A counterexample to a proposition is an example that satisfies all its hypotheses but makes its conclusion false. For a proposition asserting that is true for every , a counterexample is an element for which is false. Equivalently, a counterexample to must satisfy and not . Finding one counterexample therefore provides a proof that a claim containing a universal quantifier is false (Velleman 2019).

Conjectures are often based on extensive numerical or structural evidence. Confirming examples can strengthen the evidence for a conjecture but cannot prove a universal claim, whereas a single verified counterexample refutes it. A purported counterexample that violates one of the conjecture's hypotheses does not refute the conjecture. Counterexamples can also suggest improved conjectures by exposing a missing hypothesis or an overly strong conclusion (Lakatos 2015).

In a setting governed by the well ordering principle, one can also prove a universal claim by assuming a counterexample exists and choosing a least one. If the least counterexample yields a smaller counterexample, the resulting contradiction proves that no counterexample exists. This is a proof by contradiction.

Systematic collections of counterexamples are especially common in analysis and topology (Steen and Seebach 1995, Gelbaum and Olmsted 2003).

See also

Conclusion, Conjecture, Contradiction, Experimental Mathematics, Hypothesis, Proof, Proof by Contradiction, Proposition, Universal Quantifier, Well Ordering Principle

Portions of this entry contributed by Wiktor K. Macura

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References

Gelbaum, B. R. and Olmsted, J. M. H. Counterexamples in Analysis. Mineola, NY: Dover, 2003.Lakatos, I. Proofs and Refutations: The Logic of Mathematical Discovery. (Eds. J. Worrall and E. Zahar). Cambridge, England: Cambridge University Press, 2015.Steen, L. A. and Seebach, J. A. Counterexamples in Topology. New York: Dover, 1995.Velleman, D. J. How to Prove It: A Structured Approach, 3rd ed. Cambridge, England: Cambridge University Press, 2019.

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Counterexample

Cite this as:

Weisstein, Eric W., with contributions by Wiktor K. Macura. "Counterexample." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Counterexample.html

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