gen-define-proof-by-contradiction-and-give-one-short-worked-exam-f026

Define proof by contradiction and give one short worked example.

# Proof by Contradiction **Definition:** Proof by contradiction (also called *reductio ad absurdum*) is a logical technique where you assume the opposite of what you want to prove, then show this assumption leads to a logical impossibility or contradiction. Since the assumption produces a contradiction, it must be false, therefore the original statement must be true. ## Structure 1. Assume the negation of what you want to prove 2. Use logical deduction to reach a contradiction 3. Conclude the original statement must be true ## Worked Example **Claim:** √2 is irrational. **Proof:** Assume, for contradiction, that √2 is rational. Then √2 = a/b where a and b are integers with no common factors (fraction in lowest terms) and b ≠ 0. Squaring both sides: 2 = a²/b² Therefore: 2b² = a² This means a² is even, which implies a is even. So a = 2k for some integer k. Substituting: 2b² = (2k)² = 4k² Therefore: b² = 2k² This means b² is even, which implies b is also even. **Contradiction!** We now have that both a and b are even, meaning they share a common factor of 2. But we assumed a/b was in lowest terms (no common factors). Since our assumption leads to a contradiction, √2 must be irrational. ∎