Before revision: For example, the fact that "John Smith is not a lazy student" is a counterexample to the generalization “students are lazy”, and both are counterexample to, and disproof of, the generalization “all students are lazy.” In mathematics, the term "counterexample" is also used (by a slight abuse) to refer to examples which illustrate the necessity of the full hypothesis of a theorem.
After revision: For example, the fact that "John Smith is not a lazy student" is a counterexample to the generalization “students a lazy”, and both are counterexample to, and disproof of, the generalization “all students are lazy.” In mathematics, the term "counterexample" is also used (by a slight abuse) to refer to examples which illustrate the necessity of the full hypothesis of a theorem.