Direct Proof and Counterexample

Establishing and Refuting Claims

A proof is a finite chain of logically valid steps that establishes a statement beyond doubt, from axioms, definitions, and previously proven theorems. Unlike scientific evidence, a proof settles a claim for all cases at once and permanently. The most fundamental technique is the direct proof.

To prove an implication "if $p$ then $q$" directly, you assume $p$ and derive $q$ through a sequence of justified steps. Consider proving "if $n$ is even, then $n^2$ is even": assume $n$ is even, so $n = 2k$ for some integer $k$; then $n^2 = 4k^2 = 2(2k^2)$, which is even. Each step uses a definition ("even means $2k$") or basic algebra, and the conclusion follows inescapably. Proving a universal ("for all $n$, ...") means giving an argument that works for an arbitrary element, so it covers every case.

The mirror image is disproof by counterexample. To show a universal claim is false, you exhibit a single instance where it fails. To refute "every prime is odd," you name $2$ — a prime that is even. Just one counterexample demolishes a universal statement, because (from Lesson 3) its negation is an existential.

The asymmetry is fundamental: a universal statement is proved by a general argument but disproved by a single example. You can never prove a universal by checking examples — "it works for $n = 1, 2, 3$" is not a proof — but you can always disprove one with a single failure.

Common pitfall: trying to prove a universal statement by examples. Verifying a claim for several cases — even many — never establishes it for all cases; you need a general argument covering an arbitrary element. (Examples can disprove a universal, via one counterexample, but never prove it.) Confusing "true for the cases I checked" with "true in general" is a core reasoning error.