Proof Techniques

Establishing Truth Beyond Doubt

Mathematics advances by proof: an airtight chain of logical steps from accepted premises to a conclusion. A correct proof settles a claim for all cases at once.

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.

Proof by contrapositive proves $p \to q$ by instead proving the equivalent $\neg q \to \neg p$. 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."

Contradiction, Disproof, and Pitfalls

Proof by contradiction assumes the claim is false, then derives an impossibility. The classic example proves $\sqrt{2}$ is irrational by assuming it equals a fraction in lowest terms and deriving that both parts are even.

Disproof uses a counterexample to show a universal claim is false. One failure demolishes it.

Common pitfall: Confusing contradiction (assumes whole claim false, seeks any absurdity) with contrapositive (proves $\neg q \to \neg p$). A supporting example never proves a universal claim.