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# converse

Let a statement be of the form of an implication

If $p$ then $q$

i.e. 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.

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.:

*If a function is differentiable on an interval $I$, then it is continuous on $I$.*

## Mathematics Subject Classification

03B05*no label found*03F07

*no label found*

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