Proof Techniques

Establishing Truth Beyond Doubt

Mathematics advances by proof — an airtight chain of logical steps from accepted premises (axioms, definitions, prior theorems) to a conclusion. Unlike experiment, a correct proof settles a claim for all cases at once, forever. Several standard techniques cover most theorems.

Direct proof assumes the hypothesis and derives the conclusion step by step. To prove "if $n$ is even then $n^2$ is even," write $n = 2k$, so $n^2 = 4k^2 = 2(2k^2)$, which is even. Straight-line reasoning from assumption to conclusion.

Proof by contrapositive proves $p \to q$ by instead proving the equivalent $\neg q \to \neg p$. Since a statement and its contrapositive are logically equivalent (Lesson 2), this is completely valid, and sometimes the contrapositive is far easier. To prove "if $n^2$ is even then $n$ is even," it is cleaner to prove "if $n$ is odd then $n^2$ is odd."

Proof by contradiction assumes the claim is false, then derives a logical impossibility, so the assumption must have been wrong. The classic example proves $\sqrt{2}$ is irrational: assume it equals a fraction in lowest terms, and derive that both numerator and denominator are even — contradicting "lowest terms."

Disproof is different and often easier: to show a universal claim is false, exhibit a single counterexample. One example where the claim fails demolishes it; you never need to prove the negation for all cases.

Common pitfall: confusing proof by contradiction with proof by contrapositive, and misusing counterexamples. Contradiction assumes the whole statement is false and seeks any absurdity; contrapositive proves the equivalent $\neg q \to \neg p$ directly. And a counterexample disproves a universal but a single supporting example never proves one — "it works for $n=1,2,3$" is not a proof.