inverse statement

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 negated the conclusion and the premise,

If $\mathrm{\neg }p$, then $\mathrm{\neg }q$

is the inverse (or inverse statement) of the first. Note that the following constructions yield the same statement:

• the inverse of the original statement;

• the contrapositive of the converse of the original statement;

• the converse of the contrapositive of the original statement.

Therefore, just as an implication and its contrapositive are logically equivalent (proven here (http://planetmath.org/SomethingRelatedToContrapositive)), the converse of the original statement and the inverse of the original statement are also logically equivalent.

The phrase “inverse theorem” is in usage; however, it is nothing akin to the phrase “converse theorem (http://planetmath.org/ConverseTheorem)”. In the phrase “inverse theorem”, the word “inverse” typically refers to a multiplicative inverse. An example of this usage is the binomial inverse theorem (http://planetmath.org/BinomialInverseTheorem).