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Difference between revisions of "Contrary theorem"

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A theorem obtained by replacing the condition and conclusion of a given initial theorem with their negations, and interchanging them.
 
A theorem obtained by replacing the condition and conclusion of a given initial theorem with their negations, and interchanging them.
  
 
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A contrary theorem is usually called a contrapositive theorem. Formally such a theorem is of the form:  "If not B, then not A" , and is obtained by  "contrapositioncontraposition"  (i.e. by interchanging the conclusion (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025950/c0259501.png" />) and condition (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025950/c0259502.png" />) and replacing each with its negation) from a given theorem:  "If  A,  then  B" .
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A contrary theorem is usually called a contrapositive theorem. Formally such a theorem is of the form:  "If not B, then not A", and is obtained by  "contraposition"  (i.e. by interchanging the conclusion ($B$) and condition ($A$) and replacing each with its negation) from a given theorem:  "If  A,  then  B".

Latest revision as of 18:01, 30 July 2014

A theorem obtained by replacing the condition and conclusion of a given initial theorem with their negations, and interchanging them.

Comments

A contrary theorem is usually called a contrapositive theorem. Formally such a theorem is of the form: "If not B, then not A", and is obtained by "contraposition" (i.e. by interchanging the conclusion ($B$) and condition ($A$) and replacing each with its negation) from a given theorem: "If A, then B".

How to Cite This Entry:
Contrary theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Contrary_theorem&oldid=32601