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 1: irrational.
Example 2: 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: is irrational.
Suppose the statement is false. Then there is a rational number such that . Now, squaring both sides we have . This means that
Now, from example 2, is irrational, and we know that is rational since 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 is irrational.
Example 6: If a line is tangent to a circle, then it is perpendicular to the radius at the point of tangency.
Consider a circle with center , radius , and with line tangent to the circle at . We have to show that .
Suppose is not . Then there exists a segment which is . If so, it follows that (Why?). But lies outside the circle since is a tangent line. Therefore . But this contradicts the previous statement that . Hence, our supposition is false. Therefore, .
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