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For example, a Fourier series of sine and cosine functions, all continuous, may converge pointwise to a discontinuous function such as a step function. Carmichael's totient function conjecture was stated as a theorem by Robert Daniel Carmichael in 1907, but in 1922 he pointed out that his proof was incomplete. As of 2016 the problem is still open.
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 ...
Charles Akemann and Nik Weaver showed in 2003 that the statement "there exists a counterexample to Naimark's problem which is generated by ℵ 1, elements" is independent of ZFC. Miroslav Bačák and Petr Hájek proved in 2008 that the statement "every Asplund space of density character ω 1 has a renorming with the Mazur intersection property ...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
For instance, an example of a first-countable space which is not second-countable is counterexample #3, the discrete topology on an uncountable set. This particular counterexample shows that second-countability does not follow from first-countability. Several other "Counterexamples in ..." books and papers have followed, with similar motivations.
Hilbert's proof did not exhibit any explicit counterexample: only in 1967 the first explicit counterexample was constructed by Motzkin. [3] Furthermore, if the polynomial has a degree 2 d greater than two, there are significantly many more non-negative polynomials that cannot be expressed as sums of squares.
Nagata (1960) gave the following counterexample to Hilbert's problem. The field k is a field containing 48 elements a 1i, ...,a 16i, for i=1, 2, 3 that are algebraically independent over the prime field. The ring R is the polynomial ring k[x 1,...,x 16, t 1,...,t 16] in 32 variables.
The obvious way to disprove an impossibility conjecture is by providing a single counterexample. For example, Euler proposed that at least n different n th powers were necessary to sum to yet another n th power. The conjecture was disproved in 1966, with a counterexample involving a count of only four different 5th powers summing to another ...
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