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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 negated the conclusion and the premise,
If $\neg p$ then $\neg q$
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), the converse of the original statement and the inverse of the original statement are also logically equivalent.
The phrase ``inverse theorem'' is in current usage; however, it is nothing akin to the phrase ``converse theorem''. In the phrase ``inverse theorem'', the word ``inverse'' typically refers to a multiplicative inverse. An example of this usage is the binomial inverse theorem.
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