You can prove that something
is true
by showing that not something
being true
would lead to a contradictory, inconsistent, absurd conclusion.
Every such proof-by-contradiction rhetorically starts the same way:
Suppose not. Instead suppose to the contrary that …
Examples
Prove that there are infinitely many prime numbers.
If you assume the contrary, then you have a finite list of prime numbers to work with, that you can use to “break things”.
Prove that \(\sqrt{2}\) is an irrational number.
An irrational number is defined to be a number that is not rational; it’s hidden behind vocabulary, but the goal is to prove that \(\sqrt{2}\) is not a rational number. How can you directly prove something is not something else though? Suppose that it is, and reach a contradiction. “Suppose \(\sqrt{2}\) is rational, which means … ”
Prove there is no polynomial function \(f\)
of degree at least one with integer coefficients
such that the outputs \(f(0), f(1), f(2), \dotsc\) are all prime.
It’s hard to directly prove the non-existence of anything; it’s easier to assume the existence of such a thing — in this case a non-constant polynomial \(f\) with integer coefficients for which \(f(0), f(1), f(2), \dotsc\) are all prime — that you can play with, searching for some inconsistency in the qualities of \(f.\)