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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. ==In mathematics== In mathematics, counterexamples are often used to prove the boundaries of possible theorems. By using counterexamples to show that certain conjectures are false, mathematical researchers avoid going down blind alleys and learn how to modify conjectures to produce provable theorems. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「counterexample」の詳細全文を読む スポンサード リンク
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