MthT 491 Contradiction

Contradiction

If A denotes some assertion or collection of assertions, we have a contradiction if A is true and A is false - id est A is true and ¬A, the negation of A , are true.

Examples

A is the [mathematical] statement

•

All girls are good at mathematics.

The negation of A is the [mathematical] statement

•

There is some girl who is not good at mathematics.

A theorem

A ⇒ B

is proved by contradiction if we show that

¬B ⇒ ¬A.

Please note that usually the assertion A may contain within itself many definitions and properties not stated explicitly. For example, if A contains the statement

n is a natural number …,

and we proved that

¬B implies n < 0.

we would have a proof by contradiction of A ⇒B.

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On 05 Sep 2005, 17:18.