converse

Let a statement be of the form of an implication

If $p$ then $q$

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

If $q$ then $p$

is the converse 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 inverse 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 triangles:

If a triangle 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 propositions 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 differentiable on an interval $I$, then it is continuous (http://planetmath.org/ContinuousFunction) 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