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In mathematics, the adjective ''trivial'' is frequently used for objects (for example, groups or topological spaces) that have a very simple structure. The noun ''triviality'' usually refers to a simple technical aspect of some proof or definition. The origin of the term in mathematical language comes from the medieval trivium curriculum. The antonym ''nontrivial'' is commonly used by engineers and mathematicians to indicate a statement or theorem that is not obvious or easy to prove. ==Trivial and nontrivial solutions== In mathematics, the term trivial is frequently used for objects (for examples, groups or topological spaces) that have a very simple structure. Examples include: *empty set: the set containing no members *trivial group: the mathematical group containing only the identity element *trivial ring: a ring defined on a singleton set. ''Trivial'' can also be used to describe 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 solutions. For example, consider the differential equation : where ''y'' = ''f''(''x'') is a function whose derivative is ''y''′. The trivial solution is :''y'' = 0, the zero function while a nontrivial solution is :''y'' (''x'') = e''x'', the exponential function. The differential equation with boundary conditions is important in math and physics, for example describing a particle in a box in quantum mechanics, or standing waves on a string. It always has the solution . This solution is considered obvious and is called the "trivial" solution. In some cases, there may be other solutions (sinusoids), which are called "nontrivial".〔(Introduction to partial differential equations with applications, by Zachmanoglou and Thoe, p309 )〕 Similarly, mathematicians often describe Fermat's Last Theorem as asserting that there are no nontrivial integer 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'', but such solutions are all obvious and uninteresting, and hence "trivial". 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Triviality (mathematics)」の詳細全文を読む スポンサード リンク
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