CHAPTER 6
Proof by Contradiction
We now introduce
a third method of proof, called
proof by contra-
diction
. This new method is not limited to proving just conditional
statements – it can be used to prove any kind of statement whatsoever.
The basic idea is to assume that the statement we want to prove is
false
,
and then show that this assumption leads to nonsense. We are then led
to conclude that we were wrong to assume the statement was false, so
the statement must be true. As an example of this, consider the following
proposition and its proof.
Proposition If a,b∈Z, then a2−4b#=2.
Proof. Suppose this proposition is false.
This conditional statement being false means there exist numbers
a
and
b
for which a,b∈Zis true but a2−4b#=2is false.
Thus there exist integers a,b∈Zfor which a2−4b=2.
From this equation we get a2=4b+2=2(2b+1), so a2is even.
Since a2is even, it follows that ais even, so a=2cfor some integer c.
Now plug a=2cback into the boxed equation a2−4b=2.
We get (2c)2−4b=2, so 4c2−4b=2. Dividing by 2, we get 2c2−2b=1.
Therefore 1=2(c2−b), and since c2−b∈Z, it follows that 1 is even.
Since we know 1 is not even, something went wrong.
But all the logic after the first line of the proof is correct, so it must be
that the first line was incorrect. In other words, we were wrong to assume
the proposition was false. Thus the proposition is true. ■
Though you may be a bit suspicious of this line of reasoning, in the
next section we will see that it is logically sound. For now, notice that
at the end of the proof we deduced that 1 is even, which conflicts with
our knowledge that 1is odd. In essence, we have obtained the statement
(1 is odd)∧∼(1 is odd), which has the form C∧∼C. Notice that no matter
what statement
C
is, and whether or not it is true, the statement
C∧∼C
must be false. A statement—like this one—that cannot be true is called a
contradiction. Contradictions play a key role in our new technique.
