# 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 |