Generally speaking, in mathematics, statements expressed by languages, symbols or formulas that can judge truth or falsehood are called propositions. For two propositions, if the conditions and conclusions of one proposition are the conclusions and conditions of another proposition, then these two propositions are called reciprocal propositions, one of which is called the original proposition and the other is called the inverse proposition of the original proposition.

proposition

Definition of proposition: A sentence that can judge right or wrong is called a proposition. Among them, statements judged to be true are called true propositions, and statements judged to be false are called false propositions.

Every proposition has an inverse proposition. As long as the title of the original proposition is replaced by the conclusion and the conclusion is replaced by the title, the inverse proposition of the original proposition can be obtained. But the original proposition is correct, and its inverse proposition is not necessarily correct. For example, the inverse proposition of the true proposition "diagonal equality" is "equal angles are diagonal", which is a false proposition.

Reciprocal proposition

Definition of reciprocal propositions: If the conditions and conclusions of one proposition are the conclusions and conditions of another proposition, then these two propositions are called reciprocal propositions. If one of them is called the original proposition, then the other is called its inverse proposition.

Of the two propositions, if the condition of the first proposition is the conclusion of the second proposition, and the conclusion of the first proposition is the condition of the second proposition, then these two propositions are called reciprocal propositions. One of the propositions is called the inverse of the other. Exchange the conditions and conclusions of a proposition to get its inverse proposition, so every proposition has an inverse proposition.

Reverse negative proposition

The condition and conclusion of one proposition are the negation of the conclusion and the negation of the condition of another proposition respectively. Such two propositions are called mutually negative propositions; If one of the propositions is called the original proposition, the other is called the negative proposition of the original proposition.

The relationship between these four propositions is as follows:

The original proposition and the inverse proposition are reciprocal; No proposition and original proposition are mutually negative; The original proposition and the negative proposition are mutually negative; Whether the inverse proposition is reciprocal or not; Negative proposition and negative proposition are mutually negative; Negative proposition, negative proposition, negative proposition.

True-false relationship

The relationship between truth and falsehood of the four propositions is as follows: the two propositions are mutually negative and have the same truth and falsehood; Two propositions are reciprocal propositions or reciprocal propositions, and their truth values are irrelevant (the original proposition and reciprocal proposition are true and false, and reciprocal proposition is true and false).