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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 ...
Then E. T. Parker found a counterexample of order 10 using a one-hour computer search. Finally Parker, Bose, and Shrikhande showed this conjecture to be false for all n ≥ 10. In 1798 A. M. Legendre claimed that 6 is not the sum of 2 rational cubes, [9] which as Lamé pointed out in 1865 is false as 6 = (37/21) 3 + (17/21) 3.
[1] [2] More specifically, in trying to prove a proposition P, one first assumes by contradiction that it is false, and that therefore there must be at least one counterexample. With respect to some idea of size (which may need to be chosen carefully), one then concludes that there is such a counterexample C that is minimal.
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.
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
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 ...
The conjecture was disproved in 1966, with a counterexample involving a count of only four different 5th powers summing to another fifth power: 27 5 + 84 5 + 110 5 + 133 5 = 144 5. Proof by counterexample is a form of constructive proof, in that an object disproving the claim is exhibited.
1. Denotes subtraction and is read as minus; for example, 3 – 2. 2. Denotes the additive inverse and is read as minus, the negative of, or the opposite of; for example, –2. 3. Also used in place of \ for denoting the set-theoretic complement; see \ in § Set theory. × (multiplication sign) 1.
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