Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.
Concept
Triviality (mathematics)
Summary
In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., groups, topological spaces). 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, which distinguishes from the more difficult quadrivium curriculum. The opposite of trivial is nontrivial, which is commonly used to indicate that an example or a solution is not simple, or that a statement or a theorem is not easy to prove.
The judgement of whether a situation under consideration is trivial or not depends on who considers it since the situation is obviously true for someone who has sufficient knowledge or experience of it while to someone who has never seen this, it may be even hard to be understood so not trivial at all. And there can be an argument about how quickly and easily a problem should be recognized for the problem to be treated as trivial. So, triviality is not a universally agreed property in mathematics and logic.
In mathematics, the term "trivial" is often used to refer to objects (e.g., groups, topological spaces) with a very simple structure. These include, among others:
Empty set: the set containing no or null 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 is a function whose derivative is . The trivial solution is the zero function
while a nontrivial solution is the exponential function
The differential equation with boundary conditions is important in mathematics and physics, as it could be used to describe a particle in a box in quantum mechanics, or a standing wave on a string.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.