请输入您要查询的字词:

 

单词 Converse
释义

converse


Let a statement be of the form of an implicationMathworldPlanetmath

If p then q

i.e. (http://planetmath.org/Ie) it has a certain premiseMathworldPlanetmath p and a conclusion q.  The statement in which one has interchanged the conclusion and the premise,

If q then p

is the converseMathworldPlanetmath of the first.  In other words, from the former one concludes that q is necessary for p, and from the latter that p is necessary for q.

Note that the converse of an implication and the inversePlanetmathPlanetmathPlanetmath of the same implication are contrapositives of each other and thus are logically equivalent.

If there is originally a statement which is a (true) theorem and if its converse also is true, then the latter can be called the converse theorem of the original one.  Note that, if the converse of a true theorem “If p then q” is also true, then “p iff q” is a true theorem.

For example, we know the theorem on isosceles trianglesMathworldPlanetmath:

If a triangleMathworldPlanetmath contains two congruent (http://planetmath.org/Congruent2) sides, then it has two congruent angles.

There is also its converse theorem:

If a triangle contains two congruent angles, then it has two congruent sides.

Both of these propositionsPlanetmathPlanetmath are true, thus being theorems (see the entries angles of an isosceles triangle and determining from angles that a triangle is isosceles).  But there are many (true) theorems whose converses are not true, e.g. (http://planetmath.org/Eg):

If a function is differentiableMathworldPlanetmathPlanetmath on an intervalMathworldPlanetmathPlanetmath I, then it is continuousMathworldPlanetmathPlanetmath (http://planetmath.org/ContinuousFunction) on I.

Titleconverse
Canonical nameConverse
Date of creation2013-03-22 17:13:37
Last modified on2013-03-22 17:13:37
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id24
Authorpahio (2872)
Entry typeDefinition
Classificationmsc 03B05
Classificationmsc 03F07
Related topicExamplesOfContrapositive
Related topicDifferntiableFunction
Related topicInverse6
Related topicConverseOfEulersHomogeneousFunctionTheorem
Definesconverse theorem
Definesconversely
随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 23:33:38