ellenpélda

ellenpélda, commonly translated as counterexample, is an instance that disproves a general claim by showing a single case where the claim fails. In mathematics, logic, philosophy, and computer science, a counterexample demonstrates that a proposed universal statement or conjecture is false or that a given hypothesis is not sufficient by itself. The term derives from ellen- meaning against and példa meaning example in Hungarian. It is used in Hungarian mathematical writing alongside the international term counterexample.

A counterexample is not just any objection; it must be a concrete example that satisfies the conditions of the context while contradicting the claim. Counterexamples are often used to show that hypotheses in a theorem are necessary, or that a proposed rule does not hold in general. They help to refine conjectures and delineate the scope of statements.

Examples:

- In real analysis, the statement that every polynomial with real coefficients has a real root is false. The polynomial f(x) = x^2 + 1 has no real roots, so it serves as a counterexample.

- In elementary number theory, the claim that every even number is divisible by 4 is false. The number 6 is even but not divisible by 4, making it a counterexample.

Counterexamples are essential tools in mathematics and related disciplines. They illustrate the limits of general statements, motivate the search for additional hypotheses, and distinguish proofs from mere conjecture. See also: proof by counterexample, falsifiability.