What Is Proof by Contradiction?

Proof by contradiction is one of the most powerful proof techniques out there, even used by famous mathematicians such as Euler. It is a great technique to try if you are at a loss with other options.

Here is how it works: you take the statement you wish to prove true, and assume it is false. Then, you take this "proposal," and manipulate it to reach a statement we know is false (such as $1 = 0$). When that false statement is reached, we know we "made a mistake" somewhere, i.e. assuming that the statement we wanted to prove true is false. Therefore, the target statement must be true.

Note that if you wish to prove a statement to be false, simply follow the procedure above, replacing every use of "false" with "true" and every use of "true" with "false." This works because all statements are either true or false. No statement can be both true and false simultaneously.

Proof by contradiction (also known as an indirect proof) is an extremely powerful proof technique to prove a statement you can't easily prove directly.

The basic idea behind it is: First, you suppose the statement you are trying to prove is false, because that statement can either be true or false. Then, using the givens, we can come to a faulty conclusion that contradicts what we already know. For example, you could end up contradicting the fact that a median of a triangle intersects the midpoint of the opposite side, or something else. Therefore, the original statement can't be false, and it must be true.

Let's start with an intuitive, yet extremely easy example. Let's prove the following statement:

Not all computers are laptops.

It's a trivial example, but it's a good way to see how indirect proof works.

First, assume the statement is false. The resulting statement is:

All computers are laptops.

However, we know there are not only laptop computers, there are desktop as well as laptop computers. So this tells us that not all computers are laptops. But that contradicts our statement! Therefore, our original assumption has to be false since it led to a contradiction. Thus, the statement Not all computers are laptops. is true.

Well, this topic wasn't placed in the Geometry section for no reason, so let's do some geometry! This is a basic example.

Problem: In isosceles triangle $ABC$, $\overline{BC}$ is the base. A cevian from $\angle{A}$ to side $BC$ intersects $BC$ at $D$ such that $\overline{BD} \not \cong \overline{CD}$. Prove that $\overline{AD}$ does not bisect $\angle{A}$.

Proof: Using the concept of indirect proof, suppose that $\overline{AD}$ does bisect $\angle{A}$. This would mean that $\angle{BAD} \cong \angle{CAD}$. Since $\triangle{ABC}$ is isosceles with base $\overline{BC}$, $\overline{AB} \cong \overline{AC}$. Finally, by the Reflexive Property, $\overline{AD} \cong \overline{AD}$. By SAS, we have that $\triangle{ABD} \cong \triangle{ACD}$. This congruence means that $\overline{BD} \cong \overline{CD}$. But this is a contradiction, since from our giverns we have that $\overline{BD} \not \cong \overline{CD}$. Therefore, the statement that $\overline{AD}$ bisects $\angle{A}$ is false and therefore, $\overline{AD}$ does not bisect $\angle{A}$. $\Box$

Using the concept of proof by contradiction

Let's say that we want to prove two lines not parallel. (lines $s$ and $t$)

Great! We can use any of the two above facts, so we'll refer to both in the proof.

OK. Our statement that we are trying to prove is assumed to be false. The assumption is that our two lines, $s$ and $t$, are parallel.

Right! Because two angles cannot be congruent and not congruent at the same time, we have reached a contradiction.

We pointed out the contradiction, and now we state that we have proved the contrapositive (if ~original statement -> then ~given) of the original statement.

Naturally, if our contrapositive is correct, then our statement must be correct too. Thus, our two lines are not parallel and we are done. $\blacksquare{}$