inverse statement

Let a statement be of the form of an implicationMathworldPlanetmath

If p, then q

i.e. (, it has a certain premiseMathworldPlanetmath p and a conclusion q. The statement in which one has negated the conclusion and the premise,

If ¬p, then ¬q

is the inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (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 converseMathworldPlanetmath 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 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 (

Title inverse statement
Canonical name InverseStatement
Date of creation 2013-03-22 17:20:00
Last modified on 2013-03-22 17:20:00
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 10
Author Wkbj79 (1863)
Entry type Definition
Classification msc 03B05
Synonym inverse
Related topic Converse
Related topic SomethingRelatedToContrapositive
Related topic ConverseTheorem