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.

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## 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 are as follows:

*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

Similarly, mathematicians often describe Fermat's Last Theorem as asserting that there are no nontrivial integer solutions to the equation
*n* is greater than 2. Clearly, there *are* some solutions to the equation. For example,
*n*, but such solutions are all obvious and uninteresting, and hence "trivial".

## Triviality in mathematical reasoning

*Trivial* may also refer to any easy case of a proof, which for the sake of completeness cannot be ignored. For instance, proofs by mathematical induction have two parts: the "base case" that shows that the theorem is true for a particular initial value such as *n* = 0 or *n* = 1 and then an inductive step that shows that if the theorem is true for a certain value of *n*, it is also true for the value *n* + 1. The base case is often trivial and is identified as such, although there are situations where the base case is difficult but the inductive step is trivial. 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 are none. (See also Vacuous truth.)

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 judgments about triviality. The joke also applies when the first mathematician says the theorem is trivial, but is unable to prove it himself. Often, as a joke, the theorem is then referred to as "intuitively obvious." Someone experienced in calculus, for example, would consider the statement that

to be trivial. To a beginning student of calculus, though, this may not be obvious at all.

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).

## Trivial proofs

In some texts, a *trivial proof* refers to a statement involving a material implication where the consequent, or *Q*, in *P*→*Q*, is always true. Here, the proof follows simply from noting that *Q* is always true, as the implication is then true regardless of the truth value of the antecedent, *P*.

A related concept is a vacuous proof, where the antecedent, *P*, in the material implication *P*→*Q* is always false. Here, the implication is always true regardless of the truth value of the consequent, *Q*.

## Examples

*Y*is a subset of

*X*, so this type of dependence is called "trivial". All other dependences, which are less obvious, are called "nontrivial".

*other*zeros are not generally known and have important applications and involve open questions (such as the Riemann hypothesis); and so, the negative even numbers are called the trivial zeros, and any other zeros are called non-trivial.