Proof by contradiction - more than one case

Question

I am looking for some examples of when proof by contradiction is used in a problem with more than one case.

In all the elementary examples, there are only two options (eg rational/irrational, infinite/finite), so you assume the opposite, show it cannot be true and then conclude the result.

However, I remember from my undergraduate study situations where there were three or more options and proof by contradiction was used to eliminate one or more of these.

I do not remember any of these examples anymore, but I was wondering if I could have some examples of statements (ideally elementary, but more advanced works) that follow such a style in their proof. This would be good to have to show students I teach typical contexts in which this may arise.

Thank you

Answer 1

(1) Here is a $3$-case proof from Larry Cusick's webpages:

Theorem. There are no rational number solutions to the equation $x^3 + x + 1 = 0$.

Proof. (Proof by Contradiction.) Assume to the contrary there is a rational number $p/q$, in reduced form, with $p$ not equal to zero, that satisfies the equation. Then, we have $p^3/q^3 + p/q+ 1 = 0$. After multiplying each side of the equation by $q^3$, we get the equation

$$p^3 + p q^2 + q^3 = 0 \;.$$ There are three cases to consider.

(1) If $p$ and $q$ are both odd, then the left hand side of the above equation is odd. But zero is not odd, which leaves us with a contradiction.

(2) If $p$ is even and $q$ is odd, then the left hand side is odd, again a contradiction.

(3) If $p$ is odd and $q$ is even, we get the same contradiction.

The fourth case—$p$ even and $q$ even—is not possible because we assumed that $p/q$ is in reduced form. This completes the proof.

---

(2) Another $3$-case example may be found in Euclid's Elements, Book I, Prop.6, which uses the law of trichotomy for lines:

Proposition. If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.

See David Joyce's webpage:

"If $AB$ does not equal $AC$, then one of them is greater." There are three cases: $AB < AC$, $AB = AC$, $AB > AC$.