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In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. [1] For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are ...
6.1 Hyperbolic geometry. 6.2 Paradoxical spaces. 6.3 Unique. 6.4 Related or similar to manifolds. 7 Embeddings and maps between spaces. 8 Counter-examples (general ...
Appel and Haken's approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem (i.e., if they did appear, one could make a smaller counter-example).
differential geometry: Constantin Carathéodory: ... counterexample announced 2019 [20] ... and in fact examples had been found earlier of functions that were nowhere ...
This is now known to be false. The first counterexample was constructed by Atiyah & Hirzebruch (1961). Using K-theory, they constructed an example of a torsion cohomology class—that is, a cohomology class α such that nα = 0 for some positive integer n —which is not the class of an algebraic cycle. Such a class is necessarily a Hodge class.
In geometry, the Weaire–Phelan structure is a three-dimensional structure representing an idealised foam of equal-sized bubbles, with two different shapes. In 1993, Denis Weaire and Robert Phelan found that this structure was a better solution of the Kelvin problem of tiling space by equal volume cells of minimum surface area than the previous best-known solution, the Kelvin structure.
Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr.. In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) have defined a wide variety of topological properties.
Models And Counter-Examples (Mace) is a model finder. [1] Most automated theorem provers try to perform a proof by refutation on the clause normal form of the proof problem, by showing that the combination of axioms and negated conjecture can never be simultaneously true, i.e. does not have a model.
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