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Also, ''trivial'' refers to solutions (to an Equation ) that have a very simple structure, but for the sake of completeness cannot be ignored. These solutions are called the trivial solution. For example, consider the Differential Equation : where ''y'' = ''f''(''x'') is a Function whose Derivative is ''y''′. Then we have the trivial solution y and the nontrivial solution y Similarly, one often hears Fermat's Last Theorem described 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". In addition, mathematicians use ''trivial'' to 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 up the subjectivity of judgements about triviality. Someone experienced in calculus, for example, would consider the theorem that to be trivial, but to a beginning student of calculus, it might be quite difficult. |
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