Counterexample

A case that disproves a universal claim or shows that a proposed rule has an exception. In mathematics, one counterexample is enough to refute a claim asserted for every case.

Universal quantification: A logical quantifier asserting that a property holds for every object in a specified domain. A counterexample refutes a claim by finding an object in its domain where the property fails.
Euler's sum of powers conjecture: A conjecture that a perfect kth power requires at least k kth powers to sum to it. L. J. Lander and T. R. Parkin disproved it with 144^5 = 27^5 + 84^5 + 110^5 + 133^5.
Proposition: A declarative statement that is either true or false. A counterexample tests a proposition that makes a claim across cases.
Proof: A deductive argument establishing that a mathematical statement follows from accepted premises. A counterexample refutes a universal statement but does not prove an alternative theorem.
Refinement (mathematics): The process of strengthening definitions or hypotheses to make a mathematical result precise and valid. A counterexample often identifies which added condition a theorem needs.
Falsifiability: The property of a claim that it could be contradicted by some possible observation. A counterexample is a direct way to falsify a universal empirical claim.
Euler's polyhedron formula: The relation V − E + F = 2 for vertices, edges, and faces of a convex polyhedron. Nonconvex or self-intersecting polyhedra expose the need to state the formula's conditions carefully.
Predicate logic: A formal system for expressing relations and quantification over objects. Its language makes the domain and universal claims behind counterexamples explicit.
Example: A particular instance used to illustrate a concept, rule, or claim. An example may illustrate a rule; only an instance violating the claim is a counterexample.
Necessary condition: A condition that must hold whenever a specified result holds. A counterexample can show that a proposed necessary condition is not required.

Linked from 23 pages