inverse statementInverse62007-06-28 15:49:582007-06-28 18:58:24Definition\usepackage{amssymb}
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Let a statement be of the form of an implication
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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 negated the conclusion and the premise,
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If $\neg p$, then $\neg q$
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is the \emph{inverse} (or \emph{inverse statement}) of the first. Note that the following constructions yield the same statement:
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\item the inverse of the original statement;
\item the contrapositive of the converse of the original statement;
\item the converse of the contrapositive of the original statement.
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Therefore, just as an implication and its contrapositive are logically equivalent (proven \PMlinkname{here}{SomethingRelatedToContrapositive}), the converse of the original statement and the inverse of the original statement are also logically equivalent.
The phrase ``inverse theorem'' is in \PMlinkescapetext{current} usage; however, it is nothing akin to the phrase ``\PMlinkname{converse theorem}{ConverseTheorem}''. In the phrase ``inverse theorem'', the word ``inverse'' typically refers to a multiplicative inverse. An example of this usage is the \PMlinkname{binomial inverse theorem}{BinomialInverseTheorem}.