motbevis

Motbevis, in Swedish, means counterexample. In mathematics and logic, a motbevis is a specific instance that shows a general claim to be false. It is used to demonstrate that a statement cannot be true for all cases within its intended domain.

A counterexample typically targets universal claims of the form “for all x, P(x).” If one can find an x such that P(x) is false (and the premises or conditions hold for that x), the statement is disproved. Counterexamples may also challenge conditional or existential claims by revealing situations where the stated conclusion does not follow or where the claimed existence fails.

Common types of counterexamples include straightforward numerical examples, edge cases, or constructions that meet the stated hypotheses but violate the conclusion. For instance, the claim “all even numbers are divisible by 4” is contradicted by 6, which is even but not divisible by 4. The claim “every prime number is odd” is contradicted by 2, which is prime and even. Counterexamples can be simple yet powerful, often exposing flaws in conjectures, proposed generalizations, or overly broad theorems.

Counterexamples do not prove every aspect of a theory; they only show that a particular universal claim is false. They are crucial in mathematical exploration, helping to refine hypotheses, guide the search for correct formulations, and prevent overgeneralization. Related concepts include proofs by contradiction and proof by contrapositive, which use logical structure rather than a single counterexample to establish truth.