) and (4) are only negative forms of (1) and (2) respectively.

It is worth mentioning that when the original proposition is correct, its inverse proposition or negative proposition is not necessarily correct, but may be both true and false. Therefore, it is necessary to prove the correctness of two mutually inverse propositions or negative propositions respectively.

We discuss various forms of propositions, their relationships and equivalence, which is of great significance to demonstrate mathematical problems. When we prove that a proposition is difficult to take a nap, we can correct its negative proposition (equivalent proposition), which opens up a broad road for the proof of this proposition. To know whether the four related propositions are correct or not, we only need to prove two propositions that are mutually negative or mutually negative. One true and one false, it must be two true and two false. Two truths (false) must be four truths (false). As for which two to prove, of course, they are alternatives. When we study a theorem or prove a proposition to be true, we will naturally think of whether its inverse proposition (or no proposition) is correct. If it is proved to be true, a new theorem will be deduced; if it is false, the understanding of the original proposition will be deepened. Therefore, we should develop this good study habit of bringing forth the old and bringing forth the new, asking new questions and even discovering new theorems.

Is there only one inverse proposition in a proposition?

A: Suppose the original proposition is "If A is B", then the counter proposition is "If B is A". This means that both A and B contain only one item. But when a proposition has more than one condition and conclusion, it has more than one inverse proposition.