Examples and counterexamples in mathematics

From Wikibooks, open books for an open world
Jump to: navigation, search


"Any blockhead can cite generalities, but the mind-master discerns the particular cases they represent."
(George Eliot in "Daniel Deronda", Chapter 52)

Examples are inevitable for every student of mathematics. It often happens that a student sees only a series of similar simple examples and therefore cannot appreciate the volume and depth of a new notion, which leads to numerous misconceptions. As a result, the student misses the difference between general and special case, true and false statement, correct and incorrect proof.

"Counterexample is an example with a negative connotation. Whereas an example may be used to support or illustrate a claim, a counterexample is used to refute an assertion. How an example is being used often depends on the purpose or the formulation. For example, the word "nth" is an example of an English word without a vowel. It is a counterexample to the claim that every English word contains a vowel. ... In the opinion of B. R. Gelbaum and J. M. H. Olmsted - the authors of two popular books on counterexamples - much of mathematical development consists in finding (and proving) theorems and counterexamples." (Quoted from cut-the-knot)

Several books of counterexamples are pointed in "cut-the-knot" and Wikipedia:

  • Lynn Arthur Steen, J. Arthur Seebach, Jr.: Counterexamples in Topology, Springer, New York 1978, ISBN 0-486-68735-X.
  • Joseph P. Romano, Andrew F. Siegel: Counterexamples in Probability and Statistics, Chapman & Hall, New *York, London 1986, ISBN 0-412-98901-8.
  • Gary L. Wise, Eric B. Hall: Counterexamples in Probability and Real Analysis, Oxford University Press, New York 1993. ISBN 0-19-507068-2.
  • Bernard R. Gelbaum, John M. H. Olmsted: Theorems and Counterexamples in Mathematics, Springer-Verlag 1990, ISBN 978-0-387-97342-5.
  • Bernard R. Gelbaum, John M. H. Olmsted: Counterexamples in Analysis, Corrected reprint of the second (1965) edition, Dover Publications, Mineola, NY 2003, ISBN 0-486-42875-3.
  • Jordan M. Stoyanov: Counterexamples in Probability, Second edition, Wiley, Chichester 1997, ISBN 0-471-96538-3.
  • Michael Copobianco, John Mulluzzo: Examples and Counterexamples in Graph Theory, Elsevier North-Holland 1978, ISBN 0-444-00255-3.

Table of contents[edit]