More examples of proof by contradiction

We have had good discussions on mathematical proofs, so I am planning to create a mathematical proof series that will discuss the basics such as direct proof, indirect proof, and proof by mathematical induction.  But before I do that, let me continue with more examples of proof by contradiction.

Proof by contradiction, as we have discussed, is a proof strategy where you assume the opposite of a statement, and then find a contradiction somewhere in your proof. Finding a contradiction means that your assumption is false and therefore the statement is true. Below are several more examples of this proof strategy.

Example 2: $\sqrt{6}$ is irrational. The proof of this is basically the same as example 1, so it is left as an exercise.

Example 3: Proof that there are infinitely many primes.

Example 4: Knights and Liars

Example 5: $\sqrt{2} + \sqrt{3}$ is irrational.

Proof:

Suppose the statement is false. Then there is a rational number $r$ such that $\sqrt{2} + \sqrt{3} = r$. Now, squaring both sides we have $2 + 2\sqrt{6} + 3 = r^2$. This means that

$\sqrt{6} = \displaystyle\frac{r^2-5}{2}$

Now, from example 2, $\sqrt{6}$ is irrational, and we know that $(r^2-5)/2$ is rational since $r$ is rational. This makes the left hand side of the equation irrational and the right hand side rational —  a contradiction!

Therefore, our supposition is false, which implies that  $\sqrt{2} + \sqrt{3}$ is irrational.

Example 6: If a line is tangent to a circle, then it is perpendicular to the radius at the point of tangency.

Proof:

Consider a circle with center $O$, radius $OP$, and with line $m$ tangent to the circle at $P$. We have to show that $m \perp \overline{OP}$.

Suppose $\overline{OP}$ is not $\perp m$. Then there exists a segment  $OQ$ which is $\perp m$. If so, it follows that $OQ < OP$ (Why?). But $Q$ lies outside the circle since $m$ is a tangent line. Therefore $OQ > OP$. But this contradicts the previous statement that $OQ < OP$. Hence, our supposition is false. Therefore, $\overline{OP} \perp m$.

Mathematical proofs is one of the most difficult topics in mathematics, but you can only see a few examples in books. In the Introduction to Mathematical Proofs series that I am planning to write, I am going to give as many and as varied examples as possible. This series is intended to give an intuitive introduction to mathematical proofs for Grades 9-12 students.

Credit: Example 5 was adapted from Numbers: Rational and Irrational by Ivan Niven