Leaving Certificate Mathematics Definitions Practice Test

Proving by contradiction works by starting with the assumption that the statement you want to prove is false. From that assumption, you derive logical consequences using known facts until you arrive at something impossible or self-contradictory. That contradiction shows the initial assumption cannot be true, so the statement must be true.

This approach contrasts with a direct proof, which shows the statement follows straightforwardly from known truths without assuming its negation; an inductive proof builds the truth step by step, typically for all natural numbers; and an exhaustive proof checks every possible case. For example, to show that a number like the square root of 2 is irrational, you assume it can be rational and written as a fraction in lowest terms, derive that both numerator and denominator must be even, which contradicts the fraction being in lowest terms. That contradiction means the original assumption is false, so the square root of 2 is irrational.