A counterexample is an example that refutes the truth of a statement.
Building a counterexample is a common way to refute hypotheses . If there is a statement of the type “Property A holds for any X from the set M”, then a counterexample for this statement is: “ There is an object X 0 from the set M for which property A does not hold.”
Finding a counterexample manually is often very difficult. In such cases, you can use a computer . A program for finding a counterexample can simply iterate over the elements of the set M and check the execution of property A. A more complex, but also more effective, approach is to build a counterexample “in parts”. At the same time, when choosing the next “part”, options are immediately discarded, which obviously do not lead to a refutation of the statement in question. This allows you to significantly speed up the work, often by orders of magnitude.
It must be remembered that the absence of a counterexample does not serve as evidence of a hypothesis. A proof of this kind can be constructed only if the set under consideration is finite. In this case, it is enough to sort through all its elements, and if there is no counterexample among them, then the statement will be proved.
Counterexamples to common stereotypes
- The microraptor is a counterexample to the statement “All dinosaurs were scaly, because they are reptiles”
- The template that gives the scientific classification of Homo sapiens is a counterexample to the statement “People are not animals”
Classical counterexamples in mathematics
- The Dirichlet function is an example of a function that is discontinuous at each point.
- The Weierstrass function is an example of an everywhere continuous , but nowhere differentiable function .
- The Cantor function is an example of a continuous monotonic function that is not constant , but at the same time has a derivative equal to zero at almost all points.
Literature
- Gelbaum B., Olmsted J. Counterexamples in analysis . M.: Mir, 1967.
- Lakatos I. Proofs and refutations: how theorems are proved . M .: Nauka, 1967.
- Medvedev F.A. Essays on the history of the theory of functions of a real variable . M .: Nauka, 1975.
- Sekey G. Paradoxes in probability theory and mathematical statistics . M .: Mir, 1980.
- Stoyanov J. Counterexamples in probability theory . M .: Factorial, 1999.
- Schetnikov A. I. , Schetnikova A. V. The role of counterexamples in the development of the basic concepts of mathematical analysis. - Novosibirsk: ANT, 1999.
- Romano JP, Siegel AF Counterexamples in probability and statistics . Chapman & Hall, NY, 1986.
- Steen LA, Seebach JA (Jr.). Counterexamples in topology . Springer, NY, 1978.
- Wise GL, Hall EB Counterexamples in probability and real analysis . Oxford UP, 1993.