# converse

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 $q$.

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.

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):

 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