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''Trivial'' also refers to solutions to an Equation that have a very simple structure, but for the sake of completeness cannot be omitted. These solutions are called the trivial solution. For example, consider the Differential Equation : where ''y'' = ''f''(''x'') is a Function whose Derivative is ''y''′. The trivial solution is y while a nontrivial solution is y Similarly, mathematicians often describe Fermat's Last Theorem as asserting that there are no nontrivial solutions to the equation when ''n'' is greater than 2. Clearly, there ''are'' some solutions to the equation. For example, is a solution for any ''n'', as is ''a'' = 1, ''b'' = 0, ''c'' = 1. But such solutions are all obvious and uninteresting, and hence "trivial". ''Trivial'' may also refer to any easy .) A common joke in the mathematical community is to say that "trivial" is synonymous with "proved" — that is, any theorem can be considered "trivial" once it is known to be true. Another joke concerns two mathematicians who are discussing a theorem; the first mathematician says that the theorem is "trivial". In response to the other's request for an explanation, he then proceeds with twenty minutes of exposition. At the end of the explanation, the second mathematician agrees that the theorem is trivial. These jokes point out the subjectivity of judgements about triviality. Someone experienced in calculus, for example, would consider the theorem that : to be trivial. To a beginning student of calculus, though, this may not be obvious at all. Note that triviality also depends on context. A proof in Functional Analysis would probably, given a number, trivially assume the existence of a larger number. When proving basic results about the natural numbers in Elementary Number Theory though, the proof may very well hinge on the remark that any natural number has a successor (which should then in itself be proved or taken as an Axiom , see Peano's Axioms ). |
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