Let a statement be of the form of an implicationMathworldPlanetmath

If p then q

i.e. ( 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 ( 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. (

If a function is differentiableMathworldPlanetmathPlanetmath on an intervalMathworldPlanetmathPlanetmath I, then it is continuousMathworldPlanetmathPlanetmath ( on I.

Title converse
Canonical name Converse
Date of creation 2013-03-22 17:13:37
Last modified on 2013-03-22 17:13:37
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 24
Author pahio (2872)
Entry type Definition
Classification msc 03B05
Classification msc 03F07
Related topic ExamplesOfContrapositive
Related topic DifferntiableFunction
Related topic Inverse6
Related topic ConverseOfEulersHomogeneousFunctionTheorem
Defines converse theorem
Defines conversely