Direct Proof and Counterexample

Direct Proofs

A proof is a finite chain of logically valid steps establishing a statement from axioms, definitions, and theorems. A proof settles a claim for all cases permanently.

To run a direct proof of "if $p$ then $q$", assume $p$ and derive $q$ through justified algebraic and logical steps.

Example: To prove "if $n$ is even, then $n^2$ is even," assume $n = 2k$. Then $n^2 = 4k^2 = 2(2k^2)$, which is even by definition.

Refuting Claims

The mirror image is disproof by counterexample. To show a universal claim is false, exhibit a single instance where it fails.

To refute "every prime is odd," name 2, a prime that is even. A universal statement is proved by a general argument but disproved by a single example.

Common pitfall: trying to prove a universal by examples. Verifying a claim for several cases never establishes it for all cases.