# Difference between revisions of "Contrapositive"

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==Definition== | ==Definition== | ||

− | Given the statement {{M|A\implies B}} (see [[Implies|implies]]) the ''contrapositive'' of this is: {{M|¬B\implies ¬A}} | + | Given the statement {{M|A\implies B}} (see [[Implies|implies]]) the ''contrapositive'' of this is<ref>https://en.wikipedia.org/w/index.php?title=Contraposition&oldid=673682637</ref>: {{M|¬B\implies ¬A}} |

'''Proof:''' | '''Proof:''' | ||

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* <math>[A\implies B]\iff[¬B\implies ¬A]</math> | * <math>[A\implies B]\iff[¬B\implies ¬A]</math> | ||

− | For example see [[Injection]] | + | For example see the definition of: [[Injection]] |

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==See also== | ==See also== | ||

* [[Implies]] | * [[Implies]] |

## Latest revision as of 18:44, 28 August 2015

## Definition

Given the statement [ilmath]A\implies B[/ilmath] (see implies) the *contrapositive* of this is^{[1]}: [ilmath]¬B\implies ¬A[/ilmath]

**Proof:**

[ilmath]A[/ilmath] | [ilmath]B[/ilmath] | [ilmath]A\implies B[/ilmath] | [ilmath]¬A[/ilmath] | [ilmath]¬B[/ilmath] | [ilmath]¬B\implies ¬A[/ilmath] |
---|---|---|---|---|---|

T | T | T | T | ||

T | T | T | T | ||

T | T | ||||

T | T | T | T |

So as can be seen:

- [math][A\implies B]\iff[¬B\implies ¬A][/math]

For example see the definition of: Injection