Proof by contradiction and contrapositive are somewhat similar.
There are, however, a couple of defining characteristics. Proof by
contradiction proves the proposition by assuming it is false, and
when solving the proof a contradiction arises, showing that the
proposition was indeed true. Proof by contrapositive takes the
proposition and assumes that if the outcome is false, so is the
proposition. I will provide an example of proof by contrapositive
below.
Let x
be some integer.
Prove if x2is even, then x
is even.
Proof by contrapositive:
If x is not even, then x2 is not even.
If x is not even, x=2k+1for some integer k.
Plugging in 2k+1 for x, we get:
b (2k+1)2=4k2+4k+1=2(2k2+2k)+1
Since x=2k+1,
and (2k+1)2=2∗
some integer (2k2+2k)+1, x2 is not even.
Thus, having proved the contrapositive, we can infer that if x2
is
even, then x is even. b ◾
