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Pure · Proof

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Disproof by counter-example

Disproving a universal ("for all") claim with a single example where it fails — shown with the arithmetic worked out. Why one counter-example is enough to disprove, why no number of supporting examples can prove, and where to hunt for the case that breaks a statement.

Pure · Proof

Disproof by counter-example

Disproving a universal ("for all") claim with a single example where it fails — shown with the arithmetic worked out. Why one counter-example is enough to disprove, why no number of supporting examples can prove, and where to hunt for the case that breaks a statement.

Why it works

A claim that says "for all …" is a promise about every case. To disprove it you need just one case where it fails — a counter-example. The logic is clean: if the statement were true it would hold everywhere, so a single place it breaks shows it is false.

Two things follow, and both are heavily tested:
  • One counter-example is enough, and you must show the arithmetic that makes
it fail — name the value, substitute it, and state that the result breaks the claim. A bare "it doesn't work for 99" with no working earns nothing.
  • Examples can never prove a "for all" statement. Ten cases that work do not make
a proof — the eleventh might fail. (To prove such a claim you need deduction or exhaustion; examples only ever disprove.)

Where to look for the case that breaks a statement: try small values, 00 and 11, even vs odd, negatives, and primes. The counter-example is often an edge case the claim's author didn't picture.