Difference between revisions of "Converse theorem"
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A theorem in which the premises are the conclusion of the original (direct) theorem, and the conclusion is the premises of the original theorem. The converse of the converse theorem is the original (direct) theorem, so that the direct and converse theorems are mutually converse. | A theorem in which the premises are the conclusion of the original (direct) theorem, and the conclusion is the premises of the original theorem. The converse of the converse theorem is the original (direct) theorem, so that the direct and converse theorems are mutually converse. | ||
| − | The converse theorem is equivalent to the theorem that is the contrary of the direct theorem, that is, the theorem in which the premises and the conclusion of the direct theorem have been replaced by their negations. Therefore the direct theorem is equivalent to the contrary of the converse theorem, that is, the theorem asserting that if the conclusion of the direct theorem is false, then its premises are false. The well-known method of "proof | + | The converse theorem is equivalent to the theorem that is the contrary of the direct theorem, that is, the theorem in which the premises and the conclusion of the direct theorem have been replaced by their negations. Therefore the direct theorem is equivalent to the contrary of the converse theorem, that is, the theorem asserting that if the conclusion of the direct theorem is false, then its premises are false. The well-known method of "proof by contradiction" is precisely the replacement of the proof of the direct theorem by the proof of the contrary of the converse theorem. The truth of two mutually converse theorems means that the validity of the premises of one of them is not only sufficient but also necessary for the validity of the conclusion. See also [[Theorem]]; [[Necessary and sufficient conditions]]. |
Latest revision as of 10:06, 20 April 2024
A theorem in which the premises are the conclusion of the original (direct) theorem, and the conclusion is the premises of the original theorem. The converse of the converse theorem is the original (direct) theorem, so that the direct and converse theorems are mutually converse.
The converse theorem is equivalent to the theorem that is the contrary of the direct theorem, that is, the theorem in which the premises and the conclusion of the direct theorem have been replaced by their negations. Therefore the direct theorem is equivalent to the contrary of the converse theorem, that is, the theorem asserting that if the conclusion of the direct theorem is false, then its premises are false. The well-known method of "proof by contradiction" is precisely the replacement of the proof of the direct theorem by the proof of the contrary of the converse theorem. The truth of two mutually converse theorems means that the validity of the premises of one of them is not only sufficient but also necessary for the validity of the conclusion. See also Theorem; Necessary and sufficient conditions.
Converse theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Converse_theorem&oldid=11662