Contreexemples

Contreexemples, or counterexamples, are instances that disprove a general claim. In mathematics and logic, a statement of the form “for all x in a domain D, P(x) holds” is falsified by a single x0 in D for which P(x0) is false. Conversely, to establish an existential claim “there exists x in D such that P(x),” one example suffices.

Counterexamples are essential tools for testing conjectures, sharpening definitions, and clarifying the limits of theorems. They also help show that a proposed proof strategy is inadequate or that a rule has a restricted scope. The most common types are direct counterexamples—an explicit object that violates the statement—and counterexamples that arise from restricting the domain, where the statement might hold in one domain but not another. In logic, a counterexample can be viewed as a model in which the premises are true and the conclusion false.

Simple examples include the statement “All natural numbers are even,” which is false because 1 (or 3) is not even. The claim “Every real number is rational” is false; sqrt(2) or pi serve as counterexamples. The geometric assertion “All triangles are equilateral” is refuted by an isosceles or scalene triangle. Even in arithmetic, “The sum of two primes is always even” fails for 2 + 3 = 5.

A counterexample does not prove the negation of a claim in every case, but it shows the original statement is not universally valid. To avoid ambiguity, precise statements must specify the domain and the quantifiers involved.