converseConverseTheorem2007-06-08 15:53:362009-08-21 13:44:18Definitionlemmaconverse theoremconversely% this is the default PlanetMath preamble. as your knowledge
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Let a statement be of the form of an implication
\begin{center}
If $p$ then $q$
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\PMlinkname{i.e.}{Ie} it has a certain premise $p$ and a conclusion $q$.\, The statement in which one has interchanged the conclusion and the premise,
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If $q$ then $p$
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is the \emph{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 \emph{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:
\emph{If a triangle contains two \PMlinkname{congruent}{Congruent2} sides, then it has two congruent angles.}
There is also its converse theorem:
\emph{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, \PMlinkname{e.g.}{Eg}:
\emph{If a function is differentiable on an interval $I$, then it is \PMlinkname{continuous}{ContinuousFunction} on $I$.}