Contraposition

In logic and math, contraposition is the right way to reverse "if-then" statements. For example, the contrapositive of the sentence "If it is raining, then I wear my coat" is the sentence "If I don't wear my coat, then it isn't raining." If the original sentence is correct, the contrapositive is always correct.

It is easier to see how contraposition is the right way to reverse an if-then statement by looking at two other reversals that can sometimes look similar, but are bad logic:

Inversion

"If it is not raining, then I don't wear my coat." This is wrong as one can wear a coat for other reasons (eg. It is snowing). In this statement, we have made both the "if" and the "then" statements negative.

Conversion

"If I wear my coat, then it is raining." This is wrong as wearing a coat doesn't cause it to rain. In this statement we have switched the "if" and "then" statements.

In technical writing: the contrapositive of [math]\displaystyle{ P \rightarrow Q }[/math] is [math]\displaystyle{ \neg Q \rightarrow \neg P }[/math]. The "[math]\displaystyle{ \rightarrow }[/math]" means "if-then" (If P then Q), and the "[math]\displaystyle{ \neg P }[/math]" means "not P" (eg. "it is not raining").

Contraposition Media

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  Venn diagram for "A is a subset of B". Modification of Image:Venn A intersect B.svg based on w:en:Image:Venn A subset B.png