In mathematics, the term trivial is frequently used for objects (for examples, groups or topological spaces) that have a very simple structure. For non-mathematicians, they are sometimes more difficult to visualize or understand than other, more complicated objects. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

Examples include:

• empty set - the set containing no members
• trivial group - the mathematical group containing only the identity element

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 In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... The following list in mathematics contains the finite groups of small order up to group isomorphism. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, one often (not quite always) distinguishes between an identity, which is an assertion that two expressions are equal regardless of the values of any variables that occur within them, and an equation, which may be true for only some (or none) of the values of any such variables. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

y' = y

where y = f(x) is a function whose derivative is y′. Then we have the trivial solution In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... Jump to: navigation, search In mathematics, the derivative is one of the two central concepts of calculus. ...

y = 0, the zero function

and the nontrivial solution Jump to: navigation, search 0 (zero), alternatively called naught or nought, is both a number and a numeral. ...

y = e^x, the exponential function.

Similarly, one often hears the Fermat conjecture described as asserting that there are no nontrivial solutions to the equation aⁿ + bⁿ = cⁿ when n is greater than 2. Clearly, there are some solutions to the equation. For example, a = b = c = 0 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". Jump to: navigation, search The exponential function is one of the most important functions in mathematics. ... Pierre de Fermat Fermats last theorem (sometimes abbreviated as FLT and also called Fermats great theorem) is one of the most famous theorems in the history of mathematics. ...

In addition, mathematicians use trivial to refer to any easy case of a proof, which for the sake of completeness cannot be ignored. For instance, proofs by mathematical induction usually have two parts: a part that shows that if the theorem is true for a certain value of n, it is also true for the value n+1, and a so-called "base case" that shows that the theorem is true for the particular value n=0. The base case is often trivial and is identified as such. Similarly, one might want to prove that some property is possessed by all the members of a certain set. The main part of the proof will consider the case of a nonempty set, and examine the members in detail; in the case where the set is empty, the property is trivially possessed by all the members, since there aren't any. (See also vacuous truth.) Proof by exhaustion, also known as the brute force method or case analysis, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases, and each case is proved separately. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence. ... Informally, a logical statement is vacuously true if it is true but doesnt say anything; examples are statements of the form everything with property A also has property B, where there is nothing with property A. It is tempting to dismiss this concept as vacuous or silly. ...

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.