Counterexample

In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule or law. For example, consider the proposition 'all students are lazy'. Because this statement makes the claim that a certain property (laziness) holds for all students, even a single example of a diligent student will prove it false.Thus, any hard-working student is a counterexample to 'all students are lazy'.More precisely, a counterexample is a specific instance of the falsity of a universal quantification (a 'for all' statement). In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule or law. For example, consider the proposition 'all students are lazy'. Because this statement makes the claim that a certain property (laziness) holds for all students, even a single example of a diligent student will prove it false.Thus, any hard-working student is a counterexample to 'all students are lazy'.More precisely, a counterexample is a specific instance of the falsity of a universal quantification (a 'for all' statement). In mathematics, this term is (by a slight abuse) also sometimes used for examples illustrating the necessity of the full hypothesis of a theorem, by considering a case where a part of the hypothesis is not verified, and where one can show that the conclusion does not hold.