Proof by contrapositive
From AoPSWiki
Proof by contrapositive is a method of proof in which the contrapositive of the desired statement is proven, and thus it follows that the original statement is true. Generally, this form is only used when it is impossible to prove the original statement directly.
Contents |
Problems
Introductory
- Show that if
and 
are two integers for which 
is even, then and have the same parity.
Solution
The contrapositive of this is
- If and are two integers with opposite parity, then their sum must be odd.
So we assume and have opposite parity. Since one of these integers is even and the other odd, there is no loss of generality to suppose is even and is odd. Thus, there are integers 
and 
for which 
and 
. Now then, we compute the sum 
, which is an odd integer by definition.
- Show that if
is an odd integer, then 
is odd.
Solution
Suppose is an even integer. Then there exists and integer 
such that 
. Thus 
. Since 
is an integer, is even. Therefore is not odd.
